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Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.11, 2014
68
Explicit upper bound for the function of sum of divisors 𝛔(𝐧)
Dr. Saad A. Baddai, Khulood M. Hussein
Dept. Math ., Collere of Science for Women, Univ. of Baghdad
M-alsaedi 87 [email protected]
following theorem : where he proved the 𝑐^[6]: we developed the result proved by A.EviAbstract
then : 𝑛 ≥ 7, let :[6]Theorem
σ(n) < (2.59)n log log n,…………………..…..(1)
this proving required several Lemmas which are given to be :
,then: ≥ 39 f n:[6] iLemma1
log 𝑝𝑛 <7
5 log n
,then: ≥ 31 :[6] if nLemma2
∏ (1 +1
𝑝) <
28
15𝑝|𝑛 log log n
,then: ≠ 42, 𝑛 ≠ 210, n≥ 31 :[6] if nLemma3
∏𝑝
𝑝−1< 2.59 log log np|n .
We a dope, the procedure of Evi𝑐^by improving the lemmas and get the following results:
, then: ≥ 82If n:Lemma 2
log pn <59
43 log n
then: ≥ 31,If n ;Lemma3
∏ (1 +1
p)p|n <
236
129 log log n
, ≥ 82, n ≠ 42, n ≠ 210For n :Lemma4
∏p
p−1< 2.509038354 log log 𝑛p|n
Where our lemma give above leads us to gut the new upper bound of the function 𝜎(𝑛) which given in the
following theorem:
then: ≥ 82,If n Theorem1:
σ(n) < 2.509038354 𝑛 log log n
.𝜎(𝑛) , 𝑛 ≥ 1 : Explicit upper bound for the function of sum of divisor Key words
Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.11, 2014
69
: Introduction
In this paper, we discuss the upper bound of the multiplicative number theoretic function σ(n), where σ(n)
represent the sum of all positive divisors of n, n≥ 1.this function is written to be,
∑ 𝑑𝑑|𝑛 ,
In Hardy and whright [4] the bound given to be explicit which is
σ(n) = 0(n log log n),…………………………..(1)
Also, R.L. Duncan [3] showed that ,
𝜎(𝑛) < 𝑛
6 [7𝜔(𝑛) + 10]……………………..…(2)
Where n = p1a1 . p2
a2 … … … prar and ω(n) = r and this is an explicit bound interns of ω(n) . this bound can
not be determined for large n , in which ω(n) can not be counted.
The best explicit upper bound proved by Aleksander Ivi𝑐^[6]when he proved .
σ(n) < (2.59)n log log n,…………………..…..(3)
where n≥ 7.
Our work dealing with Aleksander Ivi𝑐^ results in (3) and we improve this bound by increasing n to be n≥
82,and proved the following result .
σ(n) < (2.5090383)n log log n,……………….(4)
, then 32. 5, represent the number of primes of n, if n=2.r ω(n) =Then= p1a1 ⋯ pr
ar.: Let n)(n 𝛚Definition
w(n)=3.
(x), π:the number of all primes less them or equal to x denoted by (x)𝛑Definition
from Definition ,we can write:
π(x) = ∑ 1
For if, x=8.5, π(8.5)=4,
x=2 , π(2)=1,
x=12 , π(12)=5.
i.e x
logx The number of primes not exceeding x is asymptotic [4]eorem1:Th
(x) ~x
log x
such that, c2and c1there exist a constants [6]Theorem2:
, c2x
log x< π(x) <
c1x
log x
We shall improve the bound given in [6]for the function σ(n).Evi𝐜, proved the following theorem.
Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.11, 2014
70
7, then ≥If n :[4]Theorem3
σ(n) < (2.59) n log log n .
ur improvment will required Some estimates given by Rosser and schcoefeld in[5].O 1, ≥If n[3]:Theorem4
then σ(n) < n
6 (7 w(n) + 10)
From J.Rosser and L.Schoefeld [5],we have the following estimates:
,then: ≥ 1If n: [3] Theorem5
σ(n)=0(log log n)
then: e3 2⁄ < x and x ≥ 67,If [5] :Theorem6
and:
x
log x − 12
< π(x)
π(x)<x
log x – 3
2
x ≥ 17, then:If :[5]Corollary 1
x
log x < π(x)
If x>1, then: :[5]Corollary 2
π (x) <(1.25556)x
log x
x>1, and B=0.261497 ………, then: If ][5 Theorem7:
Iog log x + B - 1
2log2 < ∑1
pp≤x
and for x≥280,then:
∑1
pp≤x < log log x + 𝐵 + 1
2log2x
1, then: If x>[5] :Corollary1
log log x < ∑1
pp≤x
≤ 280, then;and when xx>1,then: If[5] :orem8The
eγ(log x)(1 - 1
2log2x) < ∏
p
p−1 p≤x
and when x≤ 280, then:
∏p
p − 1< eγ (log x )(1 +
1
2log2x )
p≤x
1,then: If x> [5] Corollary1:
Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.11, 2014
71
∏p
p − 1 < eγ(log x) (1 +
1
log2x)
p≤x
Where γ=0.5772157 is Euler,s constant .
We need the following definition.
denoted the n th prime , where: pn,then ≥ 1Let n Definition:
n=1 , p1 = 2,
n=2 , p2 = 3,
n=3 , p3 = 5,
n= 4 , p4 = 7,
Now, we shall give an improvement of Evi𝐜, result by proving the following theorem.
, then ≥ 82: For n 1)Theorem(
σ(n) < 2.509038354 n log log
Now, in order to prove theorem(4),we need the following Lemmas:
then: ≥ 2, Let n[5] Lemma(1):
pn~ n Iog n
as n→ ∞. this Lemme,we can written in the from:
pn < (1+∈)n Iog n, … … . . (1)
for some ∈, ∀∈> 0 and from, we can write
log pn < (1+∈) n Iog n
where n≥ n0(∈). some fixed number n0.
Now, we shall give an improvement of Evi𝐜 , result by proving the following theorem.
, then: ≥ 82If n:Lemma 2
log pn <59
43 log n
: we shall the bound given the lemma (1) to prove lemma(2)by Proof
using the inequality given in by G.Rosser and L.Schoenfeld[5]which given in the
corollary (1)of theorem(6)
π(x) > x log x ⁄ for x > 17
with x=pn and π(x) = n, then:
Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.11, 2014
72
n > pn
logpn for n≥ 82
pn < 𝑛 log pn …………………………(1)
and so:
pn < n √pn3 for n≥ 82
pn2 3⁄
< 𝑛,
log pn2 3⁄
< log n,
2
3log pn < log n,
and this gives us that;
n log pn <3
2 n log n
there fore,
pn < n logpn <3
2 n log n,
n logpn <3
2 n log n ,
∴ pn < n3 2⁄ , for n≥ 82
3
2n log n ≤ n7 5⁄ , for n≥ 161
There fore,
pn < n7 5⁄ , for n ≥ 161
log pn <7
5 log n , for 82≤ n ≤ 161
∴ log pn <59
43log n , for n ≥ 82 ∎
then: ≥ 31,If n ;Lemma3
∏ (1 +1
p)
p|n
<236
129 log log n
, and B=0.261497……., we shall have that: x ≥ 286: for Proof
log x > Iog 286 → Iogx > 5.6559
Iog2 > 31.98920481
log2 > 2 × 31.98920481
Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.11, 2014
73
1
2log2x<
1
63.97840962= 0.015630 < 0.016
1
(2Iog2x) =
1
(2log2286)< 0.016
∑1
p < loglog 𝑥 + 𝐵 + 1 (2 log2x) <⁄p≤x log log x + log
4
3
∴ ∑1
p < loglog x + 0.261497 + 0.016p≤x
∴ ∑1
p < loglog x + 0.277497p≤x
∴ ∑1
p< 𝑙𝑜𝑔 (
4
3log x)p≤x
There fore,
log∏ (1 +1
p) = log [(1 + 1 p1⁄ ) … … … … … (1 + 1 pr⁄ )p|n ]
log∏ (1 +1
p) = p|n [ log (1 + 1 p1⁄ ) … … log(1 + 1 pr⁄ )]
= ∑ log(1 + 1 p⁄ )p|n
Also since, p≥ 2 then:
1
p≤
1
2 → 1 +
1
p≤
3
2
log (1 +1
p) ≤ log
3
2≤
1
p
∴ ∑ log (1 +1
p)
p|n
≤ ∑1
p≤ ∑
1
p < (log
4
3p≤pk
)log pk
p|n
By using Lemma(1) and pk ≥ 286 , where k=ω(n), then we have the following cases:
Case (1):
(i) If ω(n)> 62,
log ∏ (1 +1
p) < log [
4
3 log pk]p|n
< log [ 4
3 ×
59
43 log k]
< log [ 236
129 log k]
∴ log ∏ (1 +1
p) < log [
236
129 log k]p|n
∴ log ∏ (1 +1
p) < log [
236
129 log log n]p|n
(ii) when ω(n)≤ 61, and pn < 𝑛 log pn ,where n≥ 82, then by calculation, we get:
∏ (1 +1
p)p|n ≤ ∏ (1 +
1
p)p≤pn=10.22575004
Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.11, 2014
74
∴ ∏ (1 +1
p)p|n <10.22575004<
236
129 log log n
Case(2):
(i) If ω(n)>12 ,then:
∏ p > 7.42 × 1012p≤p12
,
(ii) If ω(n)≤ 11 then, by calculation, we get :
∏ (1 +1
p)p|n ≤ ∏ (1 +
1
p)p≤p11
< 6.542269709
∴ ∏ (1 +1
p)p|n < 6.542269709 <
236
129 log log n
Case(3) ;
If n >16000,and n≤ 16000,with ω(n)≤ 5 ,then by calculation we get:
∏ (1 +1
p) ≤
1152
385p|n
<236
129 log log n
Case(4);
If n>200, and n≤200,with ω(n)≤ 3,then by calculation we get:
∴ ∏ (1 +1
p) ≤
3
2∙
4
3∙
6
5p|n
≤12
5<
239
129log log n
,≥ 82, n ≠ 42, n ≠ 210For n :Lemma4
∏p
p − 1< 2.509038354 log log 𝑛
p|n
: take x>455, then: Proof
log x > log 455
log2x > log 455
log2x > 37.4580405
log−2x <1
37.4580405
< 0.026696537
1+log−2x < 1.026696537 ……………….(1)
since eγ < 1.7810727 and
∏p
p − 1 < eγ log x (1 + log−2x)
p≤ x
Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.11, 2014
75
∏p
p−1 < (p≤x 1.7810727 )(1.026696537) log x
<1.828621173 log x ,
Using Lemma (2) and k=ω(n),then if n> 82 with ω(n)<log n ,
∏ p
p − 1 ≤ ∏
p
p − 1< 1.828621173 log pk.
p≤pkp|n
≤ 1.828621173 × 59
43 log k ≤ 2.509038354 log k
∴ ∏p
p−1 ≤ 2.509038354 log k ≤ 2.509038354 log log np|n
∴ ∏p
p−1 ≤ 2.509038354 log log n p|n
Case(1):
By taking k=ω(n)≥ 82 and pk ≥ p82 = 421,then
∏p
p − 1 ≤ 2.509038354 log log n
p|n
So, if ω(n)≤ 82,then:
∏p
p−1 ≤ ∏
p
p−1 < p≤p82 p|n 10.84851192
∴ ∏p
p − 1 ≤ ∏
p
p − 1 <
p≤p82 p|n
10.84851192 < 2.509038354 log log n
: Case(2)
If n ≥ 5 × 1017,then since ∏ p > 1018p≤p17
, ω(n) ≤ 16 and n≤ 5. 1017 in this
case we get:
∴ ∏p
p − 1 ≤ ∏
p
p − 1 < 7.348238613
p≤p16 p|n
< 2.509038354 log log n
: Case(3)
If n≥ 108, then since ∏ p > 2 × 108p≤p8
, ω(n) ≤ 8 and n ≤ 108 in this case we get:
∏p
p − 1 ≤ ∏
p
p − 1< 5.8471318
p ≤ p8 p|n
∴ ∏p
p−1 ≤ ∏
p
p−1< 5.8471318 <p≤p8 p|n 2.5090383 log log n
: Case(4)
if n > 30000, then since ∏ p = 30030,p≤pk𝜔(𝑛) ≤ 5 and n≤ 30000 in this case we get:
Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.11, 2014
76
∏p
p − 1 ≤ 2 ∙
3
2∙
5
4∙
7
6∙
11
10=
77
16p|n
∴ ∏p
p−1 ≤
77
16p|n < 2.5090383 log log n
: Case(5)
if n< 300, 𝑡ℎ𝑒𝑛 𝜔(n) ≤ 3, except for n=210 =2∙3∙ 5 ∙7 and for n=210 then:
∏p
p−1 p|n < 2.5090383 log log n
which is not true, If ω(n)≤ 3,
∏p
p−1 ≤ 2 ∙
3
2∙
5
4=
15
4= 3.75p|n
∏p
p − 1 ≤
15
4= 3.75 <
p|n
2.5090383 log log n
then: ≥ 82,If n Theorem1:
σ(n) < 2.509038354 𝑛 log log n
is multiplicative function then: σ(n) ,then since p1a1 . p2
a2 ⋯ ⋯ prarn = If Proof:
σ(n) = ∏ (1 + p + ⋯ ⋯ pai) pi|n , (1 ≤ i ≤ r)
=∏ (pai+1−1
pi−1) pi|n
By lemma(4):
[ If n ≥ 82, n ≠ 42, n ≠ 210 then: ∏
p
p − 1p|n
< 2.509038354 log log n
]
σ(n) = ∏pi
ai+1
pi
=∏pai+1( 1−1 pi
ai+1⁄ )
pi−1
=∏pai.p(1−1 pi
ai+1⁄ )
pi−1
=n∏pi−pi
−ai
pi−1
To show:
p − p−a
p − 1≤
p
p − 1
Take L.H.S, than since:
1-1 pa+1 ≤ 1⁄ ,
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Vol.4, No.11, 2014
77
p(1 − 1 pa+1⁄ )
p − 1≤
p
p − 1
p − p−a
p − 1≤
p
p − 1
σ(n)=∏pi−pi
−ai
pi−1
σ(n) ≤ n ∏p
p−1p|n
σ(n) < (2.5090383)log log n. n
for n≥ 31, n ≠ 42, n ≠ 210. for n=42 and n=210 is not true in
[σ(n) < 2.5090383log log n. n ] ∎
Our result . declare for several values of 𝑛 ≥ 82 which shows the validity of our bound which given in the table
(1).
Table(1)
2.5090383 n log log n 𝜎(𝑛) n
305.1409562 126 82
309.4342252 84 83
313.7327872 129 84
318.0365693 108 85
322.3455001 132 86
326.6595105 120 87
330.9785327 135 88
335.3025008 90 89
339.6313504 138 90
343.9650188 112 91
348.3034445 141 92
352.6465679 128 93
356.9943303 144 94
361.3466748 120 95
365.7035454 147 96
370.0648877 98 97
.
.
References
[1] U.nnapurna, Inequalities for σ(n) and φ(n),Math.Magazine,(45) 1972,pp.
187-190.
Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.11, 2014
78
[2] E. Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math.
Zeitschrift,(74)1962,pp-80.
[3] R.L. Duncan, Some estimates for σ(n),Amer. Math. Monthly,(74) 1967,pp.
713-715.
[4] G.H. Hardy and E.M. Wright, An introduction to the theory of number, 4 thed. oxford univ. press,
London,1960.
[5] B.J. Rosser and L. Schoenfeld ,Approximate formulas for some function of prime numbers, Illinois Journal
of Math,(6) 1962,pp.64-94
[6] A. Ivic^ ,Two inequalities for the sum of divisor function .Univ .u Novom Sadu, zbornik Radova .prirod
mat .Fak.7(1977),17-22.
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