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Arithmetic functions From Wikipedia, the free encyclopedia
Arithmetic functionsFrom Wikipedia, the free encyclopedia
Contents
1 Additive function 11.1 Completely additive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Multiplicative functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Aliquot sequence 42.1 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Aliquot sum 73.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4 Arithmetic function 84.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2 Multiplicative and additive functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.3 Ω(n), ω(n), νp(n) – prime power decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.4 Multiplicative functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.4.1 σk(n), τ(n), d(n) – divisor sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.4.2 φ(n) – Euler totient function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.4.3 Jk(n) – Jordan totient function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.4.4 μ(n) – Möbius function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.4.5 τ(n) – Ramanujan tau function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.4.6 cq(n) – Ramanujan’s sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.5 Completely multiplicative functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.5.1 λ(n) – Liouville function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.5.2 χ(n) – characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.6 Additive functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.6.1 ω(n) – distinct prime divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
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4.7 Completely additive functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.7.1 Ω(n) – prime divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.7.2 νp(n) – prime power dividing n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.8 Neither multiplicative nor additive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.8.1 π(x), Π(x), θ(x), ψ(x) – prime count functions . . . . . . . . . . . . . . . . . . . . . . . . 124.8.2 Λ(n) – von Mangoldt function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.8.3 p(n) – partition function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.8.4 λ(n) – Carmichael function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.8.5 h(n) – Class number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.8.6 rk(n) – Sum of k squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.9 Summation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.10 Dirichlet convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.11 Relations among the functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.11.1 Dirichlet convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.11.2 Sums of squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.11.3 Divisor sum convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.11.4 Class number related . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.11.5 Prime-count related . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.11.6 Menon’s identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.11.7 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.14 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.15 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5 Average 245.1 Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.1.1 Arithmetic mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.1.2 Pythagorean means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.1.3 Statistical location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.2 Summary of types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.3 Miscellaneous types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.3.1 Average percentage return and CAGR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.4 Moving average . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.5 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6 Average order of an arithmetic function 296.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.2 Calculating mean values using Dirichlet series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
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6.2.1 The density of the k-th power free integers in N . . . . . . . . . . . . . . . . . . . . . . . 306.2.2 Visibility of lattice points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.2.3 Divisor functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.3 Better average order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.4 Mean values over F [x] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.4.2 Zeta function and Dirichlet series in F [X] . . . . . . . . . . . . . . . . . . . . . . . . . . 336.4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
7 Bell series 387.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
8 Chebyshev function 408.1 Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.2 Asymptotics and bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428.3 The exact formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438.5 Relation to primorials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448.6 Relation to the prime-counting function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448.7 The Riemann hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448.8 Smoothing function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458.9 Variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
9 Crank of a partition 489.1 Dyson’s crank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489.2 Definition of crank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499.3 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499.4 Basic result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499.5 Ramanujan and cranks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
10 Dedekind psi function 5110.1 Higher Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5110.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5210.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
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11 Dirichlet convolution 5311.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5311.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5311.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5411.4 Dirichlet inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5511.5 Dirichlet series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5511.6 Related Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5611.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5611.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
12 Divisibility sequence 5712.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5712.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
13 Divisor summatory function 5913.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5913.2 Dirichlet’s divisor problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6113.3 Piltz divisor problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6213.4 Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6313.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6313.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
14 Extremal orders of an arithmetic function 6514.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6514.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6614.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6614.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
15 Gauss circle problem 6715.1 The problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6715.2 Bounds on a solution and conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6715.3 Exact forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6815.4 Generalisations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
15.4.1 The primitive circle problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6815.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6915.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
16 Integer sequence 7016.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7016.2 Computable and definable sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7116.3 Complete sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7116.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7216.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
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17 List of OEIS sequences 7317.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7317.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
18 Mertens function 7418.1 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
18.1.1 As an integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7518.1.2 As a sum over Farey sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7718.1.3 As a determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
18.2 Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7718.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7718.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7718.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
19 Möbius inversion formula 7819.1 Statement of the formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7819.2 Series relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7919.3 Repeated transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7919.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8019.5 Multiplicative notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8119.6 Proofs of generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8119.7 Contributions of Weisner, Hall, and Rota . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8119.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8219.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8219.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
20 Normal order of an arithmetic function 8320.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8320.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8320.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8320.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
21 Partition (number theory) 8521.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8621.2 Representations of partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
21.2.1 Ferrers diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8721.2.2 Young diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
21.3 Partition function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8721.3.1 Generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8821.3.2 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8921.3.3 Partition function formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
21.4 Restricted partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9121.4.1 Conjugate and self-conjugate partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
vi CONTENTS
21.4.2 Odd parts and distinct parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9121.4.3 Restricted part size or number of parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9221.4.4 Partitions in a rectangle and Gaussian binomial coefficients . . . . . . . . . . . . . . . . . 93
21.5 Rank and Durfee square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9321.6 Young’s lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9421.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9421.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9521.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9621.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
22 Pillai’s arithmetical function 9822.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
23 Prime gap 9923.1 Simple observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9923.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10023.3 Further results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
23.3.1 Upper bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10123.3.2 Lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
23.4 Conjectures about gaps between primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10223.5 As an arithmetic function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10323.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10323.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10323.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10423.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
24 Prime-counting function 10624.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10624.2 Table of π(x), x / ln x, and li(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10724.3 Algorithms for evaluating π(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10824.4 Other prime-counting functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10924.5 Formulas for prime-counting functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10924.6 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11124.7 The Riemann hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11124.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11224.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11224.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
25 Rank of a partition 11425.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11425.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11525.3 Some basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11525.4 Ramanujan’s congruences and Dyson’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . 116
CONTENTS vii
25.5 Generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11625.6 Alternate definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11725.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11725.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11725.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
26 Von Mangoldt function 11826.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11826.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11826.3 Dirichlet series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11926.4 Chebyshev function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11926.5 Exponential series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12026.6 Riesz mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12026.7 Approximation by Riemann zeta zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12126.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12126.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12226.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12226.11Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 123
26.11.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12326.11.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12526.11.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Chapter 1
Additive function
For the algebraic meaning, see Additive map.
In number theory, an additive function is an arithmetic function f(n) of the positive integer n such that whenever aand b are coprime, the function of the product is the sum of the functions:[1]
f(ab) = f(a) + f(b).
1.1 Completely additive
An additive function f(n) is said to be completely additive if f(ab) = f(a) + f(b) holds for all positive integers a andb, even when they are not co-prime. Totally additive is also used in this sense by analogy with totally multiplicativefunctions. If f is a completely additive function then f(1) = 0.Every completely additive function is additive, but not vice versa.
1.2 Examples
Example of arithmetic functions which are completely additive are:
• The restriction of the logarithmic function to N.
• The multiplicity of a prime factor p in n, that is the largest exponent m for which pm divides n.
• a0(n) - the sum of primes dividing n counting multiplicity, sometimes called sopfr(n), the potency of n or theinteger logarithm of n (sequence A001414 in OEIS). For example:
a0(4) = 2 + 2 = 4a0(20) = a0(22 · 5) = 2 + 2+ 5 = 9a0(27) = 3 + 3 + 3 = 9a0(144) = a0(24 · 32) = a0(24) + a0(32) = 8 + 6 = 14a0(2,000) = a0(24 · 53) = a0(24) + a0(53) = 8 + 15 = 23a0(2,003) = 2003a0(54,032,858,972,279) = 1240658a0(54,032,858,972,302) = 1780417a0(20,802,650,704,327,415) = 1240681
• The function Ω(n), defined as the total number of prime factors of n, counting multiple factors multiple times,sometimes called the “Big Omega function” (sequence A001222 in OEIS). For example;
1
2 CHAPTER 1. ADDITIVE FUNCTION
Ω(1) = 0, since 1 has no prime factorsΩ(4) = 2Ω(16) = Ω(2·2·2·2) = 4Ω(20) = Ω(2·2·5) = 3Ω(27) = Ω(3·3·3) = 3Ω(144) = Ω(24 · 32) = Ω(24) + Ω(32) = 4 + 2 = 6Ω(2,000) = Ω(24 · 53) = Ω(24) + Ω(53) = 4 + 3 = 7Ω(2,001) = 3Ω(2,002) = 4Ω(2,003) = 1Ω(54,032,858,972,279) = 3Ω(54,032,858,972,302) = 6Ω(20,802,650,704,327,415) = 7
Example of arithmetic functions which are additive but not completely additive are:
• ω(n), defined as the total number of different prime factors of n (sequence A001221 in OEIS). For example:
ω(4) = 1ω(16) = ω(24) = 1ω(20) = ω(22 · 5) = 2ω(27) = ω(33) = 1ω(144) = ω(24 · 32) = ω(24) + ω(32) = 1 + 1 = 2ω(2,000) = ω(24 · 53) = ω(24) + ω(53) = 1 + 1 = 2ω(2,001) = 3ω(2,002) = 4ω(2,003) = 1ω(54,032,858,972,279) = 3ω(54,032,858,972,302) = 5ω(20,802,650,704,327,415) = 5
• a1(n) - the sum of the distinct primes dividing n, sometimes called sopf(n) (sequence A008472 in OEIS). Forexample:
a1(1) = 0a1(4) = 2a1(20) = 2 + 5 = 7a1(27) = 3a1(144) = a1(24 · 32) = a1(24) + a1(32) = 2 + 3 = 5a1(2,000) = a1(24 · 53) = a1(24) + a1(53) = 2 + 5 = 7a1(2,001) = 55a1(2,002) = 33a1(2,003) = 2003a1(54,032,858,972,279) = 1238665a1(54,032,858,972,302) = 1780410a1(20,802,650,704,327,415) = 1238677
1.3 Multiplicative functions
From any additive function f(n) it is easy to create a related multiplicative function g(n) i.e. with the property thatwhenever a and b are coprime we have:
g(ab) = g(a) × g(b).
One such example is g(n) = 2f(n).
1.4. SEE ALSO 3
1.4 See also• Sigma additivity
1.5 References[1] Erdös, P., and M. Kac. On the Gaussian Law of Errors in the Theory of Additive Functions. Proc Natl Acad Sci USA.
1939 April; 25(4): 206–207. online
1.6 Further reading• Janko Bračič, Kolobar aritmetičnih funkcij (Ring of arithmetical functions), (Obzornik mat, fiz. 49 (2002) 4,
pp. 97–108) (MSC (2000) 11A25)
Chapter 2
Aliquot sequence
In mathematics, an aliquot sequence is a recursive sequence in which each term is the sum of the proper divisorsof the previous term. The aliquot sequence starting with a positive integer k can be defined formally in terms of thesum-of-divisors function σ1 in the following way:[1]
s0 = k
s = σ1(sn₋₁) − sn₋₁.
For example, the aliquot sequence of 10 is 10, 8, 7, 1, 0 because:
σ1(10) − 10 = 5 + 2 + 1 = 8σ1(8) − 8 = 4 + 2 + 1 = 7σ1(7) − 7 = 1σ1(1) − 1 = 0
Many aliquot sequences terminate at zero (sequence A080907 in OEIS); all such sequences necessarily end with aprime number followed by 1 (since the only proper divisor of a prime is 1), followed by 0 (since 1 has no properdivisors). There are a variety of ways in which an aliquot sequence might not terminate:
• A perfect number has a repeating aliquot sequence of period 1. The aliquot sequence of 6, for example, is 6,6, 6, 6, ...
• An amicable number has a repeating aliquot sequence of period 2. For instance, the aliquot sequence of 220is 220, 284, 220, 284, ...
• A sociable number has a repeating aliquot sequence of period 3 or greater. (Sometimes the term sociablenumber is used to encompass amicable numbers as well.) For instance, the aliquot sequence of 1264460 is1264460, 1547860, 1727636, 1305184, 1264460, ...
• Some numbers have an aliquot sequence which is eventually periodic, but the number itself is not perfect,amicable, or sociable. For instance, the aliquot sequence of 95 is 95, 25, 6, 6, 6, 6, ... . Numbers like 95 thatare not perfect, but have an eventually repeating aliquot sequence of period 1 are called aspiring numbers (A063769).
The lengths of the Aliquot sequences that start at n are
1, 2, 2, 3, 2, 1, 2, 3, 4, 4, 2, 7, 2, 5, 5, 6, 2, 4, 2, 7, 3, 6, 2, 5, 1, 7, 3, 1, 2, 15, 2, 3, 6, 8, 3, 4, 2, 7, 3, 4,2, 14, 2, 5, 7, 8, 2, 6, 4, 3, ... (sequence A044050 in OEIS)
The final terms (excluding 1) of the Aliquot sequences that start at n are
4
2.1. EXTERNAL LINKS 5
1, 2, 3, 3, 5, 6, 7, 7, 3, 7, 11, 3, 13, 7, 3, 3, 17, 11, 19, 7, 11, 7, 23, 17, 6, 3, 13, 28, 29, 3, 31, 31, 3, 7,13, 17, 37, 7, 17, 43, 41, 3, 43, 43, 3, 3, 47, 41, 7, 43, ... (sequence A115350 in OEIS)
Numbers whose Aliquot sequence terminates in 1 are
1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33,34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, ... (sequence A080907 in OEIS)
Numbers whose Aliquot sequence terminates in a perfect number are
25, 95, 119, 143, 417, 445, 565, 608, 650, 652, 675, 685, 783, 790, 909, 913, ... (sequence A063769in OEIS)
Numbers whose Aliquot sequence terminates in a cycle with length at least 2 are
220, 284, 562, 1064, 1184, 1188, 1210, 1308, 1336, 1380, 1420, 1490, 1604, 1690, 1692, 1772, 1816,1898, 2008, 2122, 2152, 2172, 2362, ... (sequence A121507 in OEIS)
Numbers whose Aliquot sequence is not known to be finite or eventually periodic are
276, 306, 396, 552, 564, 660, 696, 780, 828, 888, 966, 996, 1074, 1086, 1098, 1104, 1134, 1218,1302, 1314, 1320, 1338, 1350, 1356, 1392, 1398, 1410, 1464, 1476, 1488, ... (sequence A131884 inOEIS)
An important conjecture due to Catalan with respect to aliquot sequences is that every aliquot sequence ends in one ofthe above ways–with a prime number, a perfect number, or a set of amicable or sociable numbers.[2] The alternativewould be that a number exists whose aliquot sequence is infinite, yet aperiodic. Any one of the many numbers whosealiquot sequences have not been fully determined might be such a number. The first five candidate numbers are calledthe Lehmer five (named after Dick Lehmer): 276, 552, 564, 660, and 966.[3]
As of April 2015, there were 898 positive integers less than 100,000 whose aliquot sequences have not been fullydetermined, and 9190 such integers less than 1,000,000.[4]
2.1 External links• Tables of Aliquot Cycles (J.O.M. Pedersen)
• Aliquot Page (Wolfgang Creyaufmüller)
• Aliquot sequences (Christophe Clavier)
• Forum on calculating aliquot sequences (MersenneForum)
• Aliquot sequence summary page for sequences up to 100000 (there are similar pages for higher ranges) (KarstenBonath)
• Active research site on aliquot sequences (Jean-Luc Garambois)
2.2 Notes[1] Weisstein, Eric W., “Aliquot Sequence”, MathWorld.
[2] Weisstein, Eric W., “Catalan’s Aliquot Sequence Conjecture”, MathWorld.
[3] Creyaufmüller, Wolfgang (May 24, 2014). “Lehmer Five”. Retrieved June 14, 2015.
[4] Creyaufmüller, Wolfgang (April 29, 2015). “Aliquot Pages”. Retrieved June 14, 2015.
6 CHAPTER 2. ALIQUOT SEQUENCE
2.3 References• Manuel Benito; Wolfgang Creyaufmüller; Juan Luis Varona; Paul Zimmermann. Aliquot Sequence 3630 Ends
After Reaching 100 Digits. Experimental Mathematics, vol. 11, num. 2, Natick, MA, 2002, p. 201-206.
• W. Creyaufmüller. Primzahlfamilien - Das Catalan’sche Problem und die Familien der Primzahlen im Bereich1 bis 3000 im Detail. Stuttgart 2000 (3rd ed.), 327p.
Chapter 3
Aliquot sum
In mathematics, the aliquot sum s(n) of a positive integer n is the sum of all proper divisors of n. Perfect, deficient,and abundant numbers are equal to, greater than, and less than their aliquot sums respectively. The aliquot sequenceis the sequence obtained by repeatedly applying the aliquot sum function s. The aliquot sum function is also referredto as the restricted divisor function.[1]
3.1 See also• Divisor function
3.2 References[1] Weisstein, Eric W., “Restricted Divisor Function”, MathWorld.
7
Chapter 4
Arithmetic function
In number theory, an arithmetic, arithmetical, or number-theoretic function[1][2] is a real or complex valuedfunction f(n) defined on the set of natural numbers (i.e. positive integers) that “expresses some arithmetical propertyof n".[3]
An example of an arithmetic function is the non-principal character (mod 4) defined by
χ(n) =(−4n
)=
0 ifneven is ,1 ifn ≡ 1 mod 4,
−1 ifn ≡ 3 mod 4,
where (−4n ) is the Kronecker symbol.
To emphasize that they are being thought of as functions rather than sequences, values of an arithmetic function areusually denoted by a(n) rather than an.There is a larger class of number-theoretic functions that do not fit the above definition, e.g. the prime-countingfunctions. This article provides links to functions of both classes.
4.1 Notation∑p f(p) and
∏p f(p) mean that the sum or product is over all prime numbers:
∑p f(p) = f(2) + f(3) + f(5) + · · ·
∏p f(p) = f(2)f(3)f(5) · · · .
Similarly,∑pk f(p
k) and∏pk f(p
k) mean that the sum or product is over all prime powers with strictly positiveexponent (so 1 is not included):
∑pk
f(pk) = f(2) + f(3) + f(4) + f(5) + f(7) + f(8) + f(9) + · · ·
∑d|n f(d) and
∏d|n f(d) mean that the sum or product is over all positive divisors of n, including 1 and n. E.g.,
if n = 12,
∏d|12
f(d) = f(1)f(2)f(3)f(4)f(6)f(12).
The notations can be combined:∑p|n f(p) and
∏p|n f(p) mean that the sum or product is over all prime divisors
of n. E.g., if n = 18,
8
4.2. MULTIPLICATIVE AND ADDITIVE FUNCTIONS 9
∑p|18
f(p) = f(2) + f(3),
and similarly∑pk|n f(p
k) and∏pk|n f(p
k) mean that the sum or product is over all prime powers dividing n.E.g., if n = 24,
∏pk|24
f(pk) = f(2)f(3)f(4)f(8).
4.2 Multiplicative and additive functions
An arithmetic function a is
• completely additive if a(mn) = a(m) + a(n) for all natural numbers m and n;
• completely multiplicative if a(mn) = a(m)a(n) for all natural numbers m and n;
Two whole numbers m and n are called coprime if their greatest common divisor is 1; i.e., if there is no prime numberthat divides both of them.Then an arithmetic function a is
• additive if a(mn) = a(m) + a(n) for all coprime natural numbers m and n;
• multiplicative if a(mn) = a(m)a(n) for all coprime natural numbers m and n.
4.3 Ω(n), ω(n), νp(n) – prime power decomposition
The fundamental theorem of arithmetic states that any positive integer n can be represented uniquely as a product ofpowers of primes: n = pa11 · · · pakk where p1 < p2 < ... < pk are primes and the aj are positive integers. (1 is given bythe empty product.)It is often convenient to write this as an infinite product over all the primes, where all but a finite number have a zeroexponent. Define νp(n) as the exponent of the highest power of the prime p that divides n. I.e. if p is one of the pithen νp(n) = ai, otherwise it is zero. Then
n =∏p
pνp(n).
In terms of the above the functions ω and Ω are defined by
ω(n) = k,Ω(n) = a1 + a2 + ... + ak.
To avoid repetition, whenever possible formulas for the functions listed in this article are given in terms of n and thecorresponding pi, ai, ω, and Ω.
4.4 Multiplicative functions
10 CHAPTER 4. ARITHMETIC FUNCTION
1OO 2 3 4 5
5 6 7 8 9 10 11
1 21 2 14 2 5 35
41
374 7 6
29
71
28 4
1 10
95 2
121
19
58
7 10 6
13
14
12 7 4
10
5
11 1
0
5
11 1
0
14
13
1
85
17
16 4
13
612
1
5 15
29
19
12
17 1
1
5
14 1
0
184 61
6
20
19
10 1
3 4 6 15
22
11
35
10
13 2
16 2
221
20 8
5
13
The chaotic course of Ω(n) through the natural numbers ( A001222): Beginning on the height of the red line the two least significantbinary digits of Ω(n) of all positive odd n below 1200 are represented by a line up (digit 1) or a line down (digit 0). The additionalreplacement of the “” of the semiprimes without prime factors below 5 by only one line down (here blue) even almost brings abalance between the ups and downs. The prime numbers are marked orange: orange lines for the Gaussian primes and for the otherprimes p additional the number x in orange so that p = x2 + y2 = (x + yi) (x – yi) = –i (x + yi) (y + xi) and x < y for natural xand y ( A002331).
4.4.1 σk(n), τ(n), d(n) – divisor sums
σk(n) is the sum of the kth powers of the positive divisors of n, including 1 and n, where k is a complex number.σ1(n), the sum of the (positive) divisors of n, is usually denoted by σ(n).Since a positive number to the zero power is one, σ0(n) is therefore the number of (positive) divisors of n; it is usuallydenoted by d(n) or τ(n) (for the German Teiler = divisors).
σk(n) =
ω(n)∏i=1
p(ai+1)ki − 1
pki − 1=
ω(n)∏i=1
(1 + pki + p2ki + · · ·+ paiki
).
Setting k = 0 in the second product gives
τ(n) = d(n) = (1 + a1)(1 + a2) · · · (1 + aω(n)).
4.4.2 φ(n) – Euler totient function
φ(n), the Euler totient function, is the number of positive integers not greater than n that are coprime to n.
φ(n) = n∏p|n
(1− 1
p
)= n
(p1 − 1
p1
)(p2 − 1
p2
)· · ·(pω(n) − 1
pω(n)
).
4.4.3 Jk(n) – Jordan totient function
Jk(n), the Jordan totient function, is the number of k-tuples of positive integers all less than or equal to n that forma coprime (k + 1)-tuple together with n. It is a generalization of Euler’s totient, φ(n) = J1(n).
Jk(n) = nk∏p|n
(1− 1
pk
)= nk
(pk1 − 1
pk1
)(pk2 − 1
pk2
)· · ·
(pkω(n) − 1
pkω(n)
).
4.4.4 μ(n) – Möbius function
μ(n), the Möbius function, is important because of the Möbius inversion formula. See Dirichlet convolution, below.
4.5. COMPLETELY MULTIPLICATIVE FUNCTIONS 11
µ(n) =
(−1)ω(n) = (−1)Ω(n) if ω(n) = Ω(n)
0 if ω(n) = Ω(n).
This implies that μ(1) = 1. (Because Ω(1) = ω(1) = 0.)
4.4.5 τ(n) – Ramanujan tau function
τ(n), the Ramanujan tau function, is defined by its generating function identity:
∑n≥1
τ(n)qn = q∏n≥1
(1− qn)24.
Although it is hard to say exactly what “arithmetical property of n" it “expresses”,[4] (τ(n) is (2π)−12 times the nthFourier coefficient in the q-expansion of the modular discriminant function)[5] it is included among the arithmeticalfunctions because it is multiplicative and it occurs in identities involving certain σk(n) and rk(n) functions (becausethese are also coefficients in the expansion of modular forms).
4.4.6 cq(n) – Ramanujan’s sum
cq(n), Ramanujan’s sum, is the sum of the nth powers of the primitive qth roots of unity:
cq(n) =∑
1≤a≤q
gcd(a,q)=1
e2πi
aq n.
Even though it is defined as a sum of complex numbers (irrational for most values of q), it is an integer. For a fixedvalue of n it is multiplicative in q:
If q and r are coprime, cq(n)cr(n) = cqr(n).
Many of the functions mentioned in this article have expansions as series involving these sums; see the articleRamanujan’s sum for examples.
4.5 Completely multiplicative functions
4.5.1 λ(n) – Liouville function
λ(n), the Liouville function, is defined by
λ(n) = (−1)Ω(n).
4.5.2 χ(n) – characters
All Dirichlet characters χ(n) are completely multiplicative. An example is the non-principal character (mod 4)defined in the introduction. Two characters have special notations:The principal character (mod n) is denoted by χ0(a) (or χ1(a)). It is defined as
χ0(a) =
1 if gcd(a, n) = 1,
0 if gcd(a, n) = 1.
12 CHAPTER 4. ARITHMETIC FUNCTION
The quadratic character (mod n) is denoted by the Jacobi symbol for odd n (it is not defined for even n.):
(a
n
)=
(a
p1
)a1 ( a
p2
)a2· · ·(
a
pω(n)
)aω(n)
.
In this formula (ap ) is the Legendre symbol, defined for all integers a and all odd primes p by
(a
p
)=
0 if a ≡ 0 (mod p)
+1 if a ≡ 0 (mod p) integer some for and x, a ≡ x2 (mod p)−1 such no is there if x.
Following the normal convention for the empty product,(a1
)= 1.
4.6 Additive functions
4.6.1 ω(n) – distinct prime divisors
ω(n), defined above as the number of distinct primes dividing n, is additive.
4.7 Completely additive functions
4.7.1 Ω(n) – prime divisors
Ω(n), defined above as the number of prime factors of n counted with multiplicities, is completely additive.
4.7.2 νp(n) – prime power dividing n
For a fixed prime p, νp(n), defined above as the exponent of the largest power of p dividing n, is completely additive.
4.8 Neither multiplicative nor additive
4.8.1 π(x), Π(x), θ(x), ψ(x) – prime count functions
These important functions (which are not arithmetic functions) are defined for non-negative real arguments, and areused in the various statements and proofs of the prime number theorem. They are summation functions (see the mainsection just below) of arithmetic functions which are neither multiplicative nor additive.π(x), the prime counting function, is the number of primes not exceeding x. It is the summation function of thecharacteristic function of the prime numbers.
π(x) =∑p≤x
1
A related function counts prime powers with weight 1 for primes, 1/2 for their squares, 1/3 for cubes, … It is thesummation function of the arithmetic function which takes the value 1/k on integers which are the k-th power of someprime number, and the value 0 on other integers.
Π(x) =∑pk≤x
1
k.
4.8. NEITHER MULTIPLICATIVE NOR ADDITIVE 13
θ(x) and ψ(x), the Chebyshev functions, are defined as sums of the natural logarithms of the primes not exceeding x.
ϑ(x) =∑p≤x
log p,
ψ(x) =∑pk≤x
log p.
The Chebyshev function ψ(x) is the summation function of the von Mangoldt function just below.
4.8.2 Λ(n) – von Mangoldt function
Λ(n), the von Mangoldt function, is 0 unless the argument is a prime power, in which case it is the natural log of theprime:
Λ(n) =
log p ifn = 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, . . . = pkpower prime a is0 ifn = 1, 6, 10, 12, 14, 15, 18, 20, 21, . . . power prime a not is .
4.8.3 p(n) – partition function
p(n), the partition function, is the number of ways of representing n as a sum of positive integers, where two repre-sentations with the same summands in a different order are not counted as being different:
p(n) = | (a1, a2, . . . ak) : 0 < a1 ≤ a2 ≤ · · · ≤ ak ∧ n = a1 + a2 + · · ·+ ak |.
4.8.4 λ(n) – Carmichael function
λ(n), the Carmichael function, is the smallest positive number such that aλ(n) ≡ 1 (mod n) for all a coprime ton. Equivalently, it is the least common multiple of the orders of the elements of the multiplicative group of integersmodulo n.For powers of odd primes and for 2 and 4, λ(n) is equal to the Euler totient function of n; for powers of 2 greaterthan 4 it is equal to one half of the Euler totient function of n:
λ(n) =
ϕ(n) ifn = 2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, . . .
12ϕ(n) ifn = 8, 16, 32, 64, . . .
and for general n it is the least common multiple of λ of each of the prime power factors of n:
λ(pa11 pa22 . . . p
aω(n)
ω(n) ) = lcm[λ(pa11 ), λ(pa22 ), . . . , λ(paω(n)
ω(n) )].
4.8.5 h(n) – Class number
h(n), the class number function, is the order of the ideal class group of an algebraic extension of the rationals withdiscriminant n. The notation is ambiguous, as there are in general many extensions with the same discriminant. Seequadratic field and cyclotomic field for classical examples.
14 CHAPTER 4. ARITHMETIC FUNCTION
4.8.6 rk(n) – Sum of k squares
rk(n) is the number of ways n can be represented as the sum of k squares, where representations that differ only inthe order of the summands or in the signs of the square roots are counted as different.
rk(n) = |(a1, a2, . . . , ak) : n = a21 + a22 + · · ·+ a2k|
4.9 Summation functions
Given an arithmetic function a(n), its summation function A(x) is defined by
A(x) :=∑n≤x
a(n).
A can be regarded as a function of a real variable. Given a positive integer m, A is constant along open intervals m <x < m + 1, and has a jump discontinuity at each integer for which a(m) ≠ 0.Since such functions are often represented by series and integrals, to achieve pointwise convergence it is usual todefine the value at the discontinuities as the average of the values to the left and right:
A0(m) :=1
2
∑n<m
a(n) +∑n≤m
a(n)
= A(m)− 1
2a(m).
Individual values of arithmetic functions may fluctuate wildly – as in most of the above examples. Summation func-tions “smooth out” these fluctuations. In some cases it may be possible to find asymptotic behaviour for the summationfunction for large x.A classical example of this phenomenon[6] is given by the divisor summatory function, the summation function ofd(n), the number of divisors of n:
lim infn→∞
d(n) = 2
lim supn→∞
log d(n) log lognlogn = log 2
limn→∞
d(1) + d(2) + · · ·+ d(n)
log(1) + log(2) + · · ·+ log(n) = 1.
An average order of an arithmetic function is some simpler or better-understood function which has the samesummation function asymptotically, and hence takes the same values “on average”. We say that g is an average orderof f if
∑n≤x
f(n) ∼∑n≤x
g(n)
as x tends to infinity. The example above shows that d(n) has the average order log(n).[7]
4.10 Dirichlet convolution
Given an arithmetic function a(n), let Fa(s), for complex s, be the function defined by the corresponding Dirichletseries (where it converges):[8]
4.11. RELATIONS AMONG THE FUNCTIONS 15
Fa(s) :=
∞∑n=1
a(n)
ns.
Fa(s) is called a generating function of a(n). The simplest such series, corresponding to the constant function a(n) =1 for all n, is ς(s) the Riemann zeta function.The generating function of the Möbius function is the inverse of the zeta function:
ζ(s)∞∑n=1
µ(n)
ns= 1, R s > 0.
Consider two arithmetic functions a and b and their respective generating functions Fa(s) and Fb(s). The productFa(s)Fb(s) can be computed as follows:
Fa(s)Fb(s) =
( ∞∑m=1
a(m)
ms
)( ∞∑n=1
b(n)
ns
).
It is a straightforward exercise to show that if c(n) is defined by
c(n) :=∑ij=n
a(i)b(j) =∑i|n
a(i)b(ni
),
then
Fc(s) = Fa(s)Fb(s).
This function c is called the Dirichlet convolution of a and b, and is denoted by a ∗ b .A particularly important case is convolution with the constant function a(n) = 1 for all n, corresponding to multiplyingthe generating function by the zeta function:
g(n) =∑d|n
f(d).
Multiplying by the inverse of the zeta function gives the Möbius inversion formula:
f(n) =∑d|n
µ(nd
)g(d).
If f is multiplicative, then so is g. If f is completely multiplicative, then g is multiplicative, but may or may not becompletely multiplicative.
4.11 Relations among the functions
There are a great many formulas connecting arithmetical functions with each other and with the functions of analysis,especially powers, roots, and the exponential and log functions.Here are a few examples:
16 CHAPTER 4. ARITHMETIC FUNCTION
4.11.1 Dirichlet convolutions
∑δ|n µ(δ) =
∑δ|n λ
(nδ
)|µ(δ)| =
1 if n = 1
0 if n = 1.where λ is the Liouville function. [9]
∑δ|n φ(δ) = n. [10]
φ(n) =∑δ|n µ
(nδ
)δ = n
∑δ|n
µ(δ)δ . Möbius inversion
∑d|n Jk(d) = nk. [11]
Jk(n) =∑δ|n µ
(nδ
)δk = nk
∑δ|n
µ(δ)δk. Möbius inversion
∑δ|n δ
sJr(δ)Js(nδ
)= Jr+s(n)
[12]
∑δ|n φ(δ)d
(nδ
)= σ(n). [13][14]
∑δ|n |µ(δ)| = 2ω(n). [15]
|µ(n)| =∑δ|n µ
(nδ
)2ω(δ). Möbius inversion
∑δ|n 2
ω(δ) = d(n2).
2ω(n) =∑δ|n µ
(nδ
)d(δ2). Möbius inversion
∑δ|n d(δ
2) = d2(n).
d(n2) =∑δ|n µ
(nδ
)d2(δ). Möbius inversion
∑δ|n d
(nδ
)2ω(δ) = d2(n).
∑δ|n λ(δ) =
1 if n square a is0 if nsquare. not is
where λ is the Liouville function.
∑δ|n Λ(δ) = logn. [16]
Λ(n) =∑δ|n µ
(nδ
)log(δ). Möbius inversion
4.11.2 Sums of squares
Ifk ≥ 4, rk(n) > 0. (Lagrange’s four-square theorem).
r2(n) = 4∑d|n χ(d), where χ is the non-principal character (mod 4) defined in the introduction.[17]
There is a formula for r3 in the section on class numbers below.
r4(n) = 8∑
d|n4 ∤ d
d = 8(2+ (−1)n)∑
d|n2 ∤ d
d =
8σ(n) ifn odd is24σ
(n2ν
)ifn even is
, where ν = ν2(n). [18][19][20]
4.11. RELATIONS AMONG THE FUNCTIONS 17
r6(n) = 16∑d|n χ
(nd
)d2 − 4
∑d|n χ(d)d
2. [21]
Define the function σk*(n) as[22]
σ∗k(n) = (−1)n
∑d|n
(−1)ddk =
∑d|n d
k = σk(n) ifn odd is∑d|n2 | d
dk −∑
d|n2 ∤ d
dk ifneven is .
That is, if n is odd, σk*(n) is the sum of the kth powers of the divisors of n, i.e. σk(n), and if n is even it is the sumof the kth powers of the even divisors of n minus the sum of the kth powers of the odd divisors of n.
r8(n) = 16σ∗3(n).
[21][23]
Adopt the convention that Ramanujan’s τ(x) = 0 if x is not an integer.
r24(n) =16691σ
∗11(n) +
128691
(−1)n−1259τ(n)− 512τ
(n2
) [24]
4.11.3 Divisor sum convolutions
Here “convolution” does not mean “Dirichlet convolution” but instead refers to the formula for the coefficients of theproduct of two power series:
( ∞∑n=0
anxn
)( ∞∑n=0
bnxn
)=
∞∑i=0
∞∑j=0
aibjxi+j =
∞∑n=0
(n∑i=0
aibn−i
)xn =
∞∑n=0
cnxn.
The sequence cn =∑ni=0 aibn−i is called the convolution or the Cauchy product of the sequences an and bn.
See Eisenstein series for a discussion of the series and functional identities involved in these formulas.[25]
σ3(n) =15
6nσ1(n)− σ1(n) + 12
∑0<k<n σ1(k)σ1(n− k)
. [26]
σ5(n) =121
10(3n− 1)σ3(n) + σ1(n) + 240
∑0<k<n σ1(k)σ3(n− k)
. [27]
σ7(n) =1
20
21(2n− 1)σ5(n)− σ1(n) + 504
∑0<k<n
σ1(k)σ5(n− k)
= σ3(n) + 120
∑0<k<n
σ3(k)σ3(n− k).
[27][28]
σ9(n) =1
11
10(3n− 2)σ7(n) + σ1(n) + 480
∑0<k<n
σ1(k)σ7(n− k)
=1
11
21σ5(n)− 10σ3(n) + 5040
∑0<k<n
σ3(k)σ5(n− k)
.
[26][29]
τ(n) = 65756σ11(n) +
691756σ5(n)−
6913
∑0<k<n σ5(k)σ5(n− k), where τ(n) is Ramanujan’s function.
[30][31]
Since σ (n) (for natural number k) and τ(n) are integers, the above formulas can be used to prove congruences[32] forthe functions. See Tau-function for some examples.Extend the domain of the partition function by setting p(0) = 1.
p(n) = 1n
∑1≤k≤n σ(k)p(n− k). [33] This recurrence can be used to compute p(n).
18 CHAPTER 4. ARITHMETIC FUNCTION
4.11.4 Class number related
Peter Gustav Lejeune Dirichlet discovered formulas that relate the class number h of quadratic number fields to theJacobi symbol.[34]
An integer D is called a fundamental discriminant if it is the discriminant of a quadratic number field. This isequivalent to D ≠ 1 and either a) D is squarefree and D ≡ 1 (mod 4) or b) D ≡ 0 (mod 4), D/4 is squarefree, and D/4≡ 2 or 3 (mod 4).[35]
Extend the Jacobi symbol to accept even numbers in the “denominator” by defining the Kronecker symbol:
(a2
)=
0 if aeven is
(−1)a2−1
8 if a odd. is
Then if D < −4 is a fundamental discriminant[36][37]
h(D) =1
D
|D|∑r=1
r
(D
r
)
=1
2−(D2
) |D|/2∑r=1
(D
r
).
There is also a formula relating r3 and h. Again, let D be a fundamental discriminant, D < −4. Then[38]
r3(|D|) = 12
(1−
(D
2
))h(D).
4.11.5 Prime-count related
Let Hn = 1 + 12 + 1
3 + · · ·+ 1n be the nth harmonic number. Then
σ(n) ≤ Hn + eHn logHn is true for every natural number n if and only if the Riemann hypothesis istrue. [39]
The Riemann hypothesis is also equivalent to the statement that, for all n > 5040,
σ(n) < eγn log logn (where γ is the Euler–Mascheroni constant). This is Robin’s theorem.
∑p
νp(n) = Ω(n).
ψ(x) =∑n≤x Λ(n).
[40]
Π(x) =∑n≤x
Λ(n)logn .
[41]
eθ(x) =∏p≤x p.
[42]
eψ(x) = lcm[1, 2, . . . , ⌊x⌋]. [43]
4.11. RELATIONS AMONG THE FUNCTIONS 19
4.11.6 Menon’s identity
In 1965 P. Kesava Menon proved[44]
∑1≤k≤n
gcd(k,n)=1
gcd(k − 1, n) = φ(n)d(n).
This has been generalized by a number of mathematicians, e.g.:B. Sury[45]
∑1≤k1,k2,...,ks≤n
gcd(k1,n)=1
gcd(k1 − 1, k2, . . . , ks, n) = φ(n)σs−1(n).
N. Rao[46]
∑1≤k1,k2,...,ks≤n
gcd(k1,k2,...,ks,n)=1
gcd(k1 − a1, k2 − a2, . . . , ks − as, n)s = Js(n)d(n),
where a1, a2, ..., as are integers, gcd(a1, a2, ..., as, n) = 1.L. Tóth[47]
∑1≤k≤m
gcd(k,m)=1
gcd(k2 − 1,m1) gcd(k2 − 1,m2) = φ(n)∑
d1|m1d2|m2
φ(gcd(d1, d2))2ω(lcm(d1,d2)),
where m1 and m2 are odd, m = lcm(m1, m2).In fact, if f is any arithmetical function[48][49]
∑1≤k≤n
gcd(k,n)=1
f(gcd(k − 1, n)) = φ(n)∑d|n
(µ ∗ f)(d)φ(d)
,
where * stands for Dirichlet convolution.
4.11.7 Miscellaneous
Let m and n be distinct, odd, and positive. Then the Jacobi symbol satisfies the Law of Quadratic Reciprocity:
(mn
) (nm
)= (−1)(m−1)(n−1)/4.
Let λ(n) be Liouville’s function. Then we have
|λ(n)|µ(n) = λ(n)|µ(n)| = µ(n), and
λ(n)µ(n) = |µ(n)| = µ2(n).
Let λ(n) be Carmichael’s function. Then we have
λ(n) | ϕ(n). Further,
20 CHAPTER 4. ARITHMETIC FUNCTION
λ(n) = ϕ(n) if only and if n =
1, 2, 4;
3, 5, 7, 9, 11, . . . i.e. pk where pprime odd an is ;6, 10, 14, 18, . . . i.e. 2pk where pprime odd an is .
[50]
2ω(n) ≤ d(n) ≤ 2Ω(n). [51][52]
6π2 <
ϕ(n)σ(n)n2 < 1. [53]
cq(n) =µ(
qgcd(q,n)
)ϕ(
qgcd(q,n)
)ϕ(q)=
∑δ|gcd(q,n)
µ(qδ
)δ.
[54] Note that ϕ(q) =∑δ|q µ
(qδ
)δ. [55]
cq(1) = µ(q).
cq(q) = ϕ(q).
∑δ|n d
3(δ) =(∑
δ|n d(δ))2. [56] Compare this with 13 + 23 + 33 + ... + n3 = (1 + 2 + 3 + ... +
n)2
d(uv) =∑δ|gcd(u,v) µ(δ)d
(uδ
)d(vδ
). [57]
σk(u)σk(v) =∑δ|gcd(u,v) δ
kσk(uvδ2
). [58]
τ(u)τ(v) =∑δ|gcd(u,v) δ
11τ(uvδ2
), where τ(n) is Ramanujan’s function. [59]
4.12 Notes[1] Long (1972, p. 151)
[2] Pettofrezzo & Byrkit (1970, p. 58)
[3] Hardy & Wright, intro. to Ch. XVI
[4] Hardy, Ramanujan, § 10.2
[5] Apostol, Modular Functions ..., § 1.15, Ch. 4, and ch. 6
[6] Hardy & Wright, §§ 18.1–18.2
[7] Gérald Tenenbaum (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge studies in advancedmathematics 46. Cambridge University Press. pp. 36–55. ISBN 0-521-41261-7.
[8] Hardy & Wright, § 17.6, show how the theory of generating functions can be constructed in a purely formal manner withno attention paid to convergence.
[9] Hardy & Wright, Thm. 263
[10] Hardy & Wright, Thm. 63
[11] see references at Jordan’s totient function
[12] Holden et al in external links The formula is Gegenbauer’s
[13] Hardy & Wright, Thm. 288–290
[14] Dineva in external links, prop. 4
[15] Hardy & Wright, Thm. 264
4.12. NOTES 21
[16] Hardy & Wright, Thm. 296
[17] Hardy & Wright, Thm. 278
[18] Hardy & Wright, Thm. 386
[19] Hardy, Ramanujan, eqs 9.1.2, 9.1.3
[20] Koblitz, Ex. III.5.2
[21] Hardy & Wright, § 20.13
[22] Hardy, Ramanujan, § 9.7
[23] Hardy, Ramanujan, § 9.13
[24] Hardy, Ramanujan, § 9.17
[25] The paper by Huard, Ou, Spearman, and Williams in the external links also has proofs.
[26] Ramanujan, On Certain Arithmetical Functions, Table IV; Papers, p. 146
[27] Koblitz, ex. III.2.8
[28] Koblitz, ex. III.2.3
[29] Koblitz, ex. III.2.2
[30] Koblitz, ex. III.2.4
[31] Apostol, Modular Functions ..., Ex. 6.10
[32] Apostol, Modular Functions..., Ch. 6 Ex. 10
[33] G.H. Hardy, S. Ramannujan, Asymptotic Formulæ in Combinatory Analysis, § 1.3; in Ramannujan, Papers p. 279
[34] Landau, p. 168, credits Gauss as well as Dirichlet
[35] Cohen, Def. 5.1.2
[36] Cohen, Corr. 5.3.13
[37] see Edwards, § 9.5 exercises for more complicated formulas.
[38] Cohen, Prop 5.10.3
[39] See Divisor function.
[40] Hardy & Wright, eq. 22.1.2
[41] See prime counting functions.
[42] Hardy & Wright, eq. 22.1.1
[43] Hardy & Wright, eq. 22.1.3
[44] László Tóth, Menon’s Identity and Arithmetical Sums ..., #External links, eq. 1
[45] Tóth, eq. 5
[46] Tóth, eq. 3
[47] Tóth, eq. 35
[48] Tóth, eq. 2
[49] Tóth states that Menon proved this for multiplicative f in 1965 and V. Sita Ramaiah for general f.
[50] See Multiplicative group of integers modulo n and Primitive root modulo n.
[51] Hardy Ramanujan, eq. 3.10.3
[52] Hardy & Wright, § 22.13
[53] Hardy & Wright, Thm. 329
22 CHAPTER 4. ARITHMETIC FUNCTION
[54] Hardy & Wright, Thms. 271, 272
[55] Hardy & Wright, eq. 16.3.1
[56] Ramanujan, Some Formulæ in the Analytic Theory of Numbers, eq. (C); Papers p.133. A footnote says that Hardy toldRamanujan it also appears in an 1857 paper by Liouville.
[57] Ramanujan, Some Formulæ in the Analytic Theory of Numbers, eq. (F); Papers p.134
[58] Apostol, Modular Functions ..., ch. 6 eq. 4
[59] Apostol, Modular Functions ..., ch. 6 eq. 3
4.13 References
• Tom M. Apostol (1976), Introduction to Analytic Number Theory, Springer Undergraduate Texts in Mathe-matics, ISBN 0-387-90163-9
• Apostol, Tom M. (1989), Modular Functions and Dirichlet Series in Number Theory (2nd Edition), New York:Springer, ISBN 0-387-97127-0
• Cohen, Henri (1993), A Course in Computational Algebraic Number Theory, Berlin: Springer, ISBN 3-540-55640-0
• Edwards, Harold (1977). Fermat’s Last Theorem. New York: Springer. ISBN 0-387-90230-9.
• Hardy, G. H. (1999), Ramanujan: Twelve Lectures on Subjects Suggested by his Life and work, Providence RI:AMS / Chelsea, ISBN 978-0-8218-2023-0
• Hardy, G. H.; Wright, E. M. (1979) [1938], An Introduction to the Theory of Numbers (5th ed.), Oxford:Clarendon Press, ISBN 0-19-853171-0, MR 0568909, Zbl 0423.10001
• Jameson, G. J. O. (2003), The Prime Number Theorem, Cambridge University Press, ISBN 0-521-89110-8
• Koblitz, Neal (1984), Introduction to Elliptic Curves and Modular Forms, New York: Springer, ISBN 0-387-97966-2
• Landau, Edmund (1966), Elementary Number Theory, New York: Chelsea
• William J. LeVeque (1996), Fundamentals of Number Theory, Courier Dover Publications, ISBN 0-486-68906-9
• Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath andCompany, LCCN 77-171950
• Elliott Mendelson (1987), Introduction to Mathematical Logic, CRC Press, ISBN 0-412-80830-7
• Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: PrenticeHall, LCCN 77-81766
• Ramanujan, Srinivasa (2000), Collected Papers, Providence RI: AMS / Chelsea, ISBN 978-0-8218-2076-6
4.14 Further reading
• Schwarz, Wolfgang; Spilker, Jürgen (1994), Arithmetical Functions. An introduction to elementary and analyticproperties of arithmetic functions and to some of their almost-periodic properties, London Mathematical SocietyLecture Note Series 184, Cambridge University Press, ISBN 0-521-42725-8, Zbl 0807.11001
4.15. EXTERNAL LINKS 23
4.15 External links• Hazewinkel, Michiel, ed. (2001), “Arithmetic function”, Encyclopedia of Mathematics, Springer, ISBN 978-
1-55608-010-4
• Matthew Holden, Michael Orrison, Michael Varble Yet another Generalization of Euler’s Totient Function
• Huard, Ou, Spearman, and Williams. Elementary Evaluation of Certain Convolution Sums Involving DivisorFunctions Elementary (i.e. not relying on the theory of modular forms) proofs of divisor sum convolutions,formulas for the number of ways of representing a number as a sum of triangular numbers, and related results.
• Dineva, Rosica, The Euler Totient, the Möbius, and the Divisor Functions
• László Tóth, Menon’s Identity and arithmetical sums representing functions of several variables
Chapter 5
Average
In colloquial language, an average is the sum of a list of numbers divided by the number of numbers in the list. Inmathematics and statistics, this would be called the arithmetic mean. However, the word average may also refer tothe median, mode, or other central or typical value. In statistics, these are all known as measures of central tendency.
5.1 Calculation
5.1.1 Arithmetic mean
Main article: Arithmetic mean
The most common type of average is the arithmetic mean. If n numbers are given, each number denoted by ai (wherei = 1,2, …, n), the arithmetic mean is the sum of the as divided by n or
AM =1
n
n∑i=1
ai =1
n(a1 + a2 + · · ·+ an)
The arithmetic mean, often simply called the mean, of two numbers, such as 2 and 8, is obtained by finding a valueA such that 2 + 8 = A + A. One may find that A = (2 + 8)/2 = 5. Switching the order of 2 and 8 to read 8 and 2does not change the resulting value obtained for A. The mean 5 is not less than the minimum 2 nor greater than themaximum 8. If we increase the number of terms in the list to 2, 8, and 11, the arithmetic mean is found by solvingfor the value of A in the equation 2 + 8 + 11 = A + A + A. One finds that A = (2 + 8 + 11)/3 = 7.
5.1.2 Pythagorean means
Main article: Pythagorean meansSee also: Mean § Pythagorean means
Along with the arithmetic mean above, the geometric mean and the harmonic mean are known collectively as thePythagorean means.
Geometric mean
The geometric mean of n non-negative numbers is obtained by multiplying them all together and then taking the nthroot. In algebraic terms, the geometric mean of a1, a2, …, an is defined as
24
5.1. CALCULATION 25
GM = n
√√√√ n∏i=1
ai = n√a1a2 · · · an
Geometric mean can be thought of as the antilog of the arithmetic mean of the logs of the numbers.Example: Geometric mean of 2 and 8 is GM =
√2 · 8 = 4
Harmonic mean
Harmonic mean for a non-empty collection of numbers a1, a2, …, an, all different from 0, is defined as the reciprocalof the arithmetic mean of the reciprocals of the ai ' s:
HM =1
1n
∑ni=1
1ai
=n
1a1
+ 1a2
+ · · ·+ 1an
One example where the harmonic mean is useful is when examining the speed for a number of fixed-distance trips.For example, if the speed for going from point A to B was 60 km/h, and the speed for returning from B to A was 40km/h, then the harmonic mean speed is given by
2160 + 1
40
= 48
Inequality concerning AM, GM, and HM
A well known inequality concerning arithmetic, geometric, and harmonic means for any set of positive numbers is
AM ≥ GM ≥ HM
It is easy to remember noting that the alphabetical order of the letters A, G, and H is preserved in the inequality. SeeInequality of arithmetic and geometric means.Thus for the above harmonic mean example: AM = 50, GM ≈ 49, and HM = 48 km/h.
5.1.3 Statistical location
The mode, the median, and the mid-range are often used in addition to the mean as estimates of central tendency indescriptive statistics.
Mode
Main article: Mode (statistics)
The most frequently occurring number in a list is called the mode. For example, the mode of the list (1, 2, 2, 3, 3, 3,4) is 3. It may happen that there are two or more numbers which occur equally often and more often than any othernumber. In this case there is no agreed definition of mode. Some authors say they are all modes and some say thereis no mode.
Median
Main article: Median
26 CHAPTER 5. AVERAGE
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.20.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
σ = 0.25
σ = 1
modemedianmean
Comparison of arithmetic mean, median and mode of two log-normal distributions with different skewness.
The median is the middle number of the group when they are ranked in order. (If there are an even number ofnumbers, the mean of the middle two is taken.)Thus to find the median, order the list according to its elements’ magnitude and then repeatedly remove the pairconsisting of the highest and lowest values until either one or two values are left. If exactly one value is left, it is themedian; if two values, the median is the arithmetic mean of these two. This method takes the list 1, 7, 3, 13 andorders it to read 1, 3, 7, 13. Then the 1 and 13 are removed to obtain the list 3, 7. Since there are two elements inthis remaining list, the median is their arithmetic mean, (3 + 7)/2 = 5.
5.2 Summary of types
See also: Mean § Other means
The table of mathematical symbols explains the symbols used below.
5.3 Miscellaneous types
Other more sophisticated averages are: trimean, trimedian, and normalized mean, with their generalizations.[1]
One can create one’s own average metric using the generalized f-mean:
y = f−1
(1
n[f(x1) + f(x2) + · · ·+ f(xn)]
)where f is any invertible function. The harmonic mean is an example of this using f(x) = 1/x, and the geometric
5.4. MOVING AVERAGE 27
mean is another, using f(x) = log x.However, this method for generating means is not general enough to capture all averages. A more general method[2]
for defining an average takes any function g(x1, x2, …, xn) of a list of arguments that is continuous, strictly increasingin each argument, and symmetric (invariant under permutation of the arguments). The average y is then the valuethat, when replacing each member of the list, results in the same function value: g(y, y, …, y) = g(x1, x2, …, xn).This most general definition still captures the important property of all averages that the average of a list of identicalelements is that element itself. The function g(x1, x2, …, xn) = x1+x2+ ··· + xn provides the arithmetic mean. Thefunction g(x1, x2, …, xn) = x1x2···xn (where the list elements are positive numbers) provides the geometric mean.The function g(x1, x2, …, xn) = −(x1−1+x2−1+ ··· + xn−1) (where the list elements are positive numbers) provides theharmonic mean.[2]
5.3.1 Average percentage return and CAGR
Main article: Compound annual growth rate
A type of average used in finance is the average percentage return. It is an example of a geometric mean. When thereturns are annual, it is called the Compound Annual Growth Rate (CAGR). For example, if we are considering aperiod of two years, and the investment return in the first year is −10% and the return in the second year is +60%,then the average percentage return or CAGR, R, can be obtained by solving the equation: (1 − 10%) × (1 + 60%) =(1 − 0.1) × (1 + 0.6) = (1 + R) × (1 + R). The value of R that makes this equation true is 0.2, or 20%. This means thatthe total return over the 2-year period is the same as if there had been 20% growth each year. Note that the orderof the years makes no difference – the average percentage returns of +60% and −10% is the same result as that for−10% and +60%.This method can be generalized to examples in which the periods are not equal. For example, consider a period of ahalf of a year for which the return is −23% and a period of two and a half years for which the return is +13%. Theaverage percentage return for the combined period is the single year return, R, that is the solution of the followingequation: (1 − 0.23)0.5 × (1 + 0.13)2.5 = (1 + R)0.5+2.5, giving an average percentage return R of 0.0600 or 6.00%.
5.4 Moving average
Main article: Moving average
Given a time series such as daily stock market prices or yearly temperatures people often want to create a smootherseries.[3] This helps to show underlying trends or perhaps periodic behavior. An easy way to do this is to choose anumber n and create a new series by taking the arithmetic mean of the first n values, then moving forward one placeand so on. This is the simplest form of moving average. More complicated forms involve using a weighted average.The weighting can be used to enhance or suppress various periodic behavior and there is very extensive analysis ofwhat weightings to use in the literature on filtering. In digital signal processing the term “moving average” is usedeven when the sum of the weights is not 1.0 (so the output series is a scaled version of the averages).[4] The reasonfor this is that the analyst is usually interested only in the trend or the periodic behavior. A further generalization isan “autoregressive moving average”. In this case the average also includes some of the recently calculated outputs.This allows samples from further back in the history to affect the current output.
5.5 Etymology
According to the Oxford English Dictionary, “few words have received more etymological investigation.”[5] In the16th century average meant a customs duty, or the like, and was used in the Mediterranean area. It came to mean thecost of damage sustained at sea. From that came an “average adjuster” who decided how to apportion a loss betweenthe owners and insurers of a ship and cargo.Marine damage is either particular average, which is borne only by the owner of the damaged property, or generalaverage, where the owner can claim a proportional contribution from all the parties to the marine venture. The typeof calculations used in adjusting general average gave rise to the use of “average” to mean “arithmetic mean”.
28 CHAPTER 5. AVERAGE
A second English usage, documented as early as 1674 and sometimes spelled “averish,” is as the residue and secondgrowth of field crops, which were considered suited to consumption by draught animals (“avers”).[6]
The root is found in Arabic as awar, in Italian as avaria, in French as avarie and in Dutch as averij. It is unclear inwhich language the word first appeared.There is earlier (from at least the 11th century), unrelated use of the word. It appears to be an old legal term fora tenant’s day labour obligation to a sheriff, probably anglicised from “avera” found in the English Domesday Book(1085).
5.6 See also• Average absolute deviation
• Law of averages
• Expected value
5.7 References[1] Merigo, Jose M.; Cananovas, Montserrat (2009). “The Generalized Hybrid Averaging Operator and its Application in
Decision Making”. Journal of Quantitative Methods for Economics and Business Administration 9: 69–84. ISSN 1886-516X.
[2] John Bibby (1974). “Axiomatisations of the average and a further generalisation of monotonic sequences”. GlasgowMathematical Journal, vol. 15, pp. 63–65.
[3] Box, George E.P.; Jenkins, Gwilym M. (1976). Time Series Analysis: Forecasting and Control (revised ed.). Holden-Day.ISBN 0816211043.
[4] Haykin, Simon (1986). Adaptive Filter Theory. Prentice-Hall. ISBN 0130040525.
[5] “average”. Oxford English Dictionary (3rd ed.). Oxford University Press. September 2005. (Subscription or UK publiclibrary membership required.)
[6] Ray, John (1674). A Collection of English Words Not Generally Used. London: H. Bruges. Retrieved 18 May 2015.
5.8 External links• Median as a weighted arithmetic mean of all Sample Observations
• Calculations and comparison between arithmetic and geometric mean of two values
Chapter 6
Average order of an arithmetic function
In number theory, an average order of an arithmetic function is some simpler or better-understood function whichtakes the same values “on average”.Let f be an arithmetic function. We say that an average order of f is g if
∑n≤x
f(n) ∼∑n≤x
g(n)
as x tends to infinity.It is conventional to choose an approximating function g that is continuous and monotone. But even so an averageorder is of course not unique.In cases where the limit
limN→∞
1
N
∑n≤N
f(n) = c
exists, it is said that f has a mean value (average value) c.
6.1 Examples
• An average order of d (n), the number of divisors of n, is log n;
• An average order of σ (n), the sum of divisors of n, is n π2 / 6;
• An average order of φ (n), Euler’s totient function of n, is 6 n / π2;
• An average order of r (n), the number of ways of expressing n as a sum of two squares, is π n;
• The average order of representations of a natural number as a sum of three squares is 4π n / 3;
• The average number of decompositions of a natural number into a sum of one or more consecutive primenumbers is n log 2;
• An average order of ω (n), the number of distinct prime factors of n, is log log n;
• An average order of Ω (n), the number of prime factors of n, is log log n;
• The prime number theorem is equivalent to the statement that the von Mangoldt function Λ (n) has averageorder 1;
• An average order of μ (n), the Möbius function, is zero; this is again equivalent to the prime number theorem.
29
30 CHAPTER 6. AVERAGE ORDER OF AN ARITHMETIC FUNCTION
6.2 Calculating mean values using Dirichlet series
In case F is of the form
F (n) =∑d|n
f(d),
for some arithmetic function f(n), one has,
∑n≤x
F (n) =∑d≤x
f(d)∑
n≤x,d|n
1 =∑d≤x
f(d)[x/d] = x∑d≤x
f(d)
d+O(
∑d≤x
|f(d)|). (1)
This identity often provides a practical way to calculate the mean value in terms of the Riemann zeta function. Thisis illustrated in the following example.
6.2.1 The density of the k-th power free integers in N
For an integer k ≥ 1 the set Qk of k-th-power-free integers is
Qk := n ∈ Z | n by divisible not is dk integer any for d ≥ 2.
We calculate the natural density of these numbers in N, that is, the average value of 1Qk, denoted by δ(n), in termsof the zeta function.The function δ is multiplicative, and since it is bounded by 1, its Dirichlet series converges absolutely in the half-planeRe(s)>1, and there has Euler product
∑Qk
n−s =∑n
δ(n)n−s =∏p
(1 + p−s + · · ·+ p−s(k−1)) =∏p
(1− p−sk
1− p−s
)=
ζ(s)
ζ(sk).
By the Möbius inversion formula, we get
1
ζ(ks)=∑n
µ(n)n−ks,
where µ stands for the Möbius function. Equivalently,
1
ζ(ks)=∑n
f(n)n−s,
where f(n) =
µ(d) n = dk
0 otherwise,and hence,
ζ(s)
ζ(sk)=∑n
(∑d|n
f(d))n−s.
By comparing the coefficients, we get
6.2. CALCULATING MEAN VALUES USING DIRICHLET SERIES 31
δ(n) =∑d|n
f(d)n−s.
Using (1), we get
∑d≤x
δ(d) = x∑d≤x
(f(d)/d) +O(x1/k).
We conclude that,
∑n∈Qk,n≤x
1 =x
ζ(k)+O(x1/k),
Where for this we used the relation
∑n
(f(n)/n) =∑n
f(nk)n−k =∑n
µ(n)n−k =1
ζ(k),
which follows from the Möbius inversion formula.In particular, the density of the square-free integers is ζ(2)−1 = 6
π2 .
6.2.2 Visibility of lattice points
We say that two lattice points are visible from one another if there is no lattice point on the open line segment joiningthem.Now, if gcd(a, b) = d > 1, then writing a = da2, b = db2 one observes that the point (a2, b2) is on the line segmentwhich joins (0,0) to (a, b) and hence (a, b) is not visible from the origin. Thus (a, b) is visible from the origin impliesthat (a, b) = 1. Conversely, it is also easy to see that gcd(a, b) = 1 implies that there is no other integer lattice pointin the segment joining (0,0) to (a,b). Thus, (a, b) is visible from (0,0) if and only if gcd(a, b) = 1.Notice that φ(n)n is the probability of a random point on the square (r, s) ∈ N : max(|r|, |s|) = n to be visiblefrom the origin.Thus, one can show that the natural density of the points which are visible from the origin is given by the average,
limN→∞
1
N
∑n≤N
φ(n)
n=
6
π2=
1
ζ(2).
interestingly, 1ζ(2) is also the natural density of the square-free numbers in N. In fact, this is not a coincidence.
Consider the k-dimensional lattice, Zk . The natural density of the points which are visible from the origin is 1ζ(k) ,
which is also the natural density of the k-th free integers in N.
6.2.3 Divisor functions
Consider the generalization of d(n) :
σα(n) =∑d|n
dα.
The following are true:
32 CHAPTER 6. AVERAGE ORDER OF AN ARITHMETIC FUNCTION
∑n≤x
σα(n) =
∑n≤x σα(n) =
ζ(α+1)α+1 xα+1 +O(xβ) ifα > 0,∑
n≤x σ−1(n) = ζ(2)x+O(logx) ifα = −1,∑n≤x σα(n) = ζ(−α+ 1)x+O(xmax(0,1+α)) otherwise.
where β = max(1, α) .
6.3 Better average order
This notion is best discussed through an example. From
∑n≤x
d(n) = x logx+ (2γ − 1)x+ o(x)
( γ is the Euler–Mascheroni constant) and
∑n≤x
logn = x logx− x+O(logx),
we have the asymptotic relation
∑n≤x
(d(n)− (logn+ 2γ)) = o(x) (x→ ∞),
which suggests that the function logn+ 2γ is a better choice of average order for d(n) than simply logn .
6.4 Mean values over F [x]
6.4.1 Definition
Let h(x) be a function on the set of monic polynomials over F . For n ≥ 1 we define
Aven(h) =1
qn
∑fmonic ,deg(f)=n
h(f).
This is the mean value (average value) of h on the set of monic polynomials of degree n. We say that g(n) is anaverage order of h if
Aven(h) ∼ g(n)
as n tends to infinity.In cases where the limit,
limn→∞
Aven(h) = c
exists, it is said that h has a mean value (average value) c.
6.4. MEAN VALUES OVER FQ[X] 33
6.4.2 Zeta function and Dirichlet series in F [X]
Let F [X]=A be the ring of polynomials over the finite field F .Let h be a polynomial arithmetic function (i.e. a function on set of monic polynomials over A). Its correspondingDirichlet series define to be
Dh(s) =∑fmonic
h(f)|f |−s,
where for g ∈ A , set |g| = qdeg(g) if g = 0 , and |g| = 0 otherwise.The polynomial zeta function is then
ζA(s) =∑fmonic
|f |−s.
Similar to the situation in N, every Dirichlet series of a multiplicative function h has a product representation (Eulerproduct):
Dh(s) =∏P
(
∞∑n=0
h(Pn)|P |−sn),
Where the product runs over all monic irreducible polynomials P.For example, the product representation of the zeta function is as for the integers: ζA(s) =
∏P (1− |P |−s)−1 .
Unlike the classical zeta function, ζA(s) is a simple rational function:ζA(s) =
∑f (|f |−s) =
∑n
∑deg(f)=n q
−sn =∑n(q
n−sn) = (1− q1−s)−1.
In a similar way, If ƒ and g are two polynomial arithmetic functions, one defines ƒ * g, the Dirichlet convolution of ƒand g, by
(f ∗ g)(m) =∑d |m
f(m)g(md
)=∑ab=m
f(a)g(b)
where the sum extends over all monic divisors d of m, or equivalently over all pairs (a, b) of monic polynomialswhose product is m. The identity DhDg = Dh∗g still holds. Thus, like in the elementary theory, the polynomialDirichlet series and the zeta function has a connection with the notion of mean values in the context of polynomials.The following examples illustrate it.
6.4.3 Examples
The density of the k-th power free polynomials in F [X]
Define δ(f) to be 1 if f is k-th power free and 0 otherwise.We calculate the average value of δ , which is the density of the k-th power free polynomials in F [X], in the samefashion as in the integers.By multiplicativity of δ :
∑f
δ(f)
|f |s=∏P
(
k−1∑j=0
(|P |−js)) =∏P
1− |P |−sk
1− |P |−s=
ζA(s)
ζA(sk)=
1− q1−ks
1− q1−s=
ζA(s)
ζA(ks)
34 CHAPTER 6. AVERAGE ORDER OF AN ARITHMETIC FUNCTION
Denote bn the number of k-th power monic polynomials of degree n, we get
∑f
δ(f)
|f |s=∑n
∑deff=n
δ(f)|f |−s =∑n
bnq−sn.
Making the substitution u = q−s we get:
1− quk
1− qu=
∞∑n=0
bnun.
Finally, expand the left-hand side in a geometric series and compare the coefficients on un on both sides, to concludethat
bn =
qn n ≤ k − 1
qn(1− q1−k) otherwiseHence,Aven(δ) = 1− q1−k = 1
ζA(k)
And since it doesn't depend on n this is also the mean value of δ(f) .
Polynomial Divisor functions
In F [X], we define
σk(m) =∑
f |m,monic|f |k.
We will compute Aven(σk) for k ≥ 1 .First, notice that
σk(m) = h ∗ I(m)
where h(f) = |f |k and I(f) = 1 ∀f .Therefore,
∑m
σk(m)|m|−s = ζA(s)∑m
h(m)|m|−s.
Substitute q−s = u we get,
LHS =∑n(∑
deg(m)=n σk(m))un , and by Cauchy product we get,
RHS =∑n
qn(1−s)∑n
(∑
deg(m)=n
h(m))un
=∑n
qnun∑l
qlqlkul
=∑n
(n∑j=0
qn−jqjk+j)
=∑n
(qn(1− qk(n+1)
1− qk))un.
6.4. MEAN VALUES OVER FQ[X] 35
Finally we get that,
Avenσk =1− qk(n+1)
1− qk.
Notice that
qnAvenσk = qn(k+1)(1− q−k(n+1)
1− q−k) = qn(k+1)(
ζ(k + 1)
ζ(kn+ k + 1))
Thus, if we set x = qn then the above result reads
∑deg(m)=n,mmonic
σk(m) = xk+1(ζ(k + 1)
ζ(kn+ k + 1))
which resembles the analogous result for the integers:∑n≤x σk(n) =
ζ(k+1)k+1 xk+1 +O(xk)
Number of divisors
Let d(f) be the number of monic divisors of f and let D(n) be the sum of d(f) over all monics of degree n.ζA(s)
2 = (∑h |h|−s)(
∑g |g|−s) =
∑f (∑hg=f 1)|f |−s =
∑f d(f)|f |−s = Dd(s) =
∑∞n=0D(n)un
where u = q−s .Expanding the right-hand side into power series we get,
D(n) = (n+ 1)qn.
Substitute x = qn the above equation becomes:
D(n) = x logq(x)+xwhich resembles closely the analogous result for integers∑nk=1 d(k) = x logx+
(2γ − 1)x+O(√x) , where γ is Euler constant.
It is interesting to note that not a lot is known about the error term for the integers, while in the polynomials case,there is no error term! This is because of the very simple nature of the zeta function ζA(s) , and that it has NO zeros.
Polynomial von Mangoldt function
The Polynomial von Mangoldt function is defined by: ΛA(f) =
log |P | if f = |P |kmonic prime some for P integer and k ≥ 1,
0 otherwise.Where the logarithm is taken on the basis of q.Proposition. The mean value of ΛA is exactly 1.Proof. Let m be a monic polynomial, and let m =
∏li=1 P
eii be the prime decomposition of m.
We have,
∑f |m
ΛA(f) =∑
(i1,...,il)|0≤ij≤ej
ΛA(l∏
j=1
Pijj ) =
l∑j=1
ei∑i=1
ΛA(Pij ) =
l∑j=1
ei∑i=1
log |Pj |
=l∑
j=1
ej log |Pj | =l∑
j=1
log |Pj |ej = log |(l∏i=1
P eii )|
= log(m)
36 CHAPTER 6. AVERAGE ORDER OF AN ARITHMETIC FUNCTION
Hence,
I · ΛA(m) = log |m|
and we get that,
ζA(s)DΛA(s) =
∑m
log|m||m|−s.
Now,
∑m
|m|s =∑n
∑degm=n
un =∑n
qnun =∑n
qn(1−s).
Thus,
d
ds
∑m
|m|s = −∑n
log(qn)qn(1−s) = −∑n
∑deg(f)=n
log(qn)q−ns = −∑f
log |f ||f |−s.
We got that:
DΛA(s) =
−ζ ′A(s)ζA(s)
Now,
∑m
ΛA(m)|m|−s =∑n
(∑
deg(m)=n
ΛA(m)q−sm) =∑n
(∑
deg(m)=n
ΛA(m))un =−ζ ′A(s)ζA(s)
=q1−slog(q)
1− q1−s= log(q)
∞∑n=1
qnun
Hence,
∑deg(m)=n
ΛA(m) = qn log(q),
and by dividing by qn we get that,
AvenΛA(m) = log(q) = 1.
Polynomial Euler totient function
Define Euler totient function polynomial analogue, Φ , to be the number of elements in the group (A/fA)∗ . Wehave,
∑deg f=n,fmonic
Φ(f) = q2n(1− q−1).
6.5 See also• Divisor summatory function• Normal order of an arithmetic function• Extremal orders of an arithmetic function
6.6. REFERENCES 37
6.6 References• Hardy, G. H.; Wright, E. M. (2008) [1938]. An Introduction to the Theory of Numbers. Revised by D. R.
Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.). Oxford: Oxford University Press.ISBN 978-0-19-921986-5. MR 2445243. Zbl 1159.11001. Pp. 347–360
• Gérald Tenenbaum (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge studies inadvanced mathematics 46. Cambridge University Press. pp. 36–55. ISBN 0-521-41261-7. Zbl 0831.11001.
• Tom M. Apostol (1976), Introduction to Analytic Number Theory, Springer Undergraduate Texts in Mathe-matics, ISBN 0-387-90163-9
• Michael Rosen (2000), Number Theory in Function Fields, Springer Graduate Texts In Mathematics, ISBN0-387-95335-3
• Hugh L. Montgomery , Robert C. Vaughan (2006), Multiplicative Number Theory, Cambridge University Press,ISBN 978-0521849036
• Michael Baakea, Robert V. Moodyb, Peter A.B. Pleasantsc (2000), Diffraction from visible lattice points andkth power free integers, Discrete Mathematics- Journal
Chapter 7
Bell series
In mathematics, the Bell series is a formal power series used to study properties of arithmetical functions. Bell serieswere introduced and developed by Eric Temple Bell.Given an arithmetic function f and a prime p , define the formal power series fp(x) , called the Bell series of fmodulo p as:
fp(x) =∞∑n=0
f(pn)xn.
Two multiplicative functions can be shown to be identical if all of their Bell series are equal; this is sometimes calledthe uniqueness theorem: given multiplicative functions f and g , one has f = g if and only if:
fp(x) = gp(x) for all primes p .
Two series may be multiplied (sometimes called the multiplication theorem): For any two arithmetic functions f andg , let h = f ∗ g be their Dirichlet convolution. Then for every prime p , one has:
hp(x) = fp(x)gp(x).
In particular, this makes it trivial to find the Bell series of a Dirichlet inverse.If f is completely multiplicative, then formally:
fp(x) =1
1− f(p)x.
7.1 Examples
The following is a table of the Bell series of well-known arithmetic functions.
• The Möbius function µ has µp(x) = 1− x.
• Euler’s Totient φ has φp(x) = 1−x1−px .
• The multiplicative identity of the Dirichlet convolution δ has δp(x) = 1.
• The Liouville function λ has λp(x) = 11+x .
• The power function Id has (Idk)p(x) = 11−pkx . Here, Id is the completely multiplicative function Idk(n) =
nk .
• The divisor function σk has (σk)p(x) = 11−(1+pk)x+pkx2 .
38
7.2. SEE ALSO 39
7.2 See also• Bell numbers
7.3 References• Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New
York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001
Chapter 8
Chebyshev function
The Chebyshev function ψ(x), with x < 50
In mathematics, the Chebyshev function is either of two related functions. The first Chebyshev function ϑ(x) orθ(x) is given by
ϑ(x) =∑p≤x
log p
with the sum extending over all prime numbers p that are less than or equal to x.The second Chebyshev function ψ(x) is defined similarly, with the sum extending over all prime powers not ex-ceeding x:
ψ(x) =∑pk≤x
log p =∑n≤x
Λ(n) =∑p≤x
⌊logp x⌋ log p,
where Λ is the von Mangoldt function. The Chebyshev functions, especially the second one ψ(x), are often usedin proofs related to prime numbers, because it is typically simpler to work with them than with the prime-counting
40
8.1. RELATIONSHIPS 41
-50
-40
-30
-20
-10
0
10
20
30
40
50
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
psi(n
)-n+
log(
pi)
n
Chebyshev (summatory von Mangoldt) function
The function ψ(x) − x, for x < 10,000
function, π(x) (See the exact formula, below.) Both Chebyshev functions are asymptotic to x, a statement equivalentto the prime number theorem.Both functions are named in honour of Pafnuty Chebyshev.
8.1 Relationships
The second Chebyshev function can be seen to be related to the first by writing it as
ψ(x) =∑p≤x
k log p
where k is the unique integer such that pk ≤ x and x < pk+1. The values k of are given in A206722. A more directrelationship is given by
ψ(x) =
∞∑n=1
ϑ(x1/n
).
Note that this last sum has only a finite number of non-vanishing terms, as
ϑ(x1/n
)= 0 for n > log2 x =
logxlog 2 , .
42 CHAPTER 8. CHEBYSHEV FUNCTION
-2000
-1500
-1000
-500
0
500
1000
1500
2000
0 1e+06 2e+06 3e+06 4e+06 5e+06 6e+06 7e+06 8e+06 9e+06 1e+07
psi(n
)-n+
log(
pi)
n
Chebyshev (summatory von Mangoldt) function
The function ψ(x) − x, for x < 10 million
The second Chebyshev function is the logarithm of the least common multiple of the integers from 1 to n.
lcm(1, 2, . . . , n) = eψ(n).
Values of lcm(1, 2, . . . , n) for the integer variable n is given at A003418.
8.2 Asymptotics and bounds
The following bounds are known for the Chebyshev functions: (in these formulas pk is the kth prime number p1 = 2,p2 = 3, etc.)
ϑ(pk) ≥ k(ln k + ln ln k − 1 + ln ln k−2.050735
ln k)
for k ≥ 1011,
ϑ(pk) ≤ k(ln k + ln ln k − 1 + ln ln k−2
ln k)
for k ≥ 198,
|ϑ(x)− x| ≤ 0.006788 xln x for x ≥ 10,544,111,
|ψ(x)− x| ≤ 0.006409 xln x for x ≥ exp(22),
0.9999√x < ψ(x)− ϑ(x) < 1.00007
√x+ 1.78 3
√x for x ≥ 121.
Further, under the Riemann hypothesis,
8.3. THE EXACT FORMULA 43
|ϑ(x)− x| = O(x1/2+ε)
|ψ(x)− x| = O(x1/2+ε)
for any ε > 0.
Upper bounds exist for both ϑ(x) and ψ(x) such that,[1]
ϑ(x) < 1.000028x
ψ(x) < 1.03883x
for any x > 0.
An explanation of the constant 1.03883 is given at A206431.
8.3 The exact formula
In 1895, Hans Carl Friedrich von Mangoldt proved an explicit expression for ψ(x) as a sum over the nontrivial zerosof the Riemann zeta function:
ψ0(x) = x−∑ρ
xρ
ρ− ζ ′(0)
ζ(0)− 1
2log(1− x−2).
(The numerical value of ζ'(0)/ζ(0) is log(2π).) Here ρ runs over the nontrivial zeros of the zeta function, and ψ0 isthe same as ψ, except that at its jump discontinuities (the prime powers) it takes the value halfway between the valuesto the left and the right:
ψ0(x) =1
2
∑n≤x
Λ(n) +∑n<x
Λ(n)
=
ψ(x)− 1
2Λ(x) x = 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, . . .
ψ(x) otherwise.
From the Taylor series for the logarithm, the last term in the explicit formula can be understood as a summation ofxω/ω over the trivial zeros of the zeta function, ω = −2,−4,−6, . . . , i.e.
∞∑k=1
x−2k
−2k=
1
2log(1− x−2).
Similarly, the first term, x = x1/1, corresponds to the simple pole of the zeta function at 1. Its being a pole rather thanzero accounts for the opposite sign of the term.
8.4 Properties
A theorem due to Erhard Schmidt states that, for some explicit positive constant K, there are infinitely many naturalnumbers x such that
ψ(x)− x < −K√x
and infinitely many natural numbers x such that
ψ(x)− x > K√x.
44 CHAPTER 8. CHEBYSHEV FUNCTION
In little-o notation, one may write the above as
ψ(x)− x = o(√x).
Hardy and Littlewood prove the stronger result, that
ψ(x)− x = o(√x log log logx
).
8.5 Relation to primorials
The first Chebyshev function is the logarithm of the primorial of x, denoted x#:
ϑ(x) =∑p≤x
log p = log∏p≤x
p = log(x#).
This proves that the primorial x# is asymptotically equal to exp((1+o(1))x), where “o” is the little-o notation (see BigO notation) and together with the prime number theorem establishes the asymptotic behavior of pn#.
8.6 Relation to the prime-counting function
The Chebyshev function can be related to the prime-counting function as follows. Define
Π(x) =∑n≤x
Λ(n)
logn .
Then
Π(x) =∑n≤x
Λ(n)
∫ x
n
dt
t log2 t+
1
logx∑n≤x
Λ(n) =
∫ x
2
ψ(t) dt
t log2 t+ψ(x)
logx .
The transition from Π to the prime-counting function, π , is made through the equation
Π(x) = π(x) +1
2π(x1/2) +
1
3π(x1/3) + · · · .
Certainly π(x) ≤ x , so for the sake of approximation, this last relation can be recast in the form
π(x) = Π(x) +O(√x).
8.7 The Riemann hypothesis
The Riemann hypothesis states that all nontrivial zeros of the zeta function have real part 1/2. In this case, |xρ| = √x
, and it can be shown that
∑ρ
xρ
ρ= O(
√x log2 x).
8.8. SMOOTHING FUNCTION 45
By the above, this implies
π(x) = li(x) +O(√x logx).
Good evidence that RH could be true comes from the fact proposed by Alain Connes and others, that if we differentiatethe von Mangoldt formula with respect to x make x = exp(u). Manipulating, we have the “Trace formula” for theexponential of the Hamiltonian operator satisfying
ζ(1/2 + iH)|n ≥ ζ(1/2 + iEn) = 0,
∑n
eiuEn = Z(u) = eu/2 − e−u/2dψ0
du− eu/2
e3u − eu= Tr(eiuH),
where the “trigonometric sum” can be considered to be the trace of the operator (statistical mechanics) eiuH ,whichis only true if ρ = 1/2 + iE(n).
Using the semiclassical approach the potential of H = T + V satisfies:
Z(u)u1/2√π
∼∫ ∞
−∞ei(uV (x)+π/4) dx
with Z(u) → 0 as u → ∞.
solution to this nonlinear integral equation can be obtained (among others) by V −1(x) ≈√(4π)d
1/2N(x)dx1/2 in order
to obtain the inverse of the potential : πN(E) = Argξ(1/2 + iE)
8.8 Smoothing function
The smoothing function is defined as
ψ1(x) =
∫ x
0
ψ(t) dt.
It can be shown that
ψ1(x) ∼x2
2.
8.9 Variational formulation
The Chebyshev function evaluated at x = exp(t) minimizes the functional
J [f ] =
∫ ∞
0
f(s)ζ ′(s+ c)
ζ(s+ c)(s+ c)ds−
∫ ∞
0
∫ ∞
0
e−stf(s)f(t) ds dt,
so
f(t) = ψ(et)e−ct,
for c > 0.
46 CHAPTER 8. CHEBYSHEV FUNCTION
-1e+07
-5e+06
0
5e+06
1e+07
1.5e+07
2e+07
0 100000 200000 300000 400000 500000 600000 700000 800000 900000 1e+06
psi_
1(n)
-n*n
/2
n
Smoothed Chebyshev function
The difference of the smoothed Chebyshev function and x2/2 for x < 106
8.10 Notes
[1] Rosser, J. Barkley; Schoenfeld, Lowell (1962). “Approximate formulas for some functions of prime numbers.”. Illinois J.Math. 6: 64–94.
• ^ Pierre Dusart, “Estimates of some functions over primes without R.H.”. arXiv:1002.0442
• ^ Pierre Dusart, “Sharper bounds for ψ, θ, π, pk", Rapport de recherche n° 1998-06, Université de Limoges.An abbreviated version appeared as “The kth prime is greater than k(ln k + ln ln k − 1) for k ≥ 2”, Mathematicsof Computation, Vol. 68, No. 225 (1999), pp. 411–415.
• ^ Erhard Schmidt, "Über die Anzahl der Primzahlen unter gegebener Grenze”, Mathematische Annalen, 57(1903), pp. 195–204.
• ^ G .H. Hardy and J. E. Littlewood, “Contributions to the Theory of the Riemann Zeta-Function and theTheory of the Distribution of Primes”, Acta Mathematica, 41 (1916) pp. 119–196.
• ^ Davenport, Harold (2000). In Multiplicative Number Theory. Springer. p. 104. ISBN 0-387-95097-4.Google Book Search.
8.11 References
• Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, NewYork-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001
8.12. EXTERNAL LINKS 47
8.12 External links• Weisstein, Eric W., “Chebyshev functions”, MathWorld.
• Mangoldt summatory function at PlanetMath.org.
• Chebyshev functions at PlanetMath.org.
• Riemann’s Explicit Formula, with images and movies
Chapter 9
Crank of a partition
In number theory, the crank of a partition of an integer is a certain integer associated with the partition. The termwas first introduced without a definition by Freeman Dyson in a 1944 paper published in Eureka, a journal publishedby the Mathematics Society of Cambridge University.[1] Dyson then gave a list of properties this yet to be definedquantity should have. George E. Andrews and F.G. Garvan in 1988 discovered a definition for crank satisfying theproperties hypothesized for it by Dyson.[2]
9.1 Dyson’s crank
Let n be a non-negative integer and let p(n) denote the number of partitions of n (p(0) is defined to be 1). SrinivasaRamanujan in a paper[3] published in 1918 stated and proved the following congruences for the partition functionp(n), since known as Ramanujan congruences.
• p(5n + 4) ≡ 0 (mod 5)
• p(7n + 5) ≡ 0 (mod 7)
• p(11n + 6) ≡ 0 (mod 11)
These congruences imply that partitions of numbers of the form 5n + 4 (respectively, of the forms 7n + 5 and 11n + 6) can be divided into 5 (respectively, 7 and 11) subclasses of equal size. The then known proofs of these congruenceswere based on the ideas of generating functions and they did not specify a method for the division of the partitionsinto subclasses of equal size.In his Eureka paper Dyson proposed the concept of the rank of a partition. The rank of a partition is the integerobtained by subtracting the number of parts in the partition from the largest part in the partition. For example, therank of the partition λ = 4, 2, 1, 1, 1 of 9 is 4 − 5 = − 1. Denoting by N(m , q, n), the number of partitionsof n whose ranks are congruent to m modulo q, Dyson considered N(m , 5 , 5 n + 4) and N(m , 7 , 7 n + 5) forvarious values of n and m. Based on empirical evidences Dyson formulated the following conjectures known as rankconjectures.For all non-negative integers n we have:
• N(0 , 5 , 5n + 4) = N(1 , 5 , 5n + 4) = N(2 , 5 , 5n + 4) = N(3 , 5 , 5n + 4) = N(4 , 5 , 5n + 4).
• N(0 , 7 , 7n + 5) = N(1 , 7 , 7n + 5) = N(2 , 7 , 7n + 5) = N(3 , 7 , 7n + 5) = N(4 , 7 , 7n + 5) = N(5 , 7 , 7n +5) = N(6 , 7 , 7 n + 5)
Assuming that these conjectures are true, they provided a way of splitting up all partitions of numbers of the form5n + 4 into five classes of equal size: Put in one class all those partitions whose ranks are congruent to each othermodulo 5. The same idea can be applied to divide the partitions of integers of the from 7n + 6 into seven equallynumerous classes. But the idea fails to divide partitions of integers of the form 11n + 6 into 11 classes of the samesize, as the following table shows.
48
9.2. DEFINITION OF CRANK 49
Partitions of the integer 6 ( 11n + 6 with n = 0 ) divided into classes based on ranks
Thus the rank cannot be used to prove the theorem combinatorially. However, Dyson wrote,I hold in fact :
• that there exists an arithmetical coefficient similar to, but more recondite than, the rank of a partition; I shall callthis hypothetical coefficient the “crank” of the partition and denote by M(m , q , n) the number of partitions of nwhose crank is congruent to m modulo q;
• that M(m , q , n) = M(q − m , q , n);
• that M(0 , 11 , 11n + 6) = M(1 , 11 , 11n + 6) = M(2 , 11 , 11n + 6) = M(3 , 11 , 11n + 6) = M(4 , 11 , 11n+ 6);
• that . . .
Whether these guesses are warranted by evidence, I leave it to the reader to decide. Whatever the final verdict ofposterity may be, I believe the “crank” is unique among arithmetical functions in having been named before it wasdiscovered. May it be preserved from the ignominious fate of the planet Vulcan.
9.2 Definition of crank
In a paper[2] published in 1988 George E. Andrews and F. G. Garvan defined the crank of a partition as follows:
For a partition λ, let l(λ) denote the largest part of λ, ω(λ) denote the number of 1’s in λ, and μ(λ) denotethe number of parts of λ larger than ω (λ). The crank c(λ) is given by
c(λ) =
l(λ) if ω(λ) = 0
µ(λ)− ω(λ) if ω(λ) > 0.
The cranks of the partitions of the integers 4, 5, 6 are computed in the following tables.
Cranks of the partitions of 4Cranks of the partitions of 5Cranks of the partitions of 6
9.3 Notations
For all integers n ≥ 0 and all integers m, the number of partitions of n with crank equal to m is denoted by M(m,n)except for n=1 where M(−1,1) = -M(0,1) = M(1,1) = 1 as given by the following generating function. The numberof partitions of n with crank equal to m modulo q is denoted by M(m,q,n).The generating function for M(m,n) is given below:
∞∑n=0
∞∑m=−∞
M(m,n)zmqn =∞∏n=1
(1− qn)
(1− zqn)(1− z−1qn)
9.4 Basic result
Andrews and Garvan proved the following result[2] which shows that the crank as defined above does meet the con-ditions given by Dyson.
50 CHAPTER 9. CRANK OF A PARTITION
• M(0, 5 ,5n + 4) = M(1, 5, 5n + 4) = M(2, 5, 5n + 4) = M(3, 5, 5n + 4) = M(4, 5, 5n + 4) = p(5n + 4) / 5
• M(0, 7, 7n + 5) = M(1, 7, 7n + 5) = M(2, 7, 7n + 5) = M(3, 7, 7n + 5) = M(4, 7, 7n + 5) = M(5, 7, 7n + 5) =M(6, 7, 7n + 5) = p(7n + 5) / 7
• M(0, 11, 11n + 6) = M(1, 11, 11n + 6) = M(2, 11, 11n + 6) = M(3, 11, 11n + 6) = . . . = M(9, 11, 11n + 6) =M(10, 11, 11n + 6) = p(11n + 6) / 11
The concepts of rank and crank can both be used to classify partitions of certain integers into subclasses of equal size.However the two concepts produce different subclasses of partitions. This is illustrated in the following two tables.
Classification of the partitions of the integer 9 based on cranksClassification of the partitions of the integer 9 based on ranks
9.5 Ramanujan and cranks
Recent work by Bruce C. Berndt and his coauthors have revealed that Ramanujan knew about the crank, although notin form that Andrews and Garvan have defined. In a systematic study of the Lost Notebook of Ramanujan, Berndtand his coauthors have given substantial evidence that Ramanujan knew about the dissections of the crank generatingfunction.[4]
9.6 References[1] Freeman J. Dyson (1944). “Some Guesses in The Theory of Partitions”. Eureka (Cambridge) 8: 10–15.
[2] George E. Andrews; F.G. Garvan (April 1988). “Dyson’s crank of a partition” (PDF).Bulletin (New Series) of The AmericanMathematical Society 18 (2). Retrieved 26 November 2012.
[3] Srinivasa, Ramanujan (1919). “Some properties of p(n), number of partitions of n". Proceedings of the Cambridge Philo-sophical Society XIX: 207–210.
[4] Manjil P. Saikia (2013). “Cranks in Ramanujan’s Lost Notebook” (PDF). Journal of the Assam Academy of Mathematics6.
Chapter 10
Dedekind psi function
In number theory, the Dedekind psi function is the multiplicative function on the positive integers defined by
ψ(n) = n∏p|n
(1 +
1
p
),
where the product is taken over all primes p dividing n (by convention, ψ(1) is the empty product and so has value1). The function was introduced by Richard Dedekind in connection with modular functions.The value of ψ(n) for the first few integers n is:
1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24 ... (sequence A001615 in OEIS).
ψ(n) is greater than n for all n greater than 1, and is even for all n greater than 2. If n is a square-free number thenψ(n) = σ(n).The ψ function can also be defined by setting ψ(pn) = (p+1)pn-1 for powers of any prime p, and then extending thedefinition to all integers by multiplicativity. This also leads to a proof of the generating function in terms of theRiemann zeta function, which is
∑ ψ(n)
ns=ζ(s)ζ(s− 1)
ζ(2s).
This is also a consequence of the fact that we can write as a Dirichlet convolution of ψ = Id ∗ |µ| .
10.1 Higher Orders
The generalization to higher orders via ratios of Jordan’s totient is
ψk(n) =J2k(n)
Jk(n)
with Dirichlet series
∑n≥1
ψk(n)
ns=ζ(s)ζ(s− k)
ζ(2s)
It is also the Dirichlet convolution of a power and the square of the Möbius function,
51
52 CHAPTER 10. DEDEKIND PSI FUNCTION
ψk(n) = nk ∗ µ2(n)
If
ϵ2 = 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 . . .
is the characteristic function of the squares, another Dirichlet convolution leads to the generalized σ-function,
ϵ2(n) ∗ ψk(n) = σk(n)
10.2 References• Goro Shimura (1971). Introduction to the Arithmetic Theory of Automorphic Functions. Princeton. (page 25,
equation (1))
• Carella, N. A. (2010). “Squarefree Integers And Extreme Values Of Some Arithmetic Functions”. arXiv:1012.4817.
• Mathar, Richard J. (2011). “Survey of Dirichlet series of multiplicative arithmetic functions”. arXiv:1106.4038.Section 3.13.2
• A065958 is ψ2, A065959 is ψ3, and A065960 is ψ4
10.3 External links• Weisstein, Eric W., “Dedekind Function”, MathWorld.
Chapter 11
Dirichlet convolution
In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important innumber theory. It was developed by Peter Gustav Lejeune Dirichlet, a German mathematician.
11.1 Definition
If f and g are two arithmetic functions (i.e. functions from the positive integers to the complex numbers), one definesa new arithmetic function f ∗ g, the Dirichlet convolution of f and g, by
(f ∗ g)(n) =∑d |n
f(d)g(nd
)=∑ab=n
f(a)g(b)
where the sum extends over all positive divisors d of n, or equivalently over all distinct pairs (a, b) of positive integerswhose product is n.
11.2 Properties
The set of arithmetic functions forms a commutative ring, the Dirichlet ring, under pointwise addition (i.e. f + gis defined by (f + g)(n) = f(n) + g(n)) and Dirichlet convolution. The multiplicative identity is the unit function εdefined by ε(n) = 1 if n = 1 and ε(n) = 0 if n > 1. The units (i.e. invertible elements) of this ring are the arithmeticfunctions f with f(1) ≠ 0.Specifically, Dirichlet convolution is[1] associative,
(f ∗ g) ∗ h = f ∗ (g ∗ h),
distributes over addition
f ∗ (g + h) = f ∗ g + f ∗ h = (g + h) ∗ f,
is commutative,
f ∗ g = g ∗ f,
and has an identity element,
f ∗ ε = ε ∗ f = f.
53
54 CHAPTER 11. DIRICHLET CONVOLUTION
Furthermore, for each f for which f(1) ≠ 0 there exists a g such that f ∗ g = ε, called the Dirichlet inverse of f.The Dirichlet convolution of two multiplicative functions is again multiplicative, and every multiplicative function hasa Dirichlet inverse that is also multiplicative. The article on multiplicative functions lists several convolution relationsamong important multiplicative functions.Given a completely multiplicative function f, then f(g ∗ h) = (fg) ∗ (fh), where juxtaposition represents pointwisemultiplication.[2] The convolution of two completely multiplicative functions is a fortiori multiplicative, but not nec-essarily completely multiplicative.
11.3 Examples
In these formulas
ε is the multiplicative identity. (I.e. ε(1) = 1, all other values 0.)1 is the constant function whose value is 1 for all n. (I.e. 1(n) = 1.) Keep in mind that 1 is not theidentity.1C, where C⊂Z is a set is the indicator function. (I.e. 1C(n) = 1 if n ∈ C, 0 otherwise.)Id is the identity function whose value is n. (I.e. Id(n) = n.)Idk is the kth power function. (I.e. Idk(n) = nk.)
The other functions are defined in the article arithmetical function.
• 1 ∗ μ = ε (the Dirichlet inverse of the constant function 1 is the Möbius function.) This implies
• g = f ∗ 1 if and only if f = g ∗ μ (the Möbius inversion formula).
• λ ∗ |μ| = ε where λ is Liouville’s function.
• λ ∗ 1 = 1S where Sq = 1, 4, 9, ... is the set of squares
• Idk ∗ (Idk μ) = ε
• σk = Idk ∗ 1 definition of the divisor function σk
• σ = Id ∗ 1 definition of the function σ = σ1
• d = 1 ∗ 1 definition of the function d(n) = σ0
• Idk = σk ∗ μ Möbius inversion of the formulas for σk, σ, and d.
• Id = σ ∗ μ
• 1 = d ∗ μ
• d3 ∗ 1 = (d ∗ 1)2
• φ ∗ 1 = Id This formula is proved in the article Euler’s totient function.
• Jk ∗ 1 = Idk The Jordan’s totient function.
• (IdsJr) ∗ Js = Js ₊ r
• σ = φ ∗ d Proof: convolve 1 to both sides of Id = φ ∗ 1.
• Λ ∗ 1 = log where Λ is von Mangoldt function.
11.4. DIRICHLET INVERSE 55
11.4 Dirichlet inverse
Given an arithmetic function f its Dirichlet inverse g = f−1 may be calculated recursively (i.e. the value of g(n) is interms of g(m) for m < n) from the definition of Dirichlet inverse.For n = 1:
(f ∗ g) (1) = f(1) g(1) = ε(1) = 1, so
g(1) = 1/f(1). This implies that f does not have a Dirichlet inverse if f(1) = 0.
For n = 2
(f ∗ g) (2) = f(1) g(2) + f(2) g(1) = ε(2) = 0,g(2) = −1/f(1) (f(2) g(1)),
For n = 3
(f ∗ g) (3) = f(1) g(3) + f(3) g(1) = ε(3) = 0,g(3) = −1/f(1) (f(3) g(1)),
For n = 4
(f ∗ g) (4) = f(1) g(4) + f(2) g(2) + f(4) g(1) = ε(4) = 0,g(4) = −1/f(1) (f(4) g(1) + f(2) g(2)),
and in general for n > 1,
g(n) =−1
f(1)
∑d |nd<n
f(nd
)g(d).
Since the only division is by f(1) this shows that f has a Dirichlet inverse if and only if f(1) ≠ 0.Here is a useful table of Dirichlet inverses of common arithmetic functions:
11.5 Dirichlet series
If f is an arithmetic function, one defines its Dirichlet series generating function by
DG(f ; s) =∞∑n=1
f(n)
ns
for those complex arguments s for which the series converges (if there are any). The multiplication of Dirichlet seriesis compatible with Dirichlet convolution in the following sense:
DG(f ; s)DG(g; s) = DG(f ∗ g; s)
for all s for which both series of the left hand side converge, one of them at least converging absolutely (note thatsimple convergence of both series of the left hand side DOES NOT imply convergence of the right hand side!). Thisis akin to the convolution theorem if one thinks of Dirichlet series as a Fourier transform.
56 CHAPTER 11. DIRICHLET CONVOLUTION
11.6 Related Concepts
The restriction of the divisors in the convolution to unitary, bi-unitary or infinitary divisors defines similar commuta-tive operations which share many features with the Dirichlet convolution (existence of a Möbius inversion, persistenceof multiplicativity, definitions of totients, Euler-type product formulas over associated primes, etc.).
11.7 References[1] Proofs of all these facts are in Chan, ch. 2
[2] A proof is in the article Completely multiplicative function#Proof of pseudo-associative property.
• Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, NewYork-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001
• Chan Heng Huat (2009). Analytic Number Theory for Undergraduates. Monographs in Number Theory.World Scientific Publishing Company. ISBN 981-4271-36-5.
• Hugh L. Montgomery; Robert C. Vaughan (2007). Multiplicative number theory I. Classical theory. Cambridgetracts in advanced mathematics 97. Cambridge: Cambridge Univ. Press. p. 38. ISBN 0-521-84903-9.
• Cohen, Eckford (1959). “A class of residue systems (mod r) and related arithmetical functions. I. A general-ization of Möbius inversion”. Pacific J. Math. 9 (1). pp. 13–23. MR 0109806.
• Cohen, Eckford (1960). “Arithmetical functions associated with the unitary divisors of an integer”. MathematischeZeitschrift 74. pp. 66–80. doi:10.1007/BF01180473. MR 0112861.
• Cohen, Eckford (1960). “The number of unitary divisors of an integer”. American mathematical monthly 67(9). pp. 879–880. MR 0122790.
• Cohen, Graeme L. (1990). “On an integers’ infinitary divisors”. Math. Comp. 54 (189). pp. 395–411.doi:10.1090/S0025-5718-1990-0993927-5. MR 0993927.
• Cohen, Graeme L. (1993). “Arithmetic functions associated with infinitary divisors of an integer”. Intl. J.Math. Math. Sci. 16 (2). pp. 373–383. doi:10.1155/S0161171293000456.
• Sandor, Jozsef; Berge, Antal (2003). “The Möbius function: generalizations and extensions”. Adv. Stud.Contemp. Math. (Kyungshang) 6 (2): 77–128. MR 1962765.
• Finch, Steven (2004). “Unitarism and Infinitarism” (PDF).
11.8 External links• Hazewinkel, Michiel, ed. (2001), “Dirichlet convolution”, Encyclopedia of Mathematics, Springer, ISBN 978-
1-55608-010-4
Chapter 12
Divisibility sequence
In mathematics, a divisibility sequence is an integer sequence (an)n∈N such that for all natural numbers m, n,
ifm | n then am | an,
i.e., whenever one index is a multiple of another one, then the corresponding term also is a multiple of the other term.The concept can be generalized to sequences with values in any ring where the concept of divisibility is defined.A strong divisibility sequence is an integer sequence (an)n∈N such that for all natural numbers m, n,
gcd(am, an) = agcd(m,n).
Note that a strong divisibility sequence is immediately a divisibility sequence; ifm | n , immediately gcd(m,n) = m. Then by the strong divisibility property, gcd(am, an) = am and therefore am | an .
12.1 Examples
• Any constant sequence is a strong divisibility sequence.
• Every sequence of the form an = kn , for some nonzero integer k, is a divisibility sequence.
• Every sequence of the form an = An −Bn for integers A > B > 0 is a divisibility sequence.
• The Fibonacci numbers F = (1, 1, 2, 3, 5, 8,...) form a strong divisibility sequence.
• More generally, Lucas sequences of the first kind are divisibility sequences.
• Elliptic divisibility sequences are another class of such sequences.
12.2 References
• Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas (2003). Recurrence Sequences.American Mathematical Society. ISBN 978-0-8218-3387-2.
• Hall, Marshall (1936). “Divisibility sequences of third order”. Am. J. Math 58: 577–584. JSTOR 2370976.
• Ward, Morgan (1939). “A note on divisibility sequences”. Bull. Amer. Math. Soc 45: 334–336. doi:10.1090/s0002-9904-1939-06980-2.
• Hoggat, Jr., V. E.; Long, C. T. (1973). “Divisibility properties of generalized fibonacci polynomials” (PDF).Fibonacci Quarterly: 113.
57
58 CHAPTER 12. DIVISIBILITY SEQUENCE
• Bézivin, J.-P.; Ethö, A.; van der Porten, A. J. (1990). “A full characterization of divisibility sequences”. Am.J. Math. 112 (6): 985–1001. JSTOR 2374733.
• P. Ingram; J. H. Silverman (2012), “Primitive divisors in elliptic divisibility sequences”, in Dorian Goldfeld;Jay Jorgenson; Peter Jones; Dinakar Ramakrishnan; Kenneth A. Ribet; John Tate, Number Theory, Analysisand Geometry. In Memory of Serge Lang, Springer, pp. 243–271, ISBN 978-1-4614-1259-5
Chapter 13
Divisor summatory function
-40
-30
-20
-10
0
10
20
30
40
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
sum
_k d
ivis
or(k
) - n
log
n - n
(2 g
amm
a -1
)
n
Divisor sumatory function
The summatory function, with leading terms removed, for x < 104
In number theory, the divisor summatory function is a function that is a sum over the divisor function. It frequentlyoccurs in the study of the asymptotic behaviour of the Riemann zeta function. The various studies of the behaviourof the divisor function are sometimes called divisor problems.
13.1 Definition
The divisor summatory function is defined as
59
60 CHAPTER 13. DIVISOR SUMMATORY FUNCTION
-400
-300
-200
-100
0
100
200
300
400
0 1e+06 2e+06 3e+06 4e+06 5e+06 6e+06 7e+06 8e+06 9e+06 1e+07
sum
_k d
ivis
or(k
) - n
log
n - n
(2 g
amm
a -1
)
n
Divisor sumatory function
The summatory function, with leading terms removed, for x < 107
D(x) =∑n≤x
d(n) =∑j,k
jk≤x
1
where
d(n) = σ0(n) =∑j,k
jk=n
1
is the divisor function. The divisor function counts the number of ways that the integer n can be written as a productof two integers. More generally, one defines
Dk(x) =∑n≤x
dk(n) =∑mn≤x
dk−1(n)
where dk(n) counts the number of ways that n can be written as a product of k numbers. This quantity can bevisualized as the count of the number of lattice points fenced off by a hyperbolic surface in k dimensions. Thus, fork=2, D(x) = D2(x) counts the number of points on a square lattice bounded on the left by the vertical-axis, on thebottom by the horizontal-axis, and to the upper-right by the hyperbola jk = x. Roughly, this shape may be envisionedas a hyperbolic simplex. This allows us to provide an alternative expression for D(x), and a simple way to compute itin O(
√x) time:
D(x) =∑xk=1
⌊xk
⌋= 2
∑uk=1
⌊xk
⌋− u2 , where u = ⌊
√x⌋
If the hyperbola in this context is replaced by a circle then determining the value of the resulting function is knownas the Gauss circle problem.
13.2. DIRICHLET’S DIVISOR PROBLEM 61
The summatory function, with leading terms removed, for x < 107 , graphed as a distribution or histogram. The vertical scale isnot constant left to right; click on image for a detailed description.
13.2 Dirichlet’s divisor problem
Finding a closed form for this summed expression seems to be beyond the techniques available, but it is possible togive approximations. The leading behaviour of the series is not difficult to obtain. Peter Dirichlet demonstrated that
D(x) = x logx+ x(2γ − 1) + ∆(x)
where γ is the Euler–Mascheroni constant, and the non-leading term is
∆(x) = O(√x).
Here, O denotes Big-O notation. The Dirichlet divisor problem, precisely stated, is to find the smallest value of θfor which
∆(x) = O(xθ+ϵ
)
62 CHAPTER 13. DIVISOR SUMMATORY FUNCTION
holds true, for any ϵ > 0 . As of 2006, this problem remains unsolved. Progress has been slow. Many of the samemethods work for this problem and for Gauss’s circle problem, another lattice-point counting problem. Section F1of Unsolved Problems in Number Theory [1] surveys what is known and not known about these problems.
• In 1904, G. Voronoi proved that the error term can be improved to O(x1/3 logx). [2]:381
• In 1916, G. H. Hardy showed that inf θ ≥ 1/4 . In particular, he demonstrated that for some constant K ,there exist values of x for which ∆(x) > Kx1/4 and values of x for which ∆(x) < −Kx1/4 .[3]:69
• In 1922, J. van der Corput improved Dirichlet’s bound to inf θ ≤ 33/100. [2]:381
• In 1928, J. van der Corput proved that inf θ ≤ 27/82. [2]:381
• In 1950, Chih Tsung-tao and independently in 1953 H. E. Richert proved that inf θ ≤ 15/46. [2]:381
• In 1969, Grigori Kolesnik demonstrated that inf θ ≤ 12/37 .[2]:381
• In 1973, Grigori Kolesnik demonstrated that inf θ ≤ 346/1067 .[2]:381
• In 1982, Grigori Kolesnik demonstrated that inf θ ≤ 35/108 .[2]:381
• In 1988, H. Iwaniec and C. J. Mozzochi proved that inf θ ≤ 7/22. [4]
• In 2003, M.N. Huxley improved this to show that inf θ ≤ 131/416. [5]
So, the true value of inf θ lies somewhere between 1/4 and 131/416 (approx. 0.3149); it is widely conjectured to beexactly 1/4. Theoretical evidence lends credence to this conjecture, since ∆(x)/x1/4 has a (non-Gaussian) limitingdistribution. The value of 1/4 would also follow from a conjecture on exponent pairs.[6]
13.3 Piltz divisor problem
In the generalized case, one has
Dk(x) = xPk(logx) + ∆k(x)
where Pk is a polynomial of degree k − 1 . Using simple estimates, it is readily shown that
∆k(x) = O(x1−1/k logk−2 x
)for integer k ≥ 2 . As in the k = 2 case, the infimum of the bound is not known for any value of k . Computingthese infima is known as the Piltz divisor problem, after the name of the German mathematician Adolf Piltz (also seehis German page). Defining the order αk as the smallest value for which ∆k(x) = O (xαk+ε) holds, for any ε > 0 ,one has the following results (note that α2 is the θ of the previous section):
α2 ≤ 131416 ,
[5]
α3 ≤ 4396 ,
[7] and[8]
13.4. MELLIN TRANSFORM 63
αk ≤ 3k − 4
4k(4 ≤ k ≤ 8)
α9 ≤ 35
54, α10 ≤ 41
60, α11 ≤ 7
10
αk ≤ k − 2
k + 2(12 ≤ k ≤ 25)
αk ≤ k − 1
k + 4(26 ≤ k ≤ 50)
αk ≤ 31k − 98
32k(51 ≤ k ≤ 57)
αk ≤ 7k − 34
7k(k ≥ 58)
• E. C. Titchmarsh conjectures that αk = k−12k .
13.4 Mellin transform
Both portions may be expressed as Mellin transforms:
D(x) =1
2πi
∫ c+i∞
c−i∞ζ2(w)
xw
wdw
for c > 1 . Here, ζ(s) is the Riemann zeta function. Similarly, one has
∆(x) =1
2πi
∫ c′+i∞
c′−i∞ζ2(w)
xw
wdw
with 0 < c′ < 1 . The leading term of D(x) is obtained by shifting the contour past the double pole at w = 1 : theleading term is just the residue, by Cauchy’s integral formula. In general, one has
Dk(x) =1
2πi
∫ c+i∞
c−i∞ζk(w)
xw
wdw
and likewise for ∆k(x) , for k ≥ 2 .
13.5 Notes[1] Guy, Richard K. (2004). Unsolved Problems in Number Theory (3rd ed.). Berlin: Springer. ISBN 978-0-387-20860-2.
[2] Ivic, Aleksandar (2003). The Riemann Zeta-Function. New York: Dover Publications. ISBN 0-486-42813-3.
[3] Montgomery, Hugh; R. C. Vaughan (2007). Multiplicative Number Theory I: Classical Theory. Cambridge: CambridgeUniversity Press. ISBN 978-0-521-84903-6.
[4] Iwaniec, H.; C. J. Mozzochi (1988). “On the divisor and circle problems”. Journal of Number Theory 29: 60–93.doi:10.1016/0022-314X(88)90093-5.
[5] Huxley, M. N. (2003). “Exponential sums and lattice points III”. Proc. LondonMath. Soc. 87 (3): 591–609. doi:10.1112/S0024611503014485.ISSN 0024-6115. Zbl 1065.11079.
[6] Montgomery, Hugh L. (1994). Ten lectures on the interface between analytic number theory and harmonic analysis. Re-gional Conference Series in Mathematics 84. Providence, RI: American Mathematical Society. p. 59. ISBN 0-8218-0737-4. Zbl 0814.11001.
64 CHAPTER 13. DIVISOR SUMMATORY FUNCTION
[7] G. Kolesnik. On the estimation of multiple exponential sums, in “Recent Progress in Analytic Number Theory”, SymposiumDurham 1979 (Vol. 1), Academic, London, 1981, pp. 231–246.
[8] Aleksandar Ivić. The Theory of the Riemann Zeta-function with Applications (Theorem 13.2). John Wiley and Sons 1985.
13.6 References• H.M. Edwards, Riemann’s Zeta Function, (1974) Dover Publications, ISBN 0-486-41740-9
• E. C. Titchmarsh, The theory of the Riemann Zeta-Function, (1951) Oxford at the Clarendon Press, Oxford.(See chapter 12 for a discussion of the generalized divisor problem)
• Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, NewYork-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001 (Provides anintroductory statement of the Dirichlet divisor problem.)
• H. E. Rose. A Course in Number Theory., Oxford, 1988.
• M.N. Huxley (2003) 'Exponential Sums and Lattice Points III', Proc. London Math. Soc. (3)87: 591–609
Chapter 14
Extremal orders of an arithmetic function
In mathematics, in number theory, the extremal orders of an arithmetic function are best possible bounds of thegiven arithmetic function. Specifically, if f(n) is an arithmetic function and m(n) is a non-decreasing function that isultimately positive and
lim infn→∞
f(n)
m(n)= 1
we say that m is a minimal order for f. Similarly if M(n) is a non-decreasing function that is ultimately positive and
lim supn→∞
f(n)
M(n)= 1
we say that M is a maximal order for f.[1]:80 The subject was first studied systematically by Ramanujan starting in1915.[1]:87
14.1 Examples• For the sum-of-divisors function σ(n) we have the trivial result
lim infn→∞
σ(n)
n= 1
because always σ(n) ≥ n and for primes σ(p) = p + 1. We also have
lim supn→∞
σ(n)
n ln lnn = eγ ,
proved by Gronwall in 1913.[1]:86[2]:Theorem 323[3] Therefore n is a minimal order and e−γ n ln ln n is amaximal order for σ(n).
• For the Euler totient φ(n) we have the trivial result
lim infn→∞
ϕ(n)
n= 1
because always φ(n) ≤ n and for primes φ(p) = p − 1. We also have
lim infn→∞
ϕ(n) ln lnnn
= e−γ ,
proved by Landau in 1903.[1]:84[2]:Theorem 328
65
66 CHAPTER 14. EXTREMAL ORDERS OF AN ARITHMETIC FUNCTION
• For the number of divisors function d(n) we have the trivial lower bound 2 ≤ d(n), in which equality occurswhen n is prime, so 2 is a minimal order. For ln d(n) we have a maximal order ln 2 ln n / ln ln n, proved byWigert in 1907.[1]:82[2]:Theorem 317
• For the number of distinct prime factors ω(n) we have a trivial lower bound 1 ≤ ω(n), in which equality occurswhen n is a prime power. A maximal order for ω(n) is ln n / ln ln n.[1]:83
• For the number of prime factors counted with multiplicity Ω(n) we have a trivial lower bound 1 ≤ Ω(n), inwhich equality occurs when n is prime. A maximal order for Ω(n) is ln n / ln 2.[1]:83
14.2 See also• Average order of an arithmetic function
• Normal order of an arithmetic function
14.3 Notes[1] Tenenbaum, Gérald (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge studies in advanced
mathematics 46. Cambridge University Press. ISBN 0-521-41261-7.
[2] Hardy, G. H.; Wright, E. M. (1979). An Introduction to the Theory of Numbers (5th ed.). Oxford: Clarendon Press. ISBN0-19-853171-0.
[3] Gronwall, T. H. (1913). “Some asymptotic expressions in the theory of numbers”. Transactions of the American Mathe-matical Society 13 (4): 113–122.
14.4 Further reading• Nicolas, J.-L. (1988). “On Highly Composite Numbers”. In Andrews, G. E.; Askey, R. A.; Berndt, B. C.;
Ramanathan, K. G. Ramanujan Revisited. Academic Press. pp. 215–244. ISBN 978-0-12-058560-1. Asurvey of extremal orders, with an extensive bibliography.
Chapter 15
Gauss circle problem
In mathematics, the Gauss circle problem is the problem of determining how many integer lattice points there arein a circle centred at the origin and with radius r. The first progress on a solution was made by Carl Friedrich Gauss,hence its name.
15.1 The problem
Consider a circle in R2 with centre at the origin and radius r ≥ 0. Gauss’ circle problem asks how many points thereare inside this circle of the form (m,n) where m and n are both integers. Since the equation of this circle is given inCartesian coordinates by x2 + y2 = r2, the question is equivalently asking how many pairs of integers m and n thereare such that
m2 + n2 ≤ r2.
If the answer for a given r is denoted by N(r) then the following list shows the first few values of N(r) for r an integerbetween 0 and 12 followed by the list of values πr2 rounded to the nearest integer:
1, 5, 13, 29, 49, 81, 113, 149, 197, 253, 317, 377, 441 (sequence A000328 in OEIS)0, 3, 13, 28, 50, 79, 113, 154, 201, 254, 314, 380, 452 (sequence A075726 in OEIS)
15.2 Bounds on a solution and conjecture
The area inside a circle of radius r is given by πr2, and since a square of area 1 in R2 contains one integer point,N(r) can be expected to be roughly πr2. So it should be expected that
N(r) = πr2 + E(r)
for some error term E(r) of relatively small absolute value. Finding a correct upper bound for |E(r)| is thus the formthe problem has taken. Note that r need not be an integer. After N(4) = 49 one has N(
√17) = 57, N(
√18) =
61, N(√20) = 69, N(5) = 81. At these places E(r) increases by 8, 4, 8, 12 after which it decreases (at a rate of
2πr ) until the next time it increases.Gauss managed to prove[1] that
|E(r)| ≤ 2√2πr.
Hardy[2] and, independently, Landau found a lower bound by showing that
67
68 CHAPTER 15. GAUSS CIRCLE PROBLEM
|E(r)| = o(r1/2(log r)1/4
),
using the little o-notation. It is conjectured[3] that the correct bound is
|E(r)| = O(r1/2+ε
).
Writing |E(r)| ≤ Crt , the current bounds on t are
1
2< t ≤ 131
208= 0.6298 . . . ,
with the lower bound from Hardy and Landau in 1915, and the upper bound proved by Huxley in 2000.[4]
15.3 Exact forms
The value of N(r) can be given by several series. In terms of a sum involving the floor function it can be expressedas:[5]
N(r) = 1 + 4∞∑i=0
(⌊r2
4i+ 1
⌋−⌊
r2
4i+ 3
⌋).
A much simpler sum appears if the sum of squares function r2(n) is defined as the number of ways of writing thenumber n as the sum of two squares. Then[1]
N(r) =r2∑n=0
r2(n).
15.4 Generalisations
Although the original problem asks for integer lattice points in a circle, there is no reason not to consider other shapesor conics, indeed Dirichlet’s divisor problem is the equivalent problem where the circle is replaced by the rectangularhyperbola.[3] Similarly one could extend the question from two dimensions to higher dimensions, and ask for integerpoints within a sphere or other objects. If one ignores the geometry and merely considers the problem an algebraicone of Diophantine inequalities then there one could increase the exponents appearing in the problem from squaresto cubes, or higher.
15.4.1 The primitive circle problem
Another generalisation is to calculate the number of coprime integer solutions m, n to the equation
m2 + n2 ≤ r2.
This problem is known as the primitive circle problem, as it involves searching for primitive solutions to the originalcircle problem.[6] If the number of such solutions is denoted V(r) then the values of V(r) for r taking small integervalues are
0, 4, 8, 16, 32, 48, 72, 88, 120, 152, 192 … (sequence A175341 in OEIS).
15.5. NOTES 69
Using the same ideas as the usual Gauss circle problem and the fact that the probability that two integers are coprimeis 6/π2, it is relatively straightforward to show that
V (r) =6
πr2 +O(r1+ε).
As with the usual circle problem, the problematic part of the primitive circle problem is reducing the exponent in theerror term. At present the best known exponent is 221/304 + ε if one assumes the Riemann hypothesis.[6] Withoutassuming the Riemann hypothesis, the best known upper bound is
V (r) =6
πr2 +O(r exp(−c(log r)3/5(log log r2)−1/5))
for a positive constant c.[6] In particular, no bound on the error term of the form 1 − ε for any ε > 0 is currently knownthat does not assume the Riemann Hypothesis.
15.5 Notes[1] G.H. Hardy, Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, (1999),
p.67.
[2] G.H. Hardy, On the Expression of a Number as the Sum of Two Squares, Quart. J. Math. 46, (1915), pp.263–283.
[3] R.K. Guy, Unsolved problems in number theory, Third edition, Springer, (2004), pp.365–366.
[4] M.N. Huxley, Integer points, exponential sums and the Riemann zeta function, Number theory for the millennium, II (Ur-bana, IL, 2000) pp.275–290, A K Peters, Natick, MA, 2002, MR 1956254.
[5] D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, New York: Chelsea, (1999), pp.37–38.
[6] J. Wu, On the primitive circle problem, Monatsh. Math. 135 (2002), pp.69–81.
15.6 External links• Weisstein, Eric W., “Gauss’s circle problem”, MathWorld.
Chapter 16
Integer sequence
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers.An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a relationshipbetween its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, … (the Fibonacci sequence) is formed by startingwith 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description. The sequence0, 3, 8, 15, … is formed according to the formula n2 − 1 for the nth term: an explicit definition.Alternatively, an integer sequence may be defined by a property which members of the sequence possess and otherintegers do not possess. For example, we can determine whether a given integer is a perfect number, even though wedo not have a formula for the nth perfect number.
16.1 Examples
Integer sequences which have received their own name include:
• Abundant numbers
• Baum–Sweet sequence
• Bell numbers
• Binomial coefficients
• Carmichael numbers
• Catalan numbers
• Composite numbers
• Deficient numbers
• Euler numbers
• Even and odd numbers
• Factorial numbers
• Fibonacci numbers
• Fibonacci word
• Figurate numbers
• Golomb sequence
• Happy numbers
70
16.2. COMPUTABLE AND DEFINABLE SEQUENCES 71
• Highly totient numbers
• Highly composite numbers
• Home primes
• Hyperperfect numbers
• Juggler sequence
• Kolakoski sequence
• Lucky numbers
• Lucas numbers
• Padovan numbers
• Partition numbers
• Perfect numbers
• Pseudoperfect numbers
• Prime numbers
• Pseudoprime numbers
• Regular paperfolding sequence
• Rudin–Shapiro sequence
• Semiperfect numbers
• Semiprime numbers
• Superperfect numbers
• Thue-Morse sequence
• Ulam numbers
• Weird numbers
16.2 Computable and definable sequences
An integer sequence is a computable sequence, if there exists an algorithm which given n, calculates an, for alln > 0. An integer sequence is a definable sequence, if there exists some statement P(x) which is true for thatinteger sequence x and false for all other integer sequences. The set of computable integer sequences and definableinteger sequences are both countable, with the computable sequences a proper subset of the definable sequences (inother words, some sequences are definable but not computable). The set of all integer sequences is uncountable(with cardinality equal to that of the continuum); thus, almost all integer sequences are uncomputable and cannot bedefined.
16.3 Complete sequences
An integer sequence is called a complete sequence if every positive integer can be expressed as a sum of values inthe sequence, using each value at most once.
72 CHAPTER 16. INTEGER SEQUENCE
16.4 See also• On-Line Encyclopedia of Integer Sequences
• List of OEIS sequences
16.5 External links• Journal of Integer Sequences. Articles are freely available online.
• Inductive Inference of Integer Sequences
Chapter 17
List of OEIS sequences
This article provides a list of integer sequences in the On-Line Encyclopedia of Integer Sequences that have their ownWikipedia entries.
17.1 References• OEIS core sequences
17.2 External links• Index to OEIS
73
Chapter 18
Mertens function
-50
-40
-30
-20
-10
0
10
20
30
40
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
M(n
)
n
Mertens function
Mertens function to n=10,000
In number theory, the Mertens function is defined for all positive integers n as
M(n) =n∑k=1
µ(k)
where μ(k) is the Möbius function. The function is named in honour of Franz Mertens.Less formally, M(n) is the count of square-free integers up to n that have an even number of prime factors, minus thecount of those that have an odd number.The first 160 M(n) is: (sequence A002321 in OEIS)
74
18.1. REPRESENTATIONS 75
-1500
-1000
-500
0
500
1000
1500
0 1e+06 2e+06 3e+06 4e+06 5e+06 6e+06 7e+06 8e+06 9e+06 1e+07
M(n
)
n
Mertens function
Mertens function to n=10,000,000
The Mertens function slowly grows in positive and negative directions both on average and in peak value, oscillatingin an apparently chaotic manner passing through zero when n has the values
2, 39, 40, 58, 65, 93, 101, 145, 149, 150, 159, 160, 163, 164, 166, 214, 231, 232, 235, 236, 238, 254,... (sequence A028442 in OEIS).
Because the Möbius function only takes the values −1, 0, and +1, the Mertens function moves slowly and there is no nsuch that |M(n)| > n. The Mertens conjecture went further, stating that there would be no n where the absolute valueof the Mertens function exceeds the square root of n. The Mertens conjecture was proven false in 1985 by AndrewOdlyzko and Herman te Riele. However, the Riemann hypothesis is equivalent to a weaker conjecture on the growthof M(n), namely M(n) = O(n1/2 + ε). Since high values for M(n) grow at least as fast as the square root of n, this putsa rather tight bound on its rate of growth. Here, O refers to Big O notation.The above definition can be extended to real numbers as follows:
M(x) =∑
1≤k≤x
µ(k).
18.1 Representations
18.1.1 As an integral
Using the Euler product one finds that
76 CHAPTER 18. MERTENS FUNCTION
1
ζ(s)=∏p
(1− p−s) =
∞∑n=1
µ(n)
ns
where ζ(s) is the Riemann zeta function and the product is taken over primes. Then, using this Dirichlet series withPerron’s formula, one obtains:
1
2πi
∫ c+i∞
c−i∞
xs
sζ(s)ds =M(x)
where c > 1.Conversely, one has the Mellin transform
1
ζ(s)= s
∫ ∞
1
M(x)
xs+1dx
which holds for Re(s) > 1 .A curious relation given by Mertens himself involving the second Chebyshev function is
ψ(x) =M(x2
)log(2) +M
(x3
)log(3) +M
(x4
)log(4) + · · · .
A good evaluation, at least asymptotically, would be to obtain, by the method of steepest descent, the inequality
∮C
F (s)est ds ∼M(et).
Assuming that there are not multiple non-trivial roots of ζ(ρ) we have the “exact formula” by the residue theorem:
1
2πi
∮C
xs
sζ(s)ds =
∑ρ
xρ
ρζ ′(ρ)− 2 +
∞∑n=1
(−1)n−1(2π)2n
(2n)!nζ(2n+ 1)x2n.
Weyl conjectured that the Mertens function satisfied the approximate functional-differential equation
y(x)
2−
N∑r=1
B2r
(2r)!D2r−1t y
(x
t+ 1
)+ x
∫ x
0
y(u)
u2du = x−1H(logx)
where H(x) is the Heaviside step function, B are Bernoulli numbers and all derivatives with respect to t are evaluatedat t = 0.Titchmarsh(1960), and later J. Garcia provided a Trace formula involving a sum over the Möbius function and zerosof Riemann Zeta in the form
∞∑n=1
µ(n)√ng logn =
∑t
h(t)
ζ ′(1/2 + it)+ 2
∞∑n=1
(−1)n(2π)2n
(2n)!ζ(2n+ 1)
∫ ∞
−∞g(x)e−x(2n+1/2) dx,
where 't' sums over the imaginary parts of nontrivial zeros, and (g, h) are related by a Fourier transform, such that
2πg(x) =
∫ ∞
−∞h(u)eiux du.
18.2. CALCULATION 77
18.1.2 As a sum over Farey sequences
Another formula for the Mertens function is
M(n) =∑a∈Fn
e2πia where Fn is the Farey sequence of order n.
This formula is used in the proof of the Franel–Landau theorem.[1]
18.1.3 As a determinant
M(n) is the determinant of the n × n Redheffer matrix, a (0,1) matrix in which aij is 1 if either j is 1 or i divides j.
18.2 Calculation
Neither of the methods mentioned previously leads to practical algorithms to calculate the Mertens function. Usingsieve methods similar to those used in prime counting, the Mertens function has been computed for an increasingrange of n.The Mertens function for all integer values up to N may be computed in O(N2/3+ε) time, while better methods areknown. Elementary algorithms exist to compute isolated values of M(N) in O(N2/3*(ln ln(N))1/3) time.
See A084237 for values of M(N) at powers of 10.
18.3 Notes[1] Edwards, Ch. 12.2
18.4 See also• Perron’s formula
• Liouville’s function
18.5 References• Edwards, Harold (1974). Riemann’s Zeta Function. Mineola, New York: Dover. ISBN 0-486-41740-9.
• F. Mertens, "Über eine zahlentheoretische Funktion”, AkademieWissenschaftlicherWienMathematik-NaturlichKleine Sitzungsber, IIa 106, (1897) 761–830.
• A. M. Odlyzko and Herman te Riele, "Disproof of the Mertens Conjecture", Journal für die reine und ange-wandte Mathematik 357, (1985) pp. 138–160.
• Weisstein, Eric W., “Mertens function”, MathWorld.
• Jose Javier Garcia Moreta "http://www.prespacetime.com/index.php/pst/issue/view/42 Borel Resummation& the Solution of Integral Equations
• "Sloane’s A002321 : Mertens’s function", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
• Deléglise, M. and Rivat, J. “Computing the Summation of the Möbius Function.” Experiment. Math. 5, 291-295, 1996. http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.em/1047565447
Chapter 19
Möbius inversion formula
Möbius transform redirects here. It should not be confused with Möbius transformation.
In mathematics, the classic Möbius inversion formula was introduced into number theory during the 19th centuryby August Ferdinand Möbius.Other Möbius inversion formulas are obtained when different locally finite partially ordered sets replace the classiccase of the natural numbers ordered by divisibility; for an account of those, see incidence algebra.
19.1 Statement of the formula
The classic version states that if g and f are arithmetic functions satisfying
g(n) =∑d |n
f(d) integer every forn ≥ 1
then
f(n) =∑d |n
µ(d)g(n/d) integer every forn ≥ 1
where μ is the Möbius function and the sums extend over all positive divisors d of n. In effect, the original f(n) canbe determined given g(n) by using the inversion formula. The two sequences are said to be Möbius transforms ofeach other.The formula is also correct if f and g are functions from the positive integers into some abelian group (viewed as a Z-module).In the language of Dirichlet convolutions, the first formula may be written as
g = f ∗ 1
where * denotes the Dirichlet convolution, and 1 is the constant function 1(n) = 1 . The second formula is thenwritten as
f = µ ∗ g.
Many specific examples are given in the article on multiplicative functions.The theorem follows because ∗ is (commutative and) associative, and 1∗µ = ϵ , where ϵ is the identity function for theDirichlet convolution, taking values ϵ(1) = 1, ϵ(n) = 0 for alln > 1 . Thusµ∗g = µ∗(1∗f) = (µ∗1)∗f = ϵ∗f = f.
78
19.2. SERIES RELATIONS 79
19.2 Series relations
Let
an =∑d|n
bd
so that
bn =∑d|n
µ(nd
)ad
is its transform. The transforms are related by means of series: the Lambert series
∞∑n=1
anxn =
∞∑n=1
bnxn
1− xn
and the Dirichlet series:
∞∑n=1
anns
= ζ(s)
∞∑n=1
bnns
where ζ(s) is the Riemann zeta function.
19.3 Repeated transformations
Given an arithmetic function, one can generate a bi-infinite sequence of other arithmetic functions by repeatedlyapplying the first summation.For example, if one starts with Euler’s totient function φ , and repeatedly applies the transformation process, oneobtains:
1. φ the totient function
2. φ ∗ 1 = Id where Id(n) = n is the identity function
3. Id ∗1 = σ1 = σ , the divisor function
If the starting function is the Möbius function itself, the list of functions is:
1. µ , the Möbius function
2. µ ∗ 1 = ε where ε(n) =1, if n = 1
0, if n > 1is the unit function
3. ε ∗ 1 = 1 , the constant function
4. 1 ∗ 1 = σ0 = d = τ , where d = τ is the number of divisors of n, (see divisor function).
Both of these lists of functions extend infinitely in both directions. The Möbius inversion formula enables these liststo be traversed backwards.As an example the sequence starting in φ is:
80 CHAPTER 19. MÖBIUS INVERSION FORMULA
fn =
µ ∗ . . . ∗ µ︸ ︷︷ ︸−nfactors
∗φ ifn < 0
φ ifn = 0
φ ∗ 1 ∗ . . . ∗ 1︸ ︷︷ ︸nfactors
ifn > 0
The generated sequences can perhaps be more easily understood by considering the corresponding Dirichlet series:each repeated application of the transform corresponds to multiplication by the Riemann zeta function.
19.4 Generalizations
See also: Incidence algebra
A related inversion formula more useful in combinatorics is as follows: suppose F(x) and G(x) are complex-valuedfunctions defined on the interval [1,∞) such that
G(x) =∑
1≤n≤x
F (x/n) for all x ≥ 1
then
F (x) =∑
1≤n≤x
µ(n)G(x/n) for all x ≥ 1.
Here the sums extend over all positive integers n which are less than or equal to x.This in turn is a special case of a more general form. If α(n) is an arithmetic function possessing a Dirichlet inverseα−1(n) , then if one defines
G(x) =∑
1≤n≤x
α(n)F (x/n) for all x ≥ 1
then
F (x) =∑
1≤n≤x
α−1(n)G(x/n) for all x ≥ 1.
The previous formula arises in the special case of the constant function α(n) = 1 , whose Dirichlet inverse isα−1(n) = µ(n) .A particular application of the first of these extensions arises if we have (complex-valued) functions f(n) and g(n)defined on the positive integers, with
g(n) =∑
1≤m≤n
f(⌊ nm
⌋)for all n ≥ 1.
By defining F (x) = f(⌊x⌋) and G(x) = g(⌊x⌋) , we deduce that
f(n) =∑
1≤m≤n
µ(m)g(⌊ nm
⌋)for all n ≥ 1.
A simple example of the use of this formula is counting the number of reduced fractions 0 < a/b < 1, where a andb are coprime and b≤n. If we let f(n) be this number, then g(n) is the total number of fractions 0 < a/b < 1 with
19.5. MULTIPLICATIVE NOTATION 81
b≤n, where a and b are not necessarily coprime. (This is because every fraction a/b with gcd(a,b) = d and b≤n canbe reduced to the fraction (a/d)/(b/d) with b/d ≤ n/d, and vice versa.) Here it is straightforward to determine g(n) =n(n−1)/2, but f(n) is harder to compute.Another inversion formula is (where we assume that the series involved are absolutely convergent):
g(x) =
∞∑m=1
f(mx)
msfor all x ≥ 1 ⇐⇒ f(x) =
∞∑m=1
µ(m)g(mx)
msfor all x ≥ 1.
As above, this generalises to the case where α(n) is an arithmetic function possessing a Dirichlet inverse α−1(n) :
g(x) =
∞∑m=1
α(m)f(mx)
msfor all x ≥ 1 ⇐⇒ f(x) =
∞∑m=1
α−1(m)g(mx)
msfor all x ≥ 1.
19.5 Multiplicative notation
As Möbius inversion applies to any abelian group, it makes no difference whether the group operation is written asaddition or as multiplication. This gives rise to the following notational variant of the inversion formula:
If F (n) =∏d|n
f(d), then f(n) =∏d|n
F (n/d)µ(d).
19.6 Proofs of generalizations
The first generalization can be proved as follows. We use Iverson’s convention that [condition] is the indicator functionof the condition, being 1 if the condition is true and 0 if false. We use the result that
∑d|n µ(d) = i(n) , that is,
1*μ=i.We have the following:∑1≤n≤x
µ(n)g(xn
)=
∑1≤n≤x
µ(n)∑
1≤m≤x/n
f( x
mn
)=
∑1≤n≤x
µ(n)∑
1≤m≤x/n
∑1≤r≤x
[r = mn]f(xr
)=∑
1≤r≤x
f(xr
) ∑1≤n≤x
µ(n)∑
1≤m≤x/n
[m = r/n] order summation the rearranging
=∑
1≤r≤x
f(xr
)∑n|r
µ(n)
=∑
1≤r≤x
f(xr
)i(r)
= f(x) sincei(r) = 0 when except r = 1
The proof in the more general case where α(n) replaces 1 is essentially identical, as is the second generalisation.
19.7 Contributions of Weisner, Hall, and RotaThe statement of the general Möbius inversion formula was first given independently by Weisner
(1935) and Philip Hall (1936); both authors were motivated by group theory problems. Neither authorseems to have been aware of the combinatorial implications of his work and neither developed the theoryof Möbius functions. In a fundamental paper on Möbius functions, Rota showed the importance of thistheory in combinatorial mathematics and gave a deep treatment of it. He noted the relation between such
82 CHAPTER 19. MÖBIUS INVERSION FORMULA
topics as inclusion-exclusion, classical number theoretic Möbius inversion, coloring problems and flowsin networks. Since then, under the strong influence of Rota, the theory of Möbius inversion and relatedtopics has become an active area of combinatorics.[1]
19.8 See also• Farey sequence
19.9 References• Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New
York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001
• Kung, Joseph P.S. (2001), “Möbius inversion”, in Hazewinkel, Michiel, Encyclopedia ofMathematics, Springer,ISBN 978-1-55608-010-4
• K. Ireland, M. Rosen. A Classical Introduction to Modern Number Theory, (1990) Springer-Verlag.
[1] Bender, Edward A.; Goldman, J. R. (1975). “On the applications of Mö inversion in combinatorial analysis”. Amer. Math.Monthly 82: 789–803. doi:10.2307/2319793.
19.10 External links• Möbius Inversion Formula at ProofWiki
• Weisstein, Eric W., “Möbius Transform”, MathWorld.
Chapter 20
Normal order of an arithmetic function
In number theory, a normal order of an arithmetic function is some simpler or better-understood function which“usually” takes the same or closely approximate values.Let ƒ be a function on the natural numbers. We say that g is a normal order of ƒ if for every ε > 0, the inequalities
(1− ε)g(n) ≤ f(n) ≤ (1 + ε)g(n)
hold for almost all n: that is, if the proportion of n ≤ x for which this does not hold tends to 0 as x tends to infinity.It is conventional to assume that the approximating function g is continuous and monotone.
20.1 Examples• The Hardy–Ramanujan theorem: the normal order of ω(n), the number of distinct prime factors of n, is
log(log(n));• The normal order of Ω(n), the number of prime factors of n counted with multiplicity, is log(log(n));• The normal order of log(d(n)), where d(n) is the number of divisors of n, is log(2) log(log(n)).
20.2 See also• Average order of an arithmetic function• Divisor function• Extremal orders of an arithmetic function
20.3 References• Hardy, G.H.; Ramanujan, S. (1917). “The normal number of prime factors of a number n". Quart. J. Math.48: 76–92. JFM 46.0262.03.
• Hardy, G. H.; Wright, E. M. (2008) [1938]. An Introduction to the Theory of Numbers. Revised by D. R.Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.). Oxford: Oxford University Press.ISBN 978-0-19-921986-5. MR 2445243. Zbl 1159.11001.. p. 473
• Sándor, Jozsef; Crstici, Borislav (2004), Handbook of number theory II, Dordrecht: Kluwer Academic, p. 332,ISBN 1-4020-2546-7, Zbl 1079.11001
• Tenenbaum, Gérald (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge studies inadvanced mathematics 46. Translated from the 2nd French edition by C.B.Thomas. Cambridge UniversityPress. pp. 299–324. ISBN 0-521-41261-7. Zbl 0831.11001.
83
84 CHAPTER 20. NORMAL ORDER OF AN ARITHMETIC FUNCTION
20.4 External links• Weisstein, Eric W., “Normal Order”, MathWorld.
Chapter 21
Partition (number theory)
Young diagrams associated to the partitions of the positive integers 1 through 8. They are arranged so that images under the reflectionabout the main diagonal of the square are conjugate partitions.
In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way ofwriting n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the
85
86 CHAPTER 21. PARTITION (NUMBER THEORY)
Partitions of n with biggest addend k
same partition. (If order matters, the sum becomes a composition.) For example, 4 can be partitioned in five distinctways:
43 + 12 + 22 + 1 + 11 + 1 + 1 + 1
The order-dependent composition 1 + 3 is the same partition as 3 + 1, while 1 + 2 + 1 and 1 + 1 + 2 are the samepartition as 2 + 1 + 1.A summand in a partition is also called a part. The number of partitions of n is given by the partition function p(n).So p(4) = 5. The notation λ ⊢ n means that λ is a partition of n.Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number ofbranches of mathematics and physics, including the study of symmetric polynomials, the symmetric group and ingroup representation theory in general.
21.1 Examples
The seven partitions of 5 are:
• 5
21.2. REPRESENTATIONS OF PARTITIONS 87
• 4 + 1
• 3 + 2
• 3 + 1 + 1
• 2 + 2 + 1
• 2 + 1 + 1 + 1
• 1 + 1 + 1 + 1 + 1
In some sources partitions are treated as the sequence of summands, rather than as an expression with plus signs. Forexample, the partition 2 + 2 + 1 might instead be written as the tuple (2, 2, 1) or in the even more compact form (22,1) where the superscript indicates the number of repetitions of a term.
21.2 Representations of partitions
There are two common diagrammatic methods to represent partitions: as Ferrers diagrams, named after NormanMacleod Ferrers, and as Young diagrams, named after the British mathematician Alfred Young. Both have severalpossible conventions; here, we use English notation, with diagrams aligned in the upper-left corner.
21.2.1 Ferrers diagram
The partition 6 + 4 + 3 + 1 of the positive number 14 can be represented by the following diagram:The 14 circles are lined up in 4 rows, each having the size of a part of the partition. The diagrams for the 5 partitionsof the number 4 are listed below:
21.2.2 Young diagram
Main article: Young diagram
An alternative visual representation of an integer partition is its Young diagram. Rather than representing a partitionwith dots, as in the Ferrers diagram, the Young diagram uses boxes or squares. Thus, the Young diagram for thepartition 5 + 4 + 1 iswhile the Ferrers diagram for the same partition is
While this seemingly trivial variation doesn't appear worthy of separate mention, Young diagrams turn out to beextremely useful in the study of symmetric functions and group representation theory: in particular, filling the boxesof Young diagrams with numbers (or sometimes more complicated objects) obeying various rules leads to a familyof objects called Young tableaux, and these tableaux have combinatorial and representation-theoretic significance.[1]
As a type of shape made by adjacent squares joined together, Young diagrams are a special kind of polyomino.[2]
21.3 Partition function
In number theory, the partition function p(n) represents the number of possible partitions of a natural number n,which is to say the number of distinct ways of representing n as a sum of natural numbers (with order irrelevant). Byconvention p(0) = 1, p(n) = 0 for n negative.The first few values of the partition function are (starting with p(0)=1):
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255,1575, 1958, 2436, 3010, 3718, 4565, 5604, … (sequence A000041 in OEIS).
88 CHAPTER 21. PARTITION (NUMBER THEORY)
The value of p(n) has been computed for large values of n, for example p(100) = 190,569,292, p(1000) is 24,061,467,864,032,622,473,692,149,727,991or approximately 2.40615×1031.,[3] and p(10000) is 36,167,251,325,...,906,916,435,144 or approximately 3.61673×10106.As of June 2013, the largest known prime number that counts a number of partitions is p(120052058), with 12198decimal digits.[4]
21.3.1 Generating function
The generating function for p(n) is given by:[5]
∞∑n=0
p(n)xn =
∞∏k=1
(1
1− xk
).
Expanding each factor on the right-hand side as a geometric series, we can rewrite it as
(1 + x + x2 + x3 + ...)(1 + x2 + x4 + x6 + ...)(1 + x3 + x6 + x9 + ...) ....
The xn term in this product counts the number of ways to write
n = a1 + 2a2 + 3a3 + ... = (1 + 1 + ... + 1) + (2 + 2 + ... + 2) + (3 + 3 + ... + 3) + ...,
where each number i appears ai times. This is precisely the definition of a partition of n, so our product is the desiredgenerating function. More generally, the generating function for the partitions of n into numbers from a set A can befound by taking only those terms in the product where k is an element of A. This result is due to Euler.The formulation of Euler’s generating function is a special case of a q-Pochhammer symbol and is similar to theproduct formulation of many modular forms, and specifically the Dedekind eta function.The denominator of the product is Euler’s function and can be written, by the pentagonal number theorem, as
(1− x)(1− x2)(1− x3) · · · = 1− x− x2 + x5 + x7 − x12 − x15 + x22 + x26 − . . . .
where the exponents of x on the right hand side are the generalized pentagonal numbers; i.e., numbers of the form½m(3m − 1), where m is an integer. The signs in the summation alternate as (−1)m . This theorem can be used toderive a recurrence for the partition function:
21.3. PARTITION FUNCTION 89
p(k) = p(k − 1) + p(k − 2) − p(k − 5) − p(k − 7) + p(k − 12) + p(k − 15) − p(k − 22) − ...
where p(0) is taken to equal 1, and p(k) is taken to be zero for negative k.
21.3.2 Congruences
Main article: Ramanujan’s congruences
Srinivasa Ramanujan is credited with discovering that congruences in the number of partitions exist for argumentsthat are integers ending in 4 and 9.[6]
p(5k + 4) ≡ 0 (mod 5)
For instance, the number of partitions for the integer 4 is 5. For the integer 9, the number of partitions is 30; for 14there are 135 partitions. This is implied by an identity, also by Ramanujan,[7][8]
∞∑k=0
p(5k + 4)xk = 5(x5)5∞(x)6∞
where the series (x)∞ is defined as
(x)∞ =∞∏m=1
(1− xm).
He also discovered congruences related to 7 and 11:[9]
p(7k + 5) ≡ 0 (mod 7)
p(11k + 6) ≡ 0 (mod 11).
and for p=7 proved the identity[8]
∞∑k=0
p(7k + 5)xk = 7(x7)3∞(x)4∞
+ 49x(x7)7∞(x)8∞
Since 5, 7, and 11 are consecutive primes, one might think that there would be such a congruence for the next prime13, p(13k+ a) ≡ 0 (mod 13) for some a. This is, however, false. It can also be shown that there is no congruence of theform p(bk+ a) ≡ 0 (mod b) for any prime b other than 5, 7, or 11.In the 1960s, A. O. L. Atkin of the University of Illinois at Chicago discovered additional congruences for smallprime moduli. For example:
p(113 · 13 · k + 237) ≡ 0 (mod 13).
In 2000, Ken Ono of the University of Wisconsin–Madison proved that there are such congruences for every primemodulus. A few years later Ono, together with Scott Ahlgren of the University of Illinois, proved that there arepartition congruences modulo every integer coprime to 6.[10]
21.3.3 Partition function formulas
90 CHAPTER 21. PARTITION (NUMBER THEORY)
Recurrence formula
Main article: Pentagonal number theorem
Leonhard Euler's pentagonal number theorem implies the identity
p(n) = p(n− 1) + p(n− 2)− p(n− 5)− p(n− 7) + · · ·
where the numbers 1, 2, 5, 7, ... that appear on the right side of the equation are the generalized pentagonal numbersgk = k(3k−1)
2 for nonzero integers k. More formally,
p(n) =∑k =0
(−1)k−1p (n− k(3k − 1)/2)
where the summation is over all nonzero integers k (positive and negative) and p(m) is taken to be 0 if m < 0.
Approximation formulas
Approximation formulas exist that are faster to calculate than the exact formula given above.An asymptotic expression for p(n) is given by
p(n) ∼ 1
4n√3
exp(π
√2n
3
)as n→ ∞.
This asymptotic formula was first obtained by G. H. Hardy and Ramanujan in 1918 and independently by J. V.Uspensky in 1920. Considering p(1000), the asymptotic formula gives about 2.4402 × 1031, reasonably close to theexact answer given above (1.415% larger than the true value).Hardy and Ramanujan obtained an asymptotic expansion with this approximation as the first term:
p(n) =1
2√2
v∑k=1
√k Ak(n)
d
dnexp
(π
k
√2
3
(n− 1
24
)),
where
Ak(n) =∑
0≤m<k; (m, k)= 1
eπi[s(m, k) − 1k 2nm].
Here, the notation (m, n) = 1 implies that the sum should occur only over the values of m that are relatively prime ton. The function s(m, k) is a Dedekind sum.The error after v terms is of the order of the next term, and v may be taken to be of the order of √n . As an example,Hardy and Ramanujan showed that p(200) is the nearest integer to the sum of the first v=5 terms of the series.In 1937, Hans Rademacher was able to improve on Hardy and Ramanujan’s results by providing a convergent seriesexpression for p(n). It is[11]
p(n) =1
π√2
∞∑k=1
√k Ak(n)
d
dn
1√n− 1
24
sinh[π
k
√2
3
(n− 1
24
)] .
The proof of Rademacher’s formula involves Ford circles, Farey sequences, modular symmetry and the Dedekind etafunction in a central way.
21.4. RESTRICTED PARTITIONS 91
It may be shown that the k-th term of Rademacher’s series is of the order
exp(π
k
√2n
3
),
so that the first term gives the Hardy–Ramanujan asymptotic approximation.Paul Erdős published an elementary proof of the asymptotic formula for p(n) in 1942.[12][13]
Techniques for implementing the Hardy-Ramanujan-Rademacher formula efficiently on a computer are discussed inJohansson,[14] where it is shown that p(n) can be computed in softly optimal time O(n1/2+ε). The largest value of thepartition function computed exactly is p(1020), which has slightly more than 11 billion digits.[15]
21.4 Restricted partitions
In both combinatorics and number theory, families of partitions subject to various restrictions are often studied.[16]
This section surveys a few such restrictions.
21.4.1 Conjugate and self-conjugate partitions
If we now flip the diagram of the partition 6 + 4 + 3 + 1 along its main diagonal, we obtain another partition of 14:By turning the rows into columns, we obtain the partition 4 + 3 + 3 + 2 + 1 + 1 of the number 14. Such partitions aresaid to be conjugate of one another.[17] In the case of the number 4, partitions 4 and 1 + 1 + 1 + 1 are conjugate pairs,and partitions 3 + 1 and 2 + 1 + 1 are conjugate of each other. Of particular interest is the partition 2 + 2, which hasitself as conjugate. Such a partition is said to be self-conjugate.[18]
Claim: The number of self-conjugate partitions is the same as the number of partitions with distinct odd parts.Proof (outline): The crucial observation is that every odd part can be "folded" in the middle to form a self-conjugatediagram:One can then obtain a bijection between the set of partitions with distinct odd parts and the set of self-conjugatepartitions, as illustrated by the following example:
21.4.2 Odd parts and distinct parts
Among the 22 partitions of the number 8, there are 6 that contain only odd parts:
• 7 + 1
• 5 + 3
• 5 + 1 + 1 + 1
• 3 + 3 + 1 + 1
• 3 + 1 + 1 + 1 + 1 + 1
• 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
Alternatively, we could count partitions in which no number occurs more than once. If we count the partitions of 8with distinct parts, we also obtain 6:
• 8
• 7 + 1
• 6 + 2
92 CHAPTER 21. PARTITION (NUMBER THEORY)
• 5 + 3
• 5 + 2 + 1
• 4 + 3 + 1
For all positive numbers the number of partitions with odd parts equals the number of partitions with distinct parts.[19]
This result was proved by Leonhard Euler in 1748[20] and is a special case of Glaisher’s theorem.For every type of restricted partition there is a corresponding function for the number of partitions satisfying the givenrestriction. An important example is q(n), the number of partitions of n into distinct parts.[21] The first few values ofq(n) are (starting with q(0)=1):
1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, … (sequence A000009 in OEIS).
The generating function for q(n) (partitions into distinct parts) is given by[22]
∞∑n=0
q(n)xn =
∞∏k=1
(1 + xk) =
∞∏k=1
1
1− x2k−1.
The second product can be written ϕ(x2) / ϕ(x) where ϕ is Euler’s function; the pentagonal number theorem can beapplied to this as well giving a recurrence for q:[23]
q(k) = ak + q(k − 1) + q(k − 2) − q(k − 5) − q(k − 7) + q(k − 12) + q(k − 15) − q(k − 22) − ...
where ak is (−1)m if k = 3m2 − m for some integer m and is 0 otherwise.
21.4.3 Restricted part size or number of parts
Using the same conjugation trick as above, one may show that the number pk(n) of partitions of n into exactly kparts is equal to the number of partitions of n in which the largest part has size k.[24] The function pk(n) satisfies therecurrence
pk(n) = pk(n − k) + pk ₋ ₁(n− 1)
with initial values p0(0) = 1 and pk(n) = 0 if n ≤ 0 or k ≤ 0.One possible generating function for such partitions, taking k fixed and n variable, is
∑n≥0
pk(n)xn = xk ·
k∏i=1
1
1− xi.
More generally, if T is a set of positive integers then the number of partitions of n, all of whose parts belong to T,has generating function
∏t∈T
(1− xt)−1.
This can be used to solve change-making problems (where the set T specifies the available coins). As two particularcases, one has that the number of partitions of n in which all parts are 1 or 2 (or, equivalently, the number of partitionsof n into 1 or 2 parts) is
⌊n2+ 1⌋,
and the number of partitions of n in which all parts are 1, 2 or 3 (or, equivalently, the number of partitions of n intoat most three parts) is the nearest integer to (n + 3)2 / 12.[25]
21.5. RANK AND DURFEE SQUARE 93
Asymptotics
The asymptotic expression for p(n) implies that
log p(n) ∼ C√n as n→ ∞
where C = π√
23 .[26]
If A is a set of natural numbers, we let pA(n) denote the number of partitions of n into elements of A. If A possessespositive natural density α then
log pA(n) ∼ C√αn
and conversely if this asymptotic property holds for pA(n) then A has natural density α.[27] This result was stated,with a sketch of proof, by Erdős in 1942.[12][28]
If A is a finite set, this analysis does not apply (the density of a finite set is zero). If A has k elements whose greatestcommon divisor is 1, then[29]
pA(n) =
(∏a∈A
a−1
)· nk−1
(k − 1)!+O(nk−2).
21.4.4 Partitions in a rectangle and Gaussian binomial coefficients
Main article: Gaussian binomial coefficient
One may also simultaneously limit the number and size of the parts. Let p(N, M; n) denote the number of partitionsof n with at most M parts, each of size at most N. Equivalently, these are the partitions whose Young diagram fitsinside an M × N rectangle. There is a recurrence relation
p(N,M ;n) = p(N,M − 1;n) + p(N − 1,M ;n−M)
obtained by observing that p(N,M ;n) − p(N,M − 1;n) counts the partitions of n into exactly M parts of size atmost N, and subtracting 1 from each part of such a partitions yields a partition of n−M.[30]
The Gaussian binomial coefficient is defined as:
(k + ℓ
ℓ
)q
=
(k + ℓ
k
)q
=
∏k+ℓj=1(1− qj)∏k
j=1(1− qj)∏ℓj=1(1− qj)
.
The Gaussian binomial coefficient is related to the generating function of p(M, N; n) by the equality
MN∑n=0
p(M,N ;n)qn =
(M +N
M
)q
.
21.5 Rank and Durfee square
Main article: Durfee square
94 CHAPTER 21. PARTITION (NUMBER THEORY)
The rank of a partition is the largest number k such that the partition contains at least k parts of size larger than k. Forexample, the partition 4 + 3 + 3 + 2 + 1 + 1 has rank 3 because it contains 3 parts that are ≥ 3, but does not contain4 parts that are ≥ 4. In the Ferrers diagram or Young diagram of a partition of rank r, the r × r square of entries inthe upper-left is known as the Durfee square:
The Durfee square has applications within combinatorics in the proofs of various partition identities.[31] It also hassome practical significance in the form of the h-index.
21.6 Young’s lattice
Main article: Young’s lattice
There is a natural partial order on partitions given by inclusion of Young diagrams. This partially ordered set isknown as Young’s lattice. The lattice was originally defined in the context of representation theory, where it is used todescribe of the irreducible representations of symmetric groups Sn for all n, together with their branching properties,in characteristic zero. It also has received significant study for its purely combinatorial properties; notably, it is themotivating example of a differential poset.
21.7 See also
• Rank of a partition, a different notion of rank
• Crank of a partition
• Dominance order
• Factorization
• Integer factorization
• Partition of a set
• Stars and bars (combinatorics)
• Plane partition
• Polite number, defined by partitions into consecutive integers
• Multiplicative partition
• Twelvefold way
• Ewens’s sampling formula
• Faà di Bruno’s formula
• Multipartition
• Newton’s identities
• Leibniz’s distribution table for integer partitions
• Smallest-parts function
• A Goldbach partition is the partition of an even number into primes (see Goldbach’s conjecture)
• Kostant’s partition function
21.8. NOTES 95
21.8 Notes[1] Andrews (1976) p.199
[2] Josuat-Vergès, Matthieu (2010), “Bijections between pattern-avoiding fillings of Young diagrams”, Journal of Combinato-rial Theory, Series A 117 (8): 1218–1230, arXiv:0801.4928, doi:10.1016/j.jcta.2010.03.006, MR 2677686.
[3] "Sloane’s A070177 ", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
[4] http://primes.utm.edu/top20/page.php?id=54
[5] Abramowitz and Stegun p. 825, 24.2.1 eq. I(B)
[6] Hardy and Wright (2008) Theorem 359, p.380
[7] Berndt and Ono, “Ramanujan’s Unpublished Manuscript on the Partition and Tau Functions with Proofs and Commentary”
[8] Ono (2004) p.87
[9] Hardy and Wright (2008) Theorems 360,361, p.380
[10] Ono, Ken; Ahlgren, Scott (2001). “Congruence properties for the partition function” (PDF). Proceedings of the NationalAcademy of Sciences 98 (23): 12,882–12,884. doi:10.1073/pnas.191488598.
[11] Andrews (1976) p.69
[12] Erdős, Pál (1942). “On an elementary proof of some asymptotic formulas in the theory of partitions”. Ann. Math. (2) 43:437–450. doi:10.2307/1968802. Zbl 0061.07905.
[13] Nathanson (2000) p.456
[14] F. Johansson, Efficient implementation of the Hardy-Ramanujan-Rademacher formula, LMS Journal of Computation andMathematics 15 (2012), 341-359.
[15] Fredrik Johansson (March 2, 2014). “New partition function record: p(1020) computed”.
[16] Alder, Henry L. (1969). “Partition identities - from Euler to the present”. The Amer. Math. Monthly 76: 733–746.
[17] Hardy and Wright (2008) p.362
[18] Hardy and Wright (2008) p.368
[19] Hardy and Wright (2008) p.365
[20] Andrews, George E. Number Theory. W. B. Saunders Company, Philadelphia, 1971. Dover edition, page 149–150.
[21] Notation follows Abramowitz and Stegun p. 825
[22] Abramowitz and Stegun p. 825, 24.2.2 eq. I(B)
[23] Abramowitz and Stegun p. 826, 24.2.2 eq. II(A)
[24] Here the notation follows that of Stanley (1997), Section 1.
[25] Hardy, G.H. Some Famous Problems of the Theory of Numbers. Clarendon Press, 1920.
[26] Andrews (1976) pp70,97
[27] Nathanson (2000) pp.475–485
[28] Nathanson (2000) p.495
[29] Nathanson (2000) p.458–464
[30] Andrews (1976) pp.33-34
[31] see, e.g., Stanley (1997), p. 58
96 CHAPTER 21. PARTITION (NUMBER THEORY)
21.9 References
• George E. Andrews, The Theory of Partitions (1976), Cambridge University Press. ISBN 0-521-63766-X .
• Apostol, Tom M. (1990) [1976]. Modular functions and Dirichlet series in number theory. Graduate Texts inMathematics 41 (2nd ed.). New York etc.: Springer-Verlag. ISBN 0-387-97127-0. Zbl 0697.10023. (Seechapter 5 for a modern pedagogical intro to Rademacher’s formula).
• Hardy, G. H.; Wright, E. M. (2008) [1938]. An Introduction to the Theory of Numbers. Revised by D. R.Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.). Oxford: Oxford University Press.ISBN 978-0-19-921986-5. MR 2445243. Zbl 1159.11001.
• Lehmer, D. H. (1939). “On the remainder and convergence of the series for the partition function”. Trans.Amer. Math. Soc. 46: 362–373. doi:10.1090/S0002-9947-1939-0000410-9. MR 0000410. Zbl 0022.20401.Provides the main formula (no derivatives), remainder, and older form for A (n).)
• Gupta, Gwyther, Miller, Roy. Soc. Math. Tables, vol 4, Tables of partitions, (1962) (Has text, nearly completebibliography, but they (and Abramowitz) missed the Selberg formula for Ak(n), which is in Whiteman.)
• Macdonald, Ian G. (1979). Symmetric functions and Hall polynomials. Oxford Mathematical Monographs.Oxford University Press. ISBN 0-19-853530-9. Zbl 0487.20007. (See section I.1)
• Nathanson, M.B. (2000). Elementary Methods in Number Theory. Graduate Texts in Mathematics 195.Springer-Verlag. ISBN 0-387-98912-9. Zbl 0953.11002.
• Ono, Ken (2000). “Distribution of the partition function modulo m”. Ann. Math. (2) 151 (1): 293–307.doi:10.2307/121118. Zbl 0984.11050. (This paper proves congruences modulo every prime greater than 3)
• Ono, Ken (2004). The web of modularity: arithmetic of the coefficients of modular forms and q-series. CBMSRegional Conference Series in Mathematics 102. Providence, RI: American Mathematical Society. ISBN0-8218-3368-5. Zbl 1119.11026.
• Sautoy, Marcus Du. The Music of the Primes. New York: Perennial-HarperCollins, 2003.
• Richard P. Stanley, Enumerative Combinatorics, Volumes 1 and 2. Cambridge University Press, 1999 ISBN0-521-56069-1
• Whiteman, A. L. (1956). A sum connected with the series for the partition function. Pacific Journal of Math.6 (1). pp. 159–176. Zbl 0071.04004. (Provides the Selberg formula. The older form is the finite Fourierexpansion of Selberg.)
• Hans Rademacher, Collected Papers of Hans Rademacher, (1974) MIT Press; v II, p 100–107, 108–122, 460–475.
• Miklós Bóna (2002). A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory.World Scientific Publishing. ISBN 981-02-4900-4. (qn elementary introduction to the topic of integer parti-tion, including a discussion of Ferrers graphs)
• George E. Andrews, Kimmo Eriksson (2004). Integer Partitions. Cambridge University Press. ISBN 0-521-60090-1.
• 'A Disappearing Number', devised piece by Complicite, mention Ramanujan’s work on the Partition Function,2007
21.10 External links
• Hazewinkel, Michiel, ed. (2001), “Partition”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Partition and composition calculator
• First 4096 values of the partition function
21.10. EXTERNAL LINKS 97
• An algorithm to compute the partition function
• Weisstein, Eric W., “Partition”, MathWorld.
• Weisstein, Eric W., “Partition Function P”, MathWorld.
• Pieces of Number from Science News Online
• Lectures on Integer Partitions by Herbert S. Wilf
• Counting with partitions with reference tables to the On-Line Encyclopedia of Integer Sequences
• Integer partitions entry in the FindStat database
• Integer::Partition Perl module from CPAN
• Fast Algorithms For Generating Integer Partitions
• Generating All Partitions: A Comparison Of Two Encodings
• Amanda Folsom, Zachary A. Kent, and Ken Ono, l-adic properties of the partition function. In press.
• Jan Hendrik Bruinier and Ken Ono, An algebraic formula for the partition function. In press.
Chapter 22
Pillai’s arithmetical function
In number theory, the gcd-sum function,[1] also called Pillai’s arithmetical function,[1] is defined for every n by
P (n) =n∑k=1
gcd(k, n)
or equivalently[1]
P (n) =∑d|n
dφ(n/d)
where d is a divisor of n and φ is Euler’s totient function.it also can be written as[2]
P (n) =∑d|n
dσ(d)µ(n/d)
where, σ is the Divisor function, and µ is the Möbius function.This multiplicative arithmetical function was introduced by the Indian mathematician Subbayya SivasankaranarayanaPillai in 1933.[3]
22.1 References[1] Lászlo Tóth (2010). “A survey of gcd-sum functions”. J. Integer Sequences 13.
[2] http://math.stackexchange.com/questions/135351/sum-of-gcdk-n
[3] S. S. Pillai (1933). “On an arithmetic function”. Annamalai University Journal II: 242–248.
A018804
98
Chapter 23
Prime gap
A prime gap is the difference between two successive prime numbers. The n-th prime gap, denoted gn or g(pn) isthe difference between the (n + 1)-th and the n-th prime numbers, i.e.
gn = pn+1 − pn.
We have g1 = 1, g2 = g3 = 2, and g4 = 4. The sequence (gn) of prime gaps has been extensively studied, howevermany questions and conjectures remain unanswered.The first 60 prime gaps are:
1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6,6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, ... (sequence A001223 in OEIS).
By the definition of g the following sum can be stated as
pn+1 = 2 +n∑i=1
gi
23.1 Simple observations
The first, smallest, and only odd prime gap is 1 between the only even prime number, 2, and the first odd prime, 3.All other prime gaps are even. There is only one pair of gaps between three consecutive odd natural numbers forwhich all are prime. These gaps are g2 and g3 between the primes 3, 5, and 7.For any prime number P, we write P# for P primorial, that is, the product of all prime numbers up to and includingP. If Q is the prime number following P, then the sequence
P# + 2, P# + 3, . . . , P# + (Q− 1)
is a sequence of Q − 2 consecutive composite integers, so here there is a prime gap of at least length Q − 1. Therefore,there exist gaps between primes which are arbitrarily large, i.e., for any prime number P, there is an integer n withgn ≥ P. (This is seen by choosing n so that pn is the greatest prime number less than P# + 2.) Another way to seethat arbitrarily large prime gaps must exist is the fact that the density of primes approaches zero, according to theprime number theorem. In fact, by this theorem, P# is very roughly a number the size of exp(P), and near exp(P) theaverage distance between consecutive primes is P.In reality, prime gaps of P numbers can occur at numbers much smaller than P#. For instance, the smallest sequenceof 71 consecutive composite numbers occurs between 31398 and 31468, whereas 71# has twenty-seven digits – itsfull decimal expansion being 557940830126698960967415390.
99
100 CHAPTER 23. PRIME GAP
Although the average gap between primes increases as the natural logarithm of the integer, the ratio of the maximumprime gap to the integers involved also increases as larger and larger numbers and gaps are encountered.In the opposite direction, the twin prime conjecture asserts that gn = 2 for infinitely many integers n.
23.2 Numerical results
Prime gap function
As of 2014 the largest known prime gap with identified probable prime gap ends has length 3311852, with 97953-digit probable primes found by M. Jansen and J. K. Andersen.[1][2] The largest known prime gap with identifiedproven primes as gap ends has length 1113106, with 18662-digit primes found by P. Cami, M. Jansen and J. K.Andersen.[1][3]
We say that gn is a maximal gap if gm < gn for all m < n. As of June 2014 the largest known maximal gaphas length 1476, found by Tomás Oliveira e Silva. It is the 75th maximal gap, and it occurs after the prime1425172824437699411.[4] Other record maximal gap terms can be found at A002386.Usually the ratio of gn / ln(pn) is called the merit of the gap gn . In 1931, E. Westzynthius proved that prime gapsgrow more than logarithmically. That is,[5]
lim supn→∞
gnlog pn
= ∞.
As of January 2012, the largest known merit value, as discovered by M. Jansen, is 66520 / ln(1931*1933#/7230 -30244) ≈ 35.4244594 where 1933# indicates the primorial of 1933. This number, 1931*1933#/7230 - 30244, is an816-digit prime. The next largest known merit value is 1476 / ln(1425172824437699411) ≈ 35.31.[1][6] This primewith the gap of 1476 is also the 75th maximal gap (the last one in the table below). Other record merit terms can befound at A111870.The Cramer-Shanks-Granville ratio is the ratio of gn / (ln(pn))^2.[6] The greatest known value of this ratio is 0.9206386for the prime 1693182318746371. Other record terms can be found at A111943.
23.3. FURTHER RESULTS 101
23.3 Further results
23.3.1 Upper bounds
Bertrand’s postulate states that there is always a prime number between k and 2k, so in particular pn₊₁ < 2pn, whichmeans gn < pn.The prime number theorem says that the “average length” of the gap between a prime p and the next prime is ln p.The actual length of the gap might be much more or less than this. However, from the prime number theorem onecan also deduce an upper bound on the length of prime gaps: for every ε > 0, there is a number N such that gn < εpnfor all n > N.One can deduce that the gaps get arbitrarily smaller in proportion to the primes: the quotient
limn→∞
gnpn
= 0
Hoheisel was the first to show[7] that there exists a constant θ < 1 such that
π(x+ xθ)− π(x) ∼ xθ
log(x) as xinfinity, to tends
hence showing that
gn < pθn,
for sufficiently large n.Hoheisel obtained the possible value 32999/33000 for θ. This was improved to 249/250 by Heilbronn,[8] and to θ =3/4 + ε, for any ε > 0, by Chudakov.[9]
A major improvement is due to Ingham,[10] who showed that if
ζ(1/2 + it) = O(tc)
for some positive constant c, where O refers to the big O notation, then
π(x+ xθ)− π(x) ∼ xθ
log(x)
for any θ > (1 + 4c)/(2 + 4c). Here, as usual, ζ denotes the Riemann zeta function and π the prime-counting function.Knowing that any c > 1/6 is admissible, one obtains that θ may be any number greater than 5/8.An immediate consequence of Ingham’s result is that there is always a prime number between n3 and (n + 1)3 if nis sufficiently large.[11] The Lindelöf hypothesis would imply that Ingham’s formula holds for c any positive number:but even this would not be enough to imply that there is a prime number between n2 and (n + 1)2 for n sufficientlylarge (see Legendre’s conjecture). To verify this, a stronger result such as Cramér’s conjecture would be needed.Huxley showed that one may choose θ = 7/12.[12]
A result, due to Baker, Harman and Pintz in 2001, shows that θ may be taken to be 0.525.[13]
In 2005, Daniel Goldston, János Pintz and Cem Yıldırım proved that
lim infn→∞
gnlog pn
= 0
and later improved it[14] to
102 CHAPTER 23. PRIME GAP
lim infn→∞
gn√log pn(log log pn)2<∞.
In 2013, Yitang Zhang proved that
lim infn→∞
gn < 7 · 107
meaning that there are infinitely many gaps that do not exceed 70 million.[15] A Polymath Project collaborative effortto optimize Zhang’s bound managed to lower the bound to 4680 on July 20, 2013.[16] In November 2013, JamesMaynard introduced a new refinement of the GPY sieve, allowing him to reduce the bound to 600 and show that forany m there exists a bounded interval containing m prime numbers.[17] Using Maynard’s ideas, the Polymath projecthas since improved the bound to 246.[18][16] Further, assuming the Elliott–Halberstam conjecture and its generalizedform, the Polymath project wiki states that N has been reduced to 12 and 6, respectively.[16]
23.3.2 Lower bounds
Robert Rankin, improving results by Erik Westzynthius and Paul Erdős, proved the existence of a constant c > 0 suchthat the inequality
gn >c logn log logn log log log logn
(log log logn)2
holds for infinitely many values n: he showed that one can take any constant c < eγ, where γ is the Euler–Mascheroniconstant. The value of the constant c was later improved to any constant c < 2eγ.[19]
Paul Erdős offered a $5,000 prize for a proof or disproof that the constant c in the above inequality may be takenarbitrarily large.[20] This was proved independently by Ford-Green-Konyagin-Tao and James Maynard, in the positive,by two papers respectively sent to arXiv in 2014.[21][22]
The result was further improved to
gn ≫ logn log logn log log log lognlog log logn
(for infinitely many values of n) by Ford-Green-Konyagin-Maynard-Tao.[23]
23.4 Conjectures about gaps between primes
Even better results are possible if it is assumed that the Riemann hypothesis is true. Harald Cramér proved that,under this assumption, the gap gn satisfies
gn = O(√pn ln pn),
using the big O notation. Later, he conjectured that the gaps are even smaller. Roughly speaking he conjectured that
gn = O((ln pn)2
).
At the moment, the numerical evidence seems to point in this direction. See Cramér’s conjecture for more details.
Firoozbakht’s conjecture states that p1/nn (where pn is the nth prime) is a strictly decreasing function of n, i.e.,
p1/(n+1)n+1 < p1/nn all for n ≥ 1.
23.5. AS AN ARITHMETIC FUNCTION 103
If this conjecture is true, then the function gn = pn+1 − pn satisfies gn < (log pn)2 − log pn all for n > 4. [24]
This is one of the strongest upper bound ever conjectured for prime gaps. Moreover, this conjecture implies Cramér’sconjecture in a strong form and would be consistent with Daniel Shanks conjectured asymptotic equality of recordgaps.[25]
By using tables of maximal gaps, Firoozbakht’s conjecture has been verified for all primes below 4×1018.[26]
Mean while, the Oppermann’s conjecture is a conjecture which is weaker than Cramér’s conjecture. The expectedgap size with Oppermann’s conjecture is
gn <√pn
Andrica’s conjecture, which is a weaker conjecture to Oppermann’s, states that[20]
gn < 2√pn + 1.
This is a slight strengthening of Legendre’s conjecture that between successive square numbers there is always aprime.
23.5 As an arithmetic function
The gap gn between the nth and (n + 1)st prime numbers is an example of an arithmetic function. In this context itis usually denoted dn and called the prime difference function.[20] The function is neither multiplicative nor additive.
23.6 See also
• Bonse’s inequality
• Gaussian moat
• Twin prime
23.7 References[1] Andersen, Jens Kruse. “The Top-20 Prime Gaps”. Retrieved 2014-06-13.
[2] Largest known prime gap
[3] A proven prime gap of 1113106
[4] Maximal Prime Gaps
[5] Westzynthius, E. (1931), "Über die Verteilung der Zahlen die zu den n ersten Primzahlen teilerfremd sind”, CommentationesPhysico-Mathematicae Helsingsfors (in German) 5: 1–37, JFM 57.0186.02, Zbl 0003.24601.
[6] NEW PRIME GAP OF MAXIMUM KNOWN MERIT
[7] Hoheisel, G. (1930). “Primzahlprobleme in der Analysis”. Sitzunsberichte der Königlich Preussischen Akademie der Wis-senschaften zu Berlin 33: 3–11. JFM 56.0172.02.
[8] Heilbronn, H. A. (1933). "Über den Primzahlsatz von Herrn Hoheisel”. Mathematische Zeitschrift 36 (1): 394–423.doi:10.1007/BF01188631.
[9] Tchudakoff, N. G. (1936). “On the difference between two neighboring prime numbers”. Math. Sb. 1: 799–814.
[10] Ingham, A. E. (1937). “On the difference between consecutive primes”. Quarterly Journal of Mathematics. Oxford Series8 (1): 255–266. doi:10.1093/qmath/os-8.1.255.
104 CHAPTER 23. PRIME GAP
[11] Cheng, Yuan-You Fu-Rui (2010). “Explicit estimate on primes between consecutive cubes”. Rocky Mt. J. Math. 40:117–153. doi:10.1216/rmj-2010-40-1-117. Zbl 1201.11111.
[12] Huxley, M. N. (1972). “On the Difference between Consecutive Primes”. Inventiones Mathematicae 15 (2): 164–170.doi:10.1007/BF01418933.
[13] Baker, R. C.; Harman, G.; Pintz, J. (2001). “The difference between consecutive primes, II”. Proceedings of the LondonMathematical Society 83 (3): 532–562. doi:10.1112/plms/83.3.532.
[14] “Primes in Tuples II”. ArXiv. Retrieved 2013-11-23.
[15] Zhang, Yitang (2014). “Bounded gaps between primes”. Annals ofMathematics 179 (3): 1121–1174. doi:10.4007/annals.2014.179.3.7.MR 3171761.
[16] “Bounded gaps between primes”. Polymath. Retrieved 2013-07-21.
[17] Maynard, James (2015). “Small gaps between primes”. Annals ofMathematics 181 (1): 383–413. doi:10.4007/annals.2015.181.1.7.MR 3272929.
[18] D.H.J. Polymath (2014). “Variants of the Selberg sieve, and bounded intervals containing many primes”. Research in theMathematical Sciences 1 (12). arXiv:1407.4897. doi:10.1186/s40687-014-0012-7. MR 3373710.
[19] Pintz, J. (1997). “Very large gaps between consecutive primes”. J. Number Theory 63 (2): 286–301. doi:10.1006/jnth.1997.2081.
[20] Guy (2004) §A8
[21] Kevin Ford, Ben Green, Sergei Konyagin, and Terence Tao (2014) “Large gaps between consecutive prime numbers”
[22] James Maynard (2014) “Large gaps between primes”
[23] Ford, Kevin; Green, Ben; Konyagin, Sergei; Maynard, James; Tao, Terence (2015). “Long gaps between primes”. arXiv:1412.5029.
[24] Sinha, Nilotpal Kanti (2010), “On a new property of primes that leads to a generalization of Cramer’s conjecture”, arXiv.org> math > arXiv:1010.1399: 1–10.
[25] Shanks, Daniel (1964), “On Maximal Gaps between Successive Primes”, Mathematics of Computation (American Mathe-matical Society) 18 (88): 646–651, doi:10.2307/2002951, JSTOR 2002951, Zbl 0128.04203.
[26] Kourbatov, Alexei. “prime Gaps: Firoozbakht Conjecture”.
• Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. ISBN 978-0-387-20860-2. Zbl 1058.11001.
• Feliksiak, Jan (2013). The Symphony Of Primes, Distribution Of Primes And Riemann’s Hypothesis. Xlibris.ISBN 978-1479765584.
23.8 Further reading
• Soundararajan, Kannan (2007). “Small gaps between prime numbers: the work of Goldston-Pintz-Yıldırım”.Bull. Am. Math. Soc., New Ser. 44 (1): 1–18. doi:10.1090/s0273-0979-06-01142-6. Zbl 1193.11086.
• Mihăilescu, Preda (June 2014). “On some conjectures in additive number theory” (PDF). Newsletter of theEuropean Mathematical Society (92): 13–16. doi:10.4171/NEWS. ISSN 1027-488X.
23.9 External links
• Thomas R. Nicely, Some Results of Computational Research in Prime Numbers -- Computational NumberTheory. This reference web site includes a list of all first known occurrence prime gaps.
• Weisstein, Eric W., “Prime Difference Function”, MathWorld.
• Prime Difference Function at PlanetMath.org.
23.9. EXTERNAL LINKS 105
• Armin Shams, Re-extending Chebyshev’s theorem about Bertrand’s conjecture, does not involve an 'arbitrarilybig' constant as some other reported results.
• Chris Caldwell, Gaps Between Primes; an elementary introduction
• www.primegaps.com A study of the gaps between consecutive prime numbers
• Andrew Granville, Primes in Intervals of Bounded Length; overview of the results obtained so far up to andincluding James Maynard’s work of November 2013.
Chapter 24
Prime-counting function
In mathematics, the prime-counting function is the function counting the number of prime numbers less than orequal to some real number x.[1][2] It is denoted by π(x) (this does not refer to the number π).
10 20 30 40 50 60n
2
4
6
8
10
12
14
16
π ( n )
The values of π(n) for the first 60 integers
24.1 History
Of great interest in number theory is the growth rate of the prime-counting function.[3][4] It was conjectured in theend of the 18th century by Gauss and by Legendre to be approximately
x/ ln(x)in the sense that
limx→∞
π(x)
x/ ln(x) = 1.
106
24.2. TABLE OF Π(X), X / LN X, AND LI(X) 107
This statement is the prime number theorem. An equivalent statement is
limx→∞
π(x)/ li(x) = 1
where li is the logarithmic integral function. The prime number theorem was first proved in 1896 by JacquesHadamard and by Charles de la Vallée Poussin independently, using properties of the Riemann zeta function in-troduced by Riemann in 1859.More precise estimates of π(x)are now known; for example
π(x) = li(x) +O(xe−
√ln x/15)
where the O is big O notation. For most values of x we are interested in (i.e., when x is not unreasonably large) li(x)is greater than π(x) , but infinitely often the opposite is true. For a discussion of this, see Skewes’ number.Proofs of the prime number theorem not using the zeta function or complex analysis were found around 1948 by AtleSelberg and by Paul Erdős (for the most part independently).[5]
24.2 Table of π(x), x / ln x, and li(x)
The table shows how the three functions π(x), x / ln x and li(x) compare at powers of 10. See also,[3][6][7]and[8]
1 10 4 10 8 10 12 10 16 1020 10 24
0.9
1.0
1.1
1.2
Graph showing ratio of the prime-counting function π(x) to two of its approximations, x/ln x and Li(x). As x increases (note x axisis logarithmic), both ratios tend towards 1. The ratio for x/ln x converges from above very slowly, while the ratio for Li(x) convergesmore quickly from below.
In the On-Line Encyclopedia of Integer Sequences, the π(x) column is sequence A006880, π(x) - x / ln x is sequenceA057835, and li(x) − π(x) is sequence A057752. The value for π(1024) was originally computed by J. Buethe,
J. Franke, A. Jost, and T. Kleinjung assuming the Riemann hypothesis.[9] It has since been verified unconditionallyin a computation by D. J. Platt.[10]
108 CHAPTER 24. PRIME-COUNTING FUNCTION
24.3 Algorithms for evaluating π(x)
A simple way to find π(x) , if x is not too large, is to use the sieve of Eratosthenes to produce the primes less than orequal to x and then to count them.A more elaborate way of finding π(x) is due to Legendre: given x , if p1, p2, . . . , pn are distinct prime numbers,then the number of integers less than or equal to x which are divisible by no pi is
⌊x⌋ −∑i
⌊x
pi
⌋+∑i<j
⌊x
pipj
⌋−∑i<j<k
⌊x
pipjpk
⌋+ · · ·
(where ⌊· · · ⌋ denotes the floor function). This number is therefore equal to
π(x)− π(√x)+ 1
when the numbers p1, p2, . . . , pn are the prime numbers less than or equal to the square root of x .In a series of articles published between 1870 and 1885, Ernst Meissel described (and used) a practical combinatorialway of evaluating π(x) . Let p1 , p2, . . . , pn be the first n primes and denote by Φ(m,n) the number of naturalnumbers not greater than m which are divisible by no pi . Then
Φ(m,n) = Φ(m,n− 1)− Φ
(m
pn, n− 1
)Given a natural number m , if n = π ( 3
√m) and if µ = π (
√m)− n , then
π(m) = Φ(m,n) + n(µ+ 1) +µ2 − µ
2− 1−
µ∑k=1
π
(m
pn+k
)Using this approach, Meissel computed π(x) , for x equal to 5×105, 106, 107, and 108.In 1959, Derrick Henry Lehmer extended and simplified Meissel’s method. Define, for realm and for natural numbersn and k , Pk(m,n) as the number of numbers not greater than m with exactly k prime factors, all greater than pn .Furthermore, set P0(m,n) = 1 . Then
Φ(m,n) =+∞∑k=0
Pk(m,n)
where the sum actually has only finitely many nonzero terms. Let y denote an integer such that 3√m ≤ y ≤
√m ,
and set n = π(y) . Then P1(m,n) = π(m)− n and Pk(m,n) = 0 when k ≥ 3. Therefore
π(m) = Φ(m,n) + n− 1− P2(m,n)
The computation of P2(m,n) can be obtained this way:
P2(m,n) =∑
y<p≤√m
(π
(m
p
)− π(p) + 1
)On the other hand, the computation of Φ(m,n) can be done using the following rules:
1. Φ(m, 0) = ⌊m⌋
2. Φ(m, b) = Φ(m, b− 1)− Φ(mpb, b− 1
)Using his method and an IBM 701, Lehmer was able to compute π
(1010
).
Further improvements to this method were made by Lagarias, Miller, Odlyzko, Deléglise and Rivat.[11]
24.4. OTHER PRIME-COUNTING FUNCTIONS 109
24.4 Other prime-counting functions
Other prime-counting functions are also used because they are more convenient to work with. One is Riemann’sprime-counting function, usually denoted as Π0(x) or J0(x) . This has jumps of 1/n for prime powers pn, with ittaking a value half-way between the two sides at discontinuities. That added detail is because then it may be definedby an inverse Mellin transform. Formally, we may define Π0(x) by
Π0(x) =1
2
( ∑pn<x
1
n+∑pn≤x
1
n
)where p is a prime.We may also write
Π0(x) =x∑2
Λ(n)
lnn − 1
2
Λ(x)
lnx =∞∑n=1
1
nπ0(x
1/n)
where Λ(n) is the von Mangoldt function and
π0(x) = limε→0
π(x− ε) + π(x+ ε)
2.
Möbius inversion formula then gives
π0(x) =∞∑n=1
µ(n)
nΠ0(x
1/n)
Knowing the relationship between log of the Riemann zeta function and the von Mangoldt function Λ , and using thePerron formula we have
ln ζ(s) = s
∫ ∞
0
Π0(x)x−s−1 dx
The Chebyshev function weights primes or prime powers pn by ln(p):
θ(x) =∑p≤x
ln p
ψ(x) =∑pn≤x
ln p =∞∑n=1
θ(x1/n) =∑n≤x
Λ(n).
Riemann’s prime-counting function has an ordinary generating function that can be expressed in terms of formalpower series as:
∞∑n=1
Π0(n)xn =
∞∑a=2
xa
1− x− 1
2
∞∑a=2
∞∑b=2
xab
1− x+
1
3
∞∑a=2
∞∑b=2
∞∑c=2
xabc
1− x− 1
4
∞∑a=2
∞∑b=2
∞∑c=2
∞∑d=2
xabcd
1− x+ · · ·
24.5 Formulas for prime-counting functions
Formulas for prime-counting functions come in two kinds: arithmetic formulas and analytic formulas. Analyticformulas for prime-counting were the first used to prove the prime number theorem. They stem from the work ofRiemann and von Mangoldt, and are generally known as explicit formulas.[12]
110 CHAPTER 24. PRIME-COUNTING FUNCTION
We have the following expression for ψ:
ψ0(x) = x−∑ρ
xρ
ρ− ln 2π − 1
2ln(1− x−2)
where
ψ0(x) = limε→0
ψ(x− ε) + ψ(x+ ε)
2.
Here ρ are the zeros of the Riemann zeta function in the critical strip, where the real part of ρ is between zero andone. The formula is valid for values of x greater than one, which is the region of interest. The sum over the roots isconditionally convergent, and should be taken in order of increasing absolute value of the imaginary part. Note thatthe same sum over the trivial roots gives the last subtrahend in the formula.For Π0(x) we have a more complicated formula
Π0(x) = li(x)−∑ρ
li(xρ)− ln 2 +∫ ∞
x
dt
t(t2 − 1) ln t .
Again, the formula is valid for x > 1, while ρ are the nontrivial zeros of the zeta function ordered according totheir absolute value, and, again, the latter integral, taken with minus sign, is just the same sum, but over the trivialzeros. The first term li(x) is the usual logarithmic integral function; the expression li(xρ) in the second term shouldbe considered as Ei(ρ ln x), where Ei is the analytic continuation of the exponential integral function from positivereals to the complex plane with branch cut along the negative reals.Thus, Möbius inversion formula gives us[13]
π0(x) = R(x)−∑ρ
R(xρ)− 1
lnx +1
πarctan π
lnx
valid for x > 1, where
R(x) =∞∑n=1
µ(n)
nli(x1/n) = 1 +
∞∑k=1
(lnx)kk!kζ(k + 1)
is so-called Riemann’s R-function.[14] The latter series for it is known as Gram series [15] and converges for all positivex.The sum over non-trivial zeta zeros in the formula for π0(x) describes the fluctuations of π0(x) , while the remainingterms give the “smooth” part of prime-counting function,[16] so one can use
R(x)− 1
lnx +1
πarctan π
lnxas the best estimator of π(x) for x > 1.The amplitude of the “noisy” part is heuristically about √
x/ ln x , so the fluctuations of the distribution of primes maybe clearly represented with the Δ-function:
∆(x) =
(π0(x)− R(x) + 1
lnx − 1
πarctan π
lnx
) lnx√x.
An extensive table of the values of Δ(x) is available.[7]
24.6. INEQUALITIES 111
Δ-function (red line) on log scale
24.6 Inequalities
Here are some useful inequalities for π(x).
xln x < π(x) < 1.25506 x
ln x for x ≥ 17.[17]
The left inequality holds for x ≥ 17 and the right inequality holds for x > 1.An explanation of the constant 1.25506 is given at (sequence A209883 in OEIS).Pierre Dusart proved in 2010:
xln x−1 < π(x) for x ≥ 5393 , and
π(x) < xln x−1.1 for x ≥ 60184 .[18]
Here are some inequalities for the nth prime, pn.[19]
n(ln(n lnn)− 1) < pn < nln(n lnn)for n ≥ 6.
The left inequality holds for n ≥ 1 and the right inequality holds for n ≥ 6.An approximation for the nth prime number is
pn = n(ln(n lnn)− 1) +n(ln lnn− 2)
lnn +O
(n(ln lnn)2(lnn)2
).
24.7 The Riemann hypothesis
The Riemann hypothesis is equivalent to a much tighter bound on the error in the estimate for π(x) , and hence to amore regular distribution of prime numbers,
π(x) = li(x) +O(√x logx).
112 CHAPTER 24. PRIME-COUNTING FUNCTION
Specifically,[20]
|π(x)− li(x)| < 1
8π
√x logx, all forx ≥ 2657.
24.8 See also
• Bertrand’s postulate
• Oppermann’s conjecture
• Foias constant
24.9 References[1] Bach, Eric; Shallit, Jeffrey (1996). Algorithmic Number Theory. MIT Press. volume 1 page 234 section 8.8. ISBN
0-262-02405-5.
[2] Weisstein, Eric W., “Prime Counting Function”, MathWorld.
[3] “How many primes are there?". Chris K. Caldwell. Retrieved 2008-12-02.
[4] Dickson, Leonard Eugene (2005). History of the Theory of Numbers, Vol. I: Divisibility and Primality. Dover Publications.ISBN 0-486-44232-2.
[5] Ireland, Kenneth; Rosen, Michael (1998). A Classical Introduction to Modern Number Theory (Second ed.). Springer.ISBN 0-387-97329-X.
[6] “Tables of values of pi(x) and of pi2(x)". Tomás Oliveira e Silva. Retrieved 2008-09-14.
[7] “Values of π(x) and Δ(x) for various x’s”. Andrey V. Kulsha. Retrieved 2008-09-14.
[8] “A table of values of pi(x)". Xavier Gourdon, Pascal Sebah, Patrick Demichel. Retrieved 2008-09-14.
[9] “Conditional Calculation of pi(1024)". Chris K. Caldwell. Retrieved 2010-08-03.
[10] “Computing π(x) Analytically)". Retrieved Jul 25, 2012.
[11] “Computing π(x): The Meissel, Lehmer, Lagarias, Miller, Odlyzko method” (PDF). Marc Deléglise and Jöel Rivat, Math-ematics of Computation, vol. 65, number 33, January 1996, pages 235–245. Retrieved 2008-09-14.
[12] Titchmarsh, E.C. (1960). The Theory of Functions, 2nd ed. Oxford University Press.
[13] Riesel, Hans; Göhl, Gunnar (1970). “Some calculations related to Riemann’s prime number formula”. Mathematics ofComputation (American Mathematical Society) 24 (112): 969–983. doi:10.2307/2004630. ISSN 0025-5718. JSTOR2004630. MR 0277489.
[14] Weisstein, Eric W., “Riemann Prime Counting Function”, MathWorld.
[15] Weisstein, Eric W., “Gram Series”, MathWorld.
[16] “The encoding of the prime distribution by the zeta zeros”. Matthew Watkins. Retrieved 2008-09-14.
[17] Rosser, J. Barkley; Schoenfeld, Lowell (1962). “Approximate formulas for some functions of prime numbers”. Illinois J.Math. 6: 64–94. ISSN 0019-2082. Zbl 0122.05001.
[18] Dusart, Pierre. ""ESTIMATES OF SOME FUNCTIONS OVER PRIMES WITHOUT R.H."" (PDF). arxiv.org. Re-trieved 22 April 2014.
[19] Inequalities for the n-th prime number at function.wolfram, retrieved March 22, 2013
[20] Schoenfeld, Lowell (1976). “Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II”. Mathematics of Computation(American Mathematical Society) 30 (134): 337–360. doi:10.2307/2005976. ISSN 0025-5718. JSTOR 2005976. MR0457374.
24.10. EXTERNAL LINKS 113
24.10 External links• Chris Caldwell, The Nth Prime Page at The Prime Pages.
• Tomás Oliveira e Silva, Tables of prime-counting functions.
Chapter 25
Rank of a partition
11 parts
largest part = 21
rank = 10The rank of a partition, shown as its Young diagram
In mathematics, particularly in the fields of number theory and combinatorics, the rank of a partition of a positiveinteger is a certain integer associated with the partition. In fact at least two different definitions of rank appear in theliterature. The first definition, with which most of this article is concerned, is that the rank of a partition is the numberobtained by subtracting the number of parts in the partition from the largest part in the partition. The concept wasintroduced by Freeman Dyson in a paper published in the journal Eureka.[1] It was presented in the context of a studyof certain congruence properties of the partition function discovered by the Indian mathematical genius SrinivasaRamanujan. A different concept, sharing the same name, is used in combinatorics, where the rank is taken to be thesize of the Durfee square of the partition.
25.1 Definition
By a partition of a positive integer n we mean a finite multiset λ = λk, λk ₋ ₁, . . . , λ1 of positive integers satisfyingthe following two conditions:
114
25.2. NOTATIONS 115
• λ ≥ . . . ≥ λ2 ≥ λ1 > 0.
• λk + . . . + λ2 + λ1 = n.
If λk, . . . , λ2, λ1 are distinct, that is, if
• λk > . . . > λ2 > λ1 > 0
the partition λ is called a strict partition of n. The integers λk, λk ₋ ₁, ..., λ1 are the parts of the partition. The numberof parts in the partition λ is k and the largest part in the partition is λk. The rank of the partition λ (whether ordinaryor strict) is defined as λk − k.[1]
The ranks of the partitions of n take the following values and no others:[1]
n − 1, n −3, n −4, . . . , 2, 1, 0, −1, −2, . . . , −(n − 4), −(n − 3), −(n − 1).
The following table gives the ranks of the various partitions of the number 5.
Ranks of the partitions of the integer 5
25.2 Notations
The following notations are used to specify how many partitions have a given rank. Let n, q be a positive integers andm be any integer.
• The total number of partitions of n is denoted by p(n).
• The number of partitions of n with rank m is denoted by N(m, n).
• The number of partitions of n with rank congruent to m modulo q is denoted by N(m, q, n).
• The number of strict partitions of n is denoted by Q(n).
• The number of strict partitions of n with rank m is denoted by R(m, n).
• The number of strict partitions of n with rank congruent to m modulo q is denoted by T(m, q, n).
For example,
p(5) = 7 , N(2, 5) = 1 , N(3, 5) = 0 , N(2, 2, 5) = 5 .Q(5) = 3 , R(2, 5) = 1 , R(3, 5) = 0 , T(2, 2, 5) = 2.
25.3 Some basic results
Let n, q be a positive integers and m be any integer.[1]
• N(m,n) = N(−m,n)
• N(m, q, n) = N(q −m, q, n)
• N(m, q, n) =∑∞r=−∞N(m+ rq, n)
116 CHAPTER 25. RANK OF A PARTITION
25.4 Ramanujan’s congruences and Dyson’s conjecture
Srinivasa Ramanujan in a paper published in 1919 proved the following congruences involving the partition functionp(n):[2]
• p(5 n + 4) ≡ 0 (mod 5)
• p(7n + 5) ≡ 0 (mod 7)
• p(11n + 6) ≡ 0 (mod 11)
In commenting on this result, Dyson noted that " . . . although we can prove that the partitions of 5n + 4 canbe divided into five equally numerous subclasses, it is unsatisfactory to receive from the proofs no concrete idea ofhow the division is to be made. We require a proof which will not appeal to generating functions, . . . ".[1] Dysonintroduced the idea of rank of a partition to accomplish the task he set for himself. Using this new idea, he made thefollowing conjectures:
• N(0, 5, 5n + 4) = N(1, 5, 5n + 4) = N(2, 5, 5n + 4) = N(3, 5, 5n + 4) = N(4, 5, 5n + 4)
• N(0, 7, 7n + 5) = N(1, 7, 7n + 5) = N(2, 7, 7n + 5) = . . . = N(6, 7, 7n + 5)
These conjectures were proved by Atkin and Swinnerton-Dyer in 1954.[3]
The following tables show how the partitions of the integers 4 (5 × n + 4 with n = 0) and 9 (5 × n + 4 with n = 1 ) getdivided into five equally numerous subclasses.
Partitions of the integer 4Partitions of the integer 9
25.5 Generating functions• The generating function of p(n) was discovered by Euler and is well-known.[4]
∞∑n=0
p(n)xn =∞∏k=1
1
(1− xk)
• The generating function for N(m, n) is given below:[5]
∞∑m=−∞
∞∑n=0
N(m,n)zmqn = 1 +∞∑n=1
qn2∏n
k=1(1− zqk)(1− z−1qk)
• The generating function for Q ( n ) is given below:[6]
∞∑n=0
Q(n)xn =∞∏k=0
1
(1− x2k−1)
• The generating function for Q ( m , n ) is given below:[6]
∞∑m,n=0
Q(m,n)zmqn = 1 +∞∑s=1
qs(s+1)/2
(1− zq)(1− zq2) · · · (1− zqs)
25.6. ALTERNATE DEFINITION 117
25.6 Alternate definition
Main article: Durfee square
In combinatorics, the phrase rank of a partition is sometimes used to describe a different concept: the rank of apartition λ is the largest integer i such that λ has at least i parts each of which is no smaller than i.[7] Equivalently, thisis the length of the main diagonal in the Young diagram or Ferrers diagram for λ, or the side-length of the Durfeesquare of λ.The table of ranks of partitions of 5 is given below.
Ranks of the partitions of the integer 5
25.7 Further reading• Asymptotic formulas for the rank partition function:[8]
• Congruences for rank function:[9]
• Generalisation of rank to BG-rank:[10]
25.8 See also
Crank of a partition
25.9 References[1] F. Dyson (1944). “Some guesses in the theory of partitions”. Eureka (Cambridge) 8: 10–15.
[2] Srinivasa, Ramanujan (1919). “Some properties of p(n), number of partitions of n". Proceedings of the Cambridge Philo-sophical Society XIX: 207–210.
[3] A. O. L. Atkin; H. P. F. Swinnerton-Dyer (1954). “Some properties of partitions,”. Proceedings of the London Mathemat-ical Society 66 (4): 84–106.
[4] G.H. Hardy and E.W. Wright (1938). An introduction to the theory of numbers. London: Oxford University Press. p. 274.
[5] Bringmann, Kathrin (2009). “Congruences for Dyson’s ranks”. International Journal of Number Theory 5 (4). Retrieved24 November 2012.
[6] Maria Monks (2010). “Number theoretic properties of generating functions related to Dyson’s rank for partitions intodistinct parts”. Proceedings of the American Mathematical Society 138: 481–494. doi:10.1090/s0002-9939-09-10076-x.Retrieved 24 November 2012.
[7] Stanley, Richard P. (1999) Enumerative Combinatorics, Volume 2, p. 289. Cambridge University Press. ISBN 0-521-56069-1.
[8] Bringman, Kathrin (July 2009). “Asymptotics For Rank Partition Functions”. Transactions Of The AmericanMathematicalSociety 361 (7): 3483–3500. doi:10.1090/s0002-9947-09-04553-x. Retrieved 21 November 2012.
[9] Bringmann, Kathrin. “Congruences for Dyson’s rank”. Retrieved 21 November 2012.
[10] Alexander Berkovich and Frank Garvan. “The BG-rank of a partition and its applications”. Retrieved 21 November 2012.
Chapter 26
Von Mangoldt function
In mathematics, the vonMangoldt function is an arithmetic function named after German mathematician Hans vonMangoldt. It is an example of an important arithmetic function that is neither multiplicative nor additive.
26.1 Definition
The von Mangoldt function, denoted by Λ(n), is defined as
Λ(n) =
log p ifn = pk prime some for p integer and k ≥ 1,
0 otherwise.
The values of Λ(n) for the first nine positive numbers are
0, log 2, log 3, log 2, log 5, 0, log 7, log 2, log 3,
which is related to (sequence A014963 in OEIS).The summatory von Mangoldt function, ψ(x), also known as the Chebyshev function, is defined as
ψ(x) =∑n≤x
Λ(n).
von Mangoldt provided a rigorous proof of an explicit formula for ψ(x) involving a sum over the non-trivial zeros ofthe Riemann zeta function. This was an important part of the first proof of the prime number theorem.
26.2 Properties
The von Mangoldt function satisfies the identity[1][2]
log(n) =∑d|n
Λ(d).
The sum is taken over all integers d that divide n. This is proved by the fundamental theorem of arithmetic, since theterms that are not powers of primes are equal to 0. For example, consider the case n = 12 = 22 × 3. Then
118
26.3. DIRICHLET SERIES 119
∑d|12
Λ(d) = Λ(1) + Λ(2) + Λ(3) + Λ(4) + Λ(6) + Λ(12)
= Λ(1) + Λ(2) + Λ(3) + Λ(22)+ Λ(2× 3) + Λ
(22 × 3
)= 0 + log(2) + log(3) + log(2) + 0 + 0
= log(2× 3× 2)
= log(12).
By Möbius inversion, we have[2][3][4]
Λ(n) = −∑d|n
µ(d) log(d) .
26.3 Dirichlet series
The von Mangoldt function plays an important role in the theory of Dirichlet series, and in particular, the Riemannzeta function. In particular, one has
log ζ(s) =∞∑n=2
Λ(n)
log(n)1
ns, Re(s) > 1.
The logarithmic derivative is then
ζ ′(s)
ζ(s)= −
∞∑n=1
Λ(n)
ns.
These are special cases of a more general relation on Dirichlet series. If one has
F (s) =∞∑n=1
f(n)
ns
for a completely multiplicative function f (n), and the series converges for Re(s) > σ0, then
F ′(s)
F (s)= −
∞∑n=1
f(n)Λ(n)
ns
converges for Re(s) > σ0.
26.4 Chebyshev function
The second Chebyshev function ψ(x) is the summatory function of the von Mangoldt function:[5]
ψ(x) =∑pk≤x
log p =∑n≤x
Λ(n) .
The Mellin transform of the Chebyshev function can be found by applying Perron’s formula:
ζ ′(s)
ζ(s)= −s
∫ ∞
1
ψ(x)
xs+1dx
which holds for Re(s) > 1.
120 CHAPTER 26. VON MANGOLDT FUNCTION
26.5 Exponential series
-0.337886
-0.337884
-0.337882
-0.33788
-0.337878
-0.337876
-0.337874
-0.337872
-0.33787
-0.337868
-0.337866
0 2e-06 4e-06 6e-06 8e-06 1e-05
sum
_n (L
ambd
a(n)
-1) e
xp (-
ny)
y
von Mangoldt Exponential Series
Hardy and Littlewood examined the series[6]
F (y) =
∞∑n=2
(Λ(n)− 1) e−ny
in the limit y → 0+. Assuming the Riemann hypothesis, they demonstrate that
F (y) = O
(1√y
).
Curiously, they also show that this function is oscillatory as well, with diverging oscillations. In particular, there existsa value K > 0 such that
F (y) < − K√y, and F (y) >
K√y
infinitely often. The graphic to the right indicates that this behaviour is not at first numerically obvious: the oscillationsare not clearly seen until the series is summed in excess of 100 million terms, and are only readily visible when y <10−5.
26.6 Riesz mean
The Riesz mean of the von Mangoldt function is given by
26.7. APPROXIMATION BY RIEMANN ZETA ZEROS 121
∑n≤λ
(1− n
λ
)δΛ(n) = − 1
2πi
∫ c+i∞
c−i∞
Γ(1 + δ)Γ(s)
Γ(1 + δ + s)
ζ ′(s)
ζ(s)λsds
=λ
1 + δ+∑ρ
Γ(1 + δ)Γ(ρ)
Γ(1 + δ + ρ)+∑n
cnλ−n.
Here, λ and δ are numbers characterizing the Riesz mean. One must take c > 1. The sum over ρ is the sum over thezeroes of the Riemann zeta function, and
∑n
cnλ−n
can be shown to be a convergent series for λ > 1.
26.7 Approximation by Riemann zeta zeros
100 200 300 400
- 4
- 2
2
4
6
The first Riemann zeta zero wave in the sum that approximates the von Mangoldt function
The real part of the sum over the zeta zeros:
−∑∞i=1 n
ρ(i) , where ρ(i) is the i-th zeta zero, peaks at primes, as can be seen in the adjoining graph, andcan also be verified through numerical computation. It does not sum up to the Von Mangoldt function.[7]
The Fourier transform of the von Mangoldt function gives a spectrum with spikes at ordinates equal to imaginary partof the Riemann zeta function zeros. This is sometimes called a duality.
26.8 See also
• Prime-counting function
122 CHAPTER 26. VON MANGOLDT FUNCTION
The Fourier transform of the von Mangoldt function gives a spectrum with imaginary parts of Riemann zeta zeros as spikes at thex-axis ordinates (right), while the von Mangoldt function can be approximated by zeta zero waves (left)
26.9 References[1] Apostol (1976) p.32
[2] Tenenbaum (1995) p.30
[3] Apostol (1976) p.33
[4] Schroeder, Manfred R. (1997). Number theory in science and communication. With applications in cryptography, physics,digital information, computing, and self-similarity. Springer Series in Information Sciences 7 (3rd ed.). Berlin: Springer-Verlag. ISBN 3-540-62006-0. Zbl 0997.11501.
[5] Apostol (1976) p.246
[6] Hardy, G. H. & Littlewood, J. E. (1916). “Contributions to the Theory of the Riemann Zeta-Function and the Theory ofthe Distribution of Primes” (PDF). Acta Mathematica 41: 119–196. doi:10.1007/BF02422942.
[7] Conrey, J. Brian (March 2003). “The Riemann hypothesis” (PDF). Notices Am. Math. Soc. 50 (3): 341–353. Zbl1160.11341. Page 346
• Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, NewYork-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001
• Tenebaum, Gérald (1995). Introduction to analytic and probabilistic number theory. Cambridge Studies inAdvanced Mathematics 46. Translated by C.B. Thomas. Cambridge: Cambridge University Press. ISBN0-521-41261-7. Zbl 0831.11001.
26.10 External links• Allan Gut, Some remarks on the Riemann zeta distribution (2005)
• S.A. Stepanov (2001), “Mangoldt function”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4
• Chris King, Primes out of thin air (2010)
• Heike, How plot Riemann zeta zero spectrum in Mathematica? (2012)
26.11. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 123
26.11 Text and image sources, contributors, and licenses
26.11.1 Text• Additive function Source: https://en.wikipedia.org/wiki/Additive_function?oldid=655103755 Contributors: AxelBoldt, XJaM, Camem-
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• Aliquot sum Source: https://en.wikipedia.org/wiki/Aliquot_sum?oldid=670461186 Contributors: Closedmouth, David Eppstein, Geof-freyT2000 and Anonymous: 1
• Arithmetic function Source: https://en.wikipedia.org/wiki/Arithmetic_function?oldid=675930110Contributors: AxelBoldt, XJaM, MichaelHardy, David Shay, Phys, Fredrik, RedWolf, Gandalf61, Wikibot, Giftlite, Gubbubu, Nabla, Haham hanuka, Burn, Oleg Alexandrov,Linas, FlaBot, Glenn L, GangofOne, Rwalker, SmackBot, Bluebot, Colonies Chris, Richard L. Peterson, Jim.belk, RomanSpa, Mscal-culus, Madmath789, ERAGON, CRGreathouse, CBM, WillowW, Marek69, RobHar, JAnDbot, Olaf, Magioladitis, Mikhail Dvorkin,Policron, TXiKiBoT, Tomaxer, GirasoleDE, Bob1960evens, Virginia-American, LaaknorBot, Uncia, PV=nRT, Legobot, Luckas-bot,Yobot, TaBOT-zerem, 9258fahsflkh917fas, Jkelleyy, ArthurBot, Drilnoth, FrescoBot, Citation bot 1, DrilBot, WikitanvirBot, Quondum,OnePt618, Fr3aki, Sapphorain, Anita5192, Snotbot, Stapler9124, Helpful Pixie Bot, Solomon7968, Atomician, Brad7777, Glacialfox,ChrisGualtieri, Dexbot, Deltahedron, Jamesx12345, MathLine, K9re11, Mun Chan Young, SoSivr, Toanboe, KasparBot and Anonymous:31
• Average Source: https://en.wikipedia.org/wiki/Average?oldid=675569016 Contributors: Derek Ross, Tarquin, Ellmist, Zadcat, B4hand,Montrealais, Patrick, Michael Hardy, Tannin, TakuyaMurata, Ellywa, Snoyes, Den fjättrade ankan~enwiki, Andrewa, Charles Matthews,Timwi, Fuzheado, Khalad, Bevo, Donarreiskoffer, Henrygb, Dag Hovland, Wikibot, Ruakh, Giftlite, Seabhcan, Bfinn, Hokanomono, Ev-eryking, PRB, Gilgamesh~enwiki, Dmmaus, Patsup, Piotrus, Kaldari, Pmanderson, Io usurped, Mormegil, Discospinster, Michal Jurosz,Vsmith, Lulu of the Lotus-Eaters, Kndiaye, Paul August, Zenohockey, RoyBoy, Grick, O18, Smalljim, L33tminion, Jérôme, Alansohn,Stephan Leeds, Gene Nygaard, Alai, Oleg Alexandrov, Mwalcoff, Optichan, Waldir, FreplySpang, Rjwilmsi, Jake Wartenberg, Mathbot,Margosbot~enwiki, Fresheneesz, Kri, Gap, JPD, YurikBot, Wavelength, Kiscica, RussBot, DanMS, Kyorosuke, Pseudomonas, Nawl-inWiki, Dtrebbien, Syrthiss, DeadEyeArrow, Hirak 99, Chase me ladies, I'm the Cavalry, Arthur Rubin, AGToth, Bo Jacoby, Cmglee,Finell, SmackBot, RDBury, Dweller, Direvus, Hydrogen Iodide, Magic.crow, Diegotorquemada, Gilliam, Amatulic, Bluebot, ZackV, Sillyrabbit, CSWarren, Octahedron80, Nbarth, Berland, Rrburke, Gothmog.es, Savidan, G716, Mwtoews, Neutron65, Daniel.Cardenas, Lam-biam, Nishkid64, 16@r, Hypnosifl, Ft1~enwiki, Hu12, OnBeyondZebrax, Happy-melon, Wspencer11, Gco, AbsolutDan, CWY2190,Harej bot, Gogo Dodo, Tawkerbot4, Sirmylesnagopaleentheda, FrancoGG, Epbr123, Wikid77, Marek69, John254, QRS III, Majorly,Salgueiro~enwiki, JAnDbot, Belg4mit, VoABot II, JamesBWatson, PeterStJohn, Aqui, Johnbibby, Impdog, DerHexer, JaGa, ClubOranje,Infrangible, Ariel., Kostisl, CroydThoth, Userabc, J.delanoy, SharkD, Darkspots, Kfh227, Plf515, AntiSpamBot, Belovedfreak, Robroot,CardinalDan, Idioma-bot, TXiKiBoT, Anonymous Dissident, Anna Lincoln, Raymondwinn, Justinfr, Jeff.U, Hadleywickham, Jshtz4, In-sanity Incarnate, Dmcq, Monty845, SieBot, Luboogers25, Scottqwerty123, Kjtobo, Melcombe, DEMcAdams, Bigvic318, Martarius,ClueBot, The Thing That Should Not Be, Smithpith, AirdishStraus, Amirab, Seanwal111111, DragonBot, Arjayay, Gizurr, Wildscop,Hans Adler, Al-Andalusi, Qwfp, Herrsheng, Stickee, WikHead, Frood, Thatguyflint, Averagejef, Averagejefpet, Tayste, Addbot, Hg-berman, Fgnievinski, SunDragon34, Chamal N, Glane23, Nickylame2, Ozob, West.andrew.g, Dingo1729, Ehrenkater, VASANTH S.N.,Tide rolls, Kiril Simeonovski, Gail, Yobot, Moosetraxx, 2D, Aphillabaum, Mmxx, Amicon, AnomieBOT, Rubinbot, Jim1138, Htim,Giants27, Carlsotr, Xqbot, Molotron, Jffootball133, Jonas AGX, VladimirReshetnikov, Dougofborg, RTFVerterra, Sławomir Biały, Ci-tation bot 1, Pinethicket, RedBot, Thinking of England, Mohsinvyr, عقیل ,کاشف Bk1 168, Heavyweight Gamer, Amal Hatun, Duo-duoduo, Lammidhania, DARTH SIDIOUS 2, RjwilmsiBot, Rjohnson92, Angsteh, EmausBot, WikitanvirBot, Srathahtars, Tommy2010,Appleptic, ZéroBot, NGPriest, TyA, Chewings72, Christopher.ursich, DASHBotAV, Rocketrod1960, Petrb, ClueBot NG, Jack Green-maven, ChristophE, Hindustanilanguage, Widr, Bogdan.bobolea, Helpful Pixie Bot, KLBot2, BG19bot, Thumani Mabwe, Mark Arsten,Op47, EdwardRussell, Loriendrew, BattyBot, Justincheng12345-bot, Riley Huntley, தமிழககுரிசில, TwoTwoHello, Lugia2453, Red-eyed demon, I am One of Many, François Robere, Nickknack00, Dr.alaagad, James Rupert Swift, Param Mudgal, Xkatex20, Michaelalc,Tegangwer, Dianauvic, Emimaclminion and Anonymous: 306
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• Crank of a partition Source: https://en.wikipedia.org/wiki/Crank_of_a_partition?oldid=673926785Contributors: Michael Hardy, R.e.b.,Krishnachandranvn, Yobot, Googolplexideas, Skyerise, Monkbot and Anonymous: 9
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124 CHAPTER 26. VON MANGOLDT FUNCTION
• Dirichlet convolution Source: https://en.wikipedia.org/wiki/Dirichlet_convolution?oldid=660192812 Contributors: AxelBoldt, MichaelHardy, Modster, TakuyaMurata, Pizza Puzzle, Dysprosia, Gandalf61, Tobias Bergemann, Giftlite, MisterSheik, Burn, Linas, Chobot,Yahya Abdal-Aziz, SmackBot, Melchoir, Khazar, Headbomb, RobHar, WinBot, Vanish2, Policron, Nousernamesleft, GirasoleDE, Virginia-American, Addbot, Petermann~enwiki, ב ,.דניאל Luckas-bot, Yobot, TaBOT-zerem, Citation bot, Ayda D, Xqbot, FrescoBot, SławomirBiały, Matsgranvik, WikitanvirBot, EleferenBot, R. J. Mathar, Quondum, Quandle, Kasirbot, Rezabot, Helpful Pixie Bot, Spectral se-quence, Mraptor0910, K9re11, Monkbot, BethNaught and Anonymous: 21
• Divisibility sequence Source: https://en.wikipedia.org/wiki/Divisibility_sequence?oldid=675006191Contributors: Michael Hardy, Bearcat,DemonThing, MFH, Rjwilmsi, JosephSilverman, Headbomb, Mpd1989, Yobot, RjwilmsiBot, R. J. Mathar, Helpful Pixie Bot, Solomon7968,Boodlepounce and Anonymous: 3
• Divisor summatory function Source: https://en.wikipedia.org/wiki/Divisor_summatory_function?oldid=674752226Contributors: MichaelHardy, Charles Matthews, Lowellian, Giftlite, Icairns, Linas, Chenxlee, Itinerant1, Arthur Rubin, Elagatis, Madmath789, Neil916, Rob-Har, Alphachimpbot, Vanish2, David Eppstein, AlleborgoBot, GirasoleDE, Tarsiks, SchreiberBike, Addbot, Uncia, Lightbot, Luckas-bot, Citation bot, Geregen2, Citation bot 1, DrilBot, Sapphorain, Snehalshekatkar, ClueBot NG, Helpful Pixie Bot, Spectral sequence,Monkbot, Teddyktchan and Anonymous: 10
• Extremal orders of an arithmetic function Source: https://en.wikipedia.org/wiki/Extremal_orders_of_an_arithmetic_function?oldid=583134141 Contributors: Michael Hardy, CRGreathouse, Uncia, Quebec99, Citation bot 1 and Helpful Pixie Bot
• Gauss circle problem Source: https://en.wikipedia.org/wiki/Gauss_circle_problem?oldid=659736091 Contributors: XJaM, MichaelHardy, Jitse Niesen, Giftlite, Chenxlee, Matt Deres, RDBury, CRGreathouse, Headbomb, David Eppstein, Gentlemath, Jsondow, Myst-Bot, Addbot, Uncia, ו ,.עוזי Luckas-bot, Cpryby, GrouchoBot, GoingBatty, NoisyJinx, KHamsun, R. J. Mathar, ChuispastonBot, Sonicy-outh86, ClueBot NG, Ansarimohammad and Anonymous: 9
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• List ofOEIS sequences Source: https://en.wikipedia.org/wiki/List_of_OEIS_sequences?oldid=670065675Contributors: Michael Hardy,Hyacinth, Gandalf61, Reddwarf2956, Wavelength, Pegship, Tamfang, Whpq, CRGreathouse, David Eppstein, Vi2, BenTels, Daniel5Ko,JohnBlackburne, Philip Trueman, Malcolmxl5, Rhubbarb, Jsondow, Yobot, Novonium, Toshio Yamaguchi, SoSivr and Anonymous: 9
• Mertens function Source: https://en.wikipedia.org/wiki/Mertens_function?oldid=620507825 Contributors: XJaM, Michael Hardy, Ge-offrey~enwiki, Frieda, Sabbut, PrimeFan, Giftlite, Bob Palin, Gene Ward Smith, Wellparp, Neilc, MuDavid, Bender235, EmilJ, Jum-buck, Oleg Alexandrov, Linas, GregorB, Dedalus, R.e.b., FlaBot, Itinerant1, Chobot, YurikBot, Redgolpe, LeonardoRob0t, SmackBot,RDBury, Unyoyega, Bluebot, PrimeHunter, Tsca.bot, TenPoundHammer, Nautical Phasmid, A. Pichler, CRGreathouse, Myasuda, Doc-tormatt, Karl-H, David Eppstein, DorganBot, GirasoleDE, Martarius, Justin W Smith, Jsondow, Virginia-American, Addbot, Luckas-bot,Yobot, Amirobot, Lucas Brown 42, Xqbot, Kstueve, JovanCormac, Mikewarbz, Khinchin’s constant, DixonDBot, Mod mmg, EmausBot,Slawekb, Shishir332, ZéroBot, TwoTwoHello, The new math, Thtffgjgh, Mutley1989, K9re11 and Anonymous: 29
• Möbius inversion formula Source: https://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula?oldid=655966483 Contributors:AxelBoldt, Andre Engels, XJaM, Chas zzz brown, Michael Hardy, Charles Matthews, Timwi, Hyacinth, Robbot, Wikibot, Giftlite,Mboverload, DefLog~enwiki, Ma'ame Michu, Rich Farmbrough, Guanabot, Nickj, EmilJ, Sjoerd visscher, 3mta3, Haham hanuka,Phils, Sl, Gene Nygaard, Oleg Alexandrov, Linas, Rjwilmsi, Bubba73, Ffaarr, Chobot, YurikBot, RDBury, Mitchan, Maksim-e~enwiki,Reedy, Nbarth, Mhym, Madmath789, RobHar, Somebody2014, AsgardBot, David Eppstein, Lantonov, DavidCBryant, VolkovBot, Alle-borgoBot, Jdgilbey, DumZiBoT, Virginia-American, Addbot, SpBot, Yobot, Xqbot, Undsoweiter, Suslindisambiguator, Ndroock1, Ao-matveev and Anonymous: 20
• Normal order of an arithmetic function Source: https://en.wikipedia.org/wiki/Normal_order_of_an_arithmetic_function?oldid=618874795Contributors: Michael Hardy, Rjwilmsi, CRGreathouse, ShelfSkewed, Vanish2, Uncia, Yobot, Citation bot 1, Helpful Pixie Bot, Deltahe-dron, Spectral sequence, Monkbot and Anonymous: 1
• Partition (number theory) Source: https://en.wikipedia.org/wiki/Partition_(number_theory)?oldid=666841125 Contributors: Tarquin,Nonenmac, Stevertigo, Michael Hardy, Pwlfong, Wshun, Eliasen, Kku, Ixfd64, TakuyaMurata, GTBacchus, Charles Matthews, Phys,McKay, Fredrik, Gandalf61, Merovingian, Henrygb, Robinh, Giftlite, Chinasaur, Jason Quinn, Macrakis, HorsePunchKid, Almit39,4pq1injbok, Sam Derbyshire, Bender235, El C, Burn, Oleg Alexandrov, Joriki, Simetrical, Linas, Thehebrewhammer, Rjwilmsi, Jive-cat, Jwmcleod, Fivemack, Maxal, Hillman, Michael Slone, Anomalocaris, Bruguiea, Tetracube, Redgolpe, Ilmari Karonen, SmackBot,RDBury, Rōnin, Timothy Clemans, Mhym, Loopology, Lambiam, Richard L. Peterson, Shoeofdeath, JRSpriggs, Timrem, Ylloh, CR-Greathouse, Thijs!bot, Wang ty87916, JoaquinFerrero, Hannes Eder, GromXXVII, Arch dude, David Eppstein, Miaers, Commons-Delinker, Lantonov, Krishnachandranvn, Daniel5Ko, Policron, Milogardner, Philip Trueman, TXiKiBoT, Mathman99, Anchor LinkBot, Rumping, Justin W Smith, EGetzler, Mild Bill Hiccup, FractalFusion, Niceguyedc, DragonBot, Kingvashy, Watchduck, Ben-der2k14, JNLII, Gciriani, Marc van Leeuwen, Jed 20012, Khunglongcon, Addbot, Zdaugherty, Download, Lightbot, ,אבינעם Blue-busy, Legobot, Yobot, Ptbotgourou, , Citation bot, Obersachsebot, Xqbot, Coretheapple, RibotBOT, FrescoBot, Robo37, Gerard-Schildberger, FoxBot, Xnn, The tree stump, EmausBot, Slawekb, R. J. Mathar, Suslindisambiguator, Vladimirdx, Ebehn, ClueBotNG, SeekingAnswers, DependableSkeleton, KiruJiwak, Mesoderm, Matt.mawson, Joel B. Lewis, Archimedes100, Partedit, BG19bot,Karun3kumar, Googleisdik, NereusAJ, Mpsimo, Alexjbest, Deltahedron, Spectral sequence, Haphaeu, Peter13542, Vesoto, Ssuben andAnonymous: 107
• Pillai’s arithmetical function Source: https://en.wikipedia.org/wiki/Pillai’s_arithmetical_function?oldid=589375309 Contributors:Michael Hardy, Gmelfi, Qwertyus and Sdipu
• Prime gap Source: https://en.wikipedia.org/wiki/Prime_gap?oldid=675865535 Contributors: XJaM, Michael Hardy, Charles Matthews,Jitse Niesen, Giftlite, Bender235, Crisófilax, EmilJ, Oleg Alexandrov, GregorB, Reddwarf2956, Graham87, Chenxlee, Rjwilmsi, Naraht,Maxal, YurikBot, SmackBot, PrimeHunter, Gutworth, Octahedron80, Timothy Clemans, Stephlar, Druseltal2005, Luokehao, Mad-math789, CRGreathouse, Mon4, Tstrobaugh, David Eppstein, Kope, VolkovBot, YohanN7, ClueBot, Terra Xin, Jsondow, Jwpat7, Ad-dbot, DOI bot, Lightbot, Luckas-bot, Hairhorn, DataWraith, Gap9551, Citation bot 1, Olivemountain, Goldenart, John of Reading,Quondum, Joel B. Lewis, Helpful Pixie Bot, BG19bot, Mark Arsten, Compfreak7, IluvatarBot, YFdyh-bot, Deltahedron, Spectral se-quence, JerGer, Gishwhes-assassin, Chumbjil000, There is a T101 in your kitchen, Ayeh19 and Anonymous: 60
26.11. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 125
• Prime-counting function Source: https://en.wikipedia.org/wiki/Prime-counting_function?oldid=671641722 Contributors: AxelBoldt,XJaM, Michael Hardy, Alodyne, Ixfd64, Eric119, Charles Matthews, Dcoetzee, Psychonaut, Gandalf61, Mattflaschen, Tobias Bergemann,Giftlite, Gene Ward Smith, Numerao, JamesHoadley, Doshell, Starblue, Alexf, Almit39, Ben-Arba, Paul August, PittBill, Crisófilax,EmilJ, Billymac00, Tos~enwiki, Oleg Alexandrov, MFH, GregorB, James Harris, Reddwarf2956, R.e.b., Bubba73, RobertG, Mathbot,Scythe33, Chobot, YurikBot, Dmharvey, Dantheox, Woscafrench, Jayamohan, Arthur Rubin, That Guy, From That Show!, SmackBot,KocjoBot~enwiki, Bluebot, JCSantos, PrimeHunter, Joerite, Rich.lewis, Mwboyer, Madmath789, CRGreathouse, Karl-H, Thijs!bot,AbcXyz, Salgueiro~enwiki, Magioladitis, EulerGamma, David Eppstein, Haseldon, Fjackson, Xenonice, Jimothy 46, Philip Trueman,Abc135246, AlleborgoBot, MathPerson, Portalian, Droog Andrey, Bender2k14, Jsondow, Katsushi, MystBot, Addbot, Xario, Lightbot,PV=nRT, Zorrobot, Yobot, Tchoř, Citation bot, Motomuku, Citation bot 1, Mikewarbz, Duoduoduo, Matsgranvik, Werner D. Sand,Faolin42, EleferenBot, Slawekb, Shishir332, Quondum, Maschen, Sapphorain, Wcherowi, Helpful Pixie Bot, Dr James Macnicol, D. B.Staple, CitationCleanerBot, Chmarkine, Qetuth, BattyBot, Cjripper, Deltahedron, DAJ NT, K9re11, Monkbot, Konstantinos Gaitanasand Anonymous: 54
• Rank of a partition Source: https://en.wikipedia.org/wiki/Rank_of_a_partition?oldid=629789765 Contributors: Michael Hardy, TobiasBergemann, Rjwilmsi, David Eppstein, Krishnachandranvn, Yobot, Cstanford.math, Joel B. Lewis, BG19bot, Anrnusna, Monkbot andAnonymous: 2
• VonMangoldt function Source: https://en.wikipedia.org/wiki/Von_Mangoldt_function?oldid=668323859Contributors: Michael Hardy,Gandalf61, Giftlite, Almit39, Bender235, EmilJ, Billymac00, PAR, Burn, Oleg Alexandrov, Linas, Josh Parris, YurikBot, Dantheox,LMSchmitt, Cícero, JHunterJ, CmdrObot, Mon4, RobHar, Strangerer, SieBot, DragonBot, Bender2k14, MystBot, Addbot, Luckas-bot,Yobot, Xqbot, GrouchoBot, Citation bot 1, Matsgranvik, Sapphorain, Quandle, ChrisGualtieri, Deltahedron, K9re11 and Anonymous:18
26.11.2 Images• File:Ambox_important.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b4/Ambox_important.svg License: Public do-
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126 CHAPTER 26. VON MANGOLDT FUNCTION
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