THE RIEMANN HYPOTHESIS
PETER MEIER & JORN STEUDING
The german mathematician Bernhard
Riemann only had a short life, nevertheless
he contributed challenging new ideas and
concepts to mathematics. His invention
of topological methods in complex analy-
sis and his foundation of Riemannian ge-
ometry made him one of the most influ-
ential mathematicians of his time. In ad-
dition he worked on differential geometry,
differential equations, and mathematical
physics. His one and only article [43] on
number theory, entitled ’On the number of
Figure 1: Bernhard Riemann, ∗ 1826/9/17,
Breselenz (Germany) - † 1866/7/20, Selasca
(Italy); professor of mathematics in Gottingen
(Germany).
primes under a given magnitude’ in Eng-
lish translation, is probably the most influ-
ential ever written in this field. This pa-
per from 1859 marks the very beginning of
analytic number theory where arithmetical
objects are studied by analytical means.∗
In this note Riemann used complex analyt-
ical methods in order to study the distri-
bution of the prime numbers. Besides, he
posed several conjectures on the so-called
Riemann zeta-function; some of these con-
jectures have been proved decades later by
new and powerful methods from function
theory. About another speculation Rie-
mann simply wrote:
”Certainly one would wish
for a stricter proof here; I
have meanwhile temporar-
ily put aside the search
for this after some fleeting
futile attempts, as it ap-
pears unnecessary for the
next objectives of my in-
vestigation.”
This statement is now known as the Rie-
mann hypothesis and it still is an open
question today. It was one of the 23 prob-
lems Hilbert posed at the International
Congress of Mathematicians in Paris in
1900 and it is one of the seven millennium
problems of the Clay Institute.
Prime numbers
A prime number is a positive integer
greater than 1 which has no proper divi-
sor. Since any integer has a unique prime
factorization, the primes are the multi-
plicative atoms of the integers. Already
Euclid had a proof for the infinitude of
prime numbers. Assuming p1, p2, . . . , pk
∗Here one could also mention Dirichlet’s work
on the finiteness of the class number from 1837.
1
2 PETER MEIER & JORN STEUDING
are prime, the number q := p1 · p2 · . . . ·pk + 1 is prime or it has a prime divisor
which must be different from all primes
p1, p2, . . . , pk (otherwise this factor would
also divide q − p1 · p2 · . . . · pk = 1). Thus,
given any finite list of primes, one can find
a new one, hence there are infinitely many
primes.
The sequence of prime numbers starts
with 2, 3, 5, 7, 11, 13, 17, . . .. This is easy to
verify and it is also not difficult to extend
this list by some primes. On the contrary,
we may ask whether 30 449 is prime or the
number 246 565 876 574 836 597 or
232 582 657 − 1 ?
Obviously, these questions become harder
to answer with increasing number of dig-
its.† Why to be interested in big primes?
In modern cryptosystems (as RSA for ex-
ample) big prime numbers are used to gen-
erate keys. These keys are often publically
known, however, the system is only safe
as long as no one is able to factor the
key, a big composite, usually, the prod-
uct of two big primes each of which hav-
ing more than one hundred digits. Al-
ready Gauss noticed that testing whether
a given large integer is prime as well as fac-
toring a given large integer into its prime
divisors are important problems in arith-
metic. Only recently, Agrawal, Kayal &
Saxena [1] found a deterministic polyno-
mial time primality test. It is widely ex-
pected that factoring large integers cannot
be done by a polynomial time algorithm
†For integers of the form Mp := 2p − 1 with
prime p, so-called Mersenne numbers, with the
Lucas-Lehmer test there is a pretty fast primal-
ity test known. The biggest known prime num-
ber is the Mersenne prime M32 582 657 found by
the GIMPS internet project in 2006. In the
course of writing it was announced that two big-
ger Mersenne primes were found, however, they
have not been independently verified before this
paper was submitted; more details can be found
at http://www.mersenne.org/.
(which would give an affirmative answer
to the open millennium problem to prove
or disprove NP 6= P). The fastest known
method is Pollard’s number field sieve (and
its extensions) which factors integers with
up to two hundred digits (see [34] for more
details).
There are a lot of interesting questions
one can ask about primes:
• Do the primes contain arithmetic
progressions of arbitrary length?
• Are there infinitely many twin
primes, that are pairs of primes of
the form p and p + 2?
• Can every even integer ≥ 4 be
written as a sum of two primes?
• Is there always a prime number in
between two consecutive squares?
Although these questions do not need any
deeper knowledge of mathematics in their
formulation, their solutions either are un-
known or are considered as deep results
in arithmetic. The last but one question
is known as Goldbach’s conjecture and we
know that there cannot be too many ex-
ceptions, but the ultimate answer seems
to be out of reach by present day meth-
ods. The same can be said about twin
primes in the second question although the
recent work of Goldston, Pintz & Yıldırım
[22] shed some new light on this deep prob-
lem. They have shown that small gaps be-
tween consecutive primes do exist. More
precisely: if pn denotes the nth prime (in
ascending order), then‡
(1) lim infn→∞
pn+1 − pn
log pn= 0.
They have further shown that if a deep but
reasonable distribution hypothesis is true
(the Elliott–Halberstam conjecture), then
there are infinitely many primes which
have distance less than or equal to 16
‡here and in the sequel log always denotes the
logarithm to base e = exp(1).
THE RIEMANN HYPOTHESIS 3
to the next prime. The first question
was recently solved by Green & Tao [23].
The proof of this celebrated theorem uses
methods from number theory, ergodic the-
ory, and harmonic analysis; it is expected
that these ideas will lead to further sig-
nificant results of comparable flavour and
depth. The last of the above listed ques-
tions is open and seems to be difficult to
answer although simply testing will con-
vince the sceptic reader.
It is a common phenomenon in number
theory, that one can ask simple questions
which are difficult to answer. We quote
from Erdos:
”Any fool can ask ques-
tions about primes, which
no wise man can answer.”
Nevertheless it may be rewarding for the
reader to exercise herself or himself in the
formulation of such questions. In the fol-
lowing, we want to consider the following
question:
• How are the prime numbers dis-
tributed among the integers?
Figure 2: This is Ulam’s spiral and not a
far galaxy: The first 40 000 numbers are listed
in a spiral. Prime numbers are coloured white
and composite numbers black.
Euler and the zeta-function
Prime numbers are very elementary ob-
jects, however, they are best understood
in the context of analysis. The Riemann
zeta-function is defined by
ζ(s) =
∞∑
n=1
1
ns.
It was Euler who made the first signifi-
cant discoveries in the first half of the eigh-
teenth century (long before Riemann as
nicely documented by Ayoub [3]). He was
studying this series as a function of a real
variable. Then it is easy to see that the se-
ries converges for s > 1 whereas for s = 1
one gets the divergent harmonic series. Be-
sides, Euler found an alternative expres-
sion which we call now Euler-product:
(2)
∞∑
n=1
1
ns=∏
p
(
1 − 1
ps
)−1
;
here the product on the right is taken over
all prime numbers p. This identity may
be regarded as an analytic version of the
unique prime factorization of integers, be-
cause rewriting each factor as an infinite
geometric series one gets equivalently
1 + 2−s + 3−s + (2 · 2)−s + 5−s+
+(2 · 3)−s + . . .
= (1 + 2−s + 2−2s + . . .) ××(1 + 3−s + 3−2s + . . .) ××(1 + 5−s + 5−2s + . . .) · . . . .
Following Euler we may use (2) to prove
the infinitude of prime numbers as follows.
If there would be only a finite number of
primes, the Euler product (2) would be fi-
nite and hence defined for s = 1. How-
ever, for this value the series on the left is
divergent and so there are infinitely many
primes. Euler’s analytic approach is su-
perior to Euclid’s classical proof since it
4 PETER MEIER & JORN STEUDING
allows to apply analytical methods in or-
der to get arithmetic information. For in-
stance, Euler found that the sum over the
reciprocals of the primes diverges, which
he wrote in the form
1
2+
1
3+
1
5+
1
7+ . . . = log log∞,
which is in modern notation§
∑
p≤x
1
p∼ log log x.
Euler was also interested in special val-
ues of the zeta-function. He discovered the
stunning formula
ζ(2) =
∞∑
n=1
1
n2=
π2
6,
and more generally
∞∑
n=1
1
n2k= (−1)k+1 (2π)2k
2(2k)!B2k,
where Bj denotes the jth Bernoulli num-
ber, defined by the power series expansion
z
exp(z) − 1=
∞∑
j=0
Bjzj
j!.
The first Bernoulli numbers are B0 = 1,
B1 = − 12 , B2 = 1
6 , B4 = − 130 and B6 =
142 . It is easily seen that Bj = 0 for all
odd j ≥ 3. One may have the idea that
Bernoulli numbers are small, however,
B50 =495057205241079648212477525
66,
and with Euler’s formula one can easily
show that |B2k| is unbounded as k → ∞.
This formula may also be used to study
the primes. By (2)
∏
p
(
1 − 1
p2
)−1
=π2
6.
§The notation f ∼ g means that the limit
limx→∞ f(x)/g(x) exists and is equal to one. The
first rigorous proof of Euler’s asymptotical for-
mula was given by Mertens in 1874 (cf. [41]).
Since π2 is irrational, the product cannot
be finite, giving a third proof that we never
run out of primes.
Not too much is known for the values
of the zeta-function at the positive odd in-
tegers. It was a sensation when Apery [2]
proved in 1978 that ζ(3) is irrational. Re-
cent work of Rivoal [45] shows that for any
ǫ > 0 the Q-vector space spanned by the
n + 1 numbers
1, ζ(3), ζ(5), . . . , ζ(2n − 1), ζ(2n + 1)
has dimension ≥ 1−ǫ1+log 2 log n whenever n
is sufficiently large. Zudilin [59] used these
ideas to prove that at least one of the four
numbers ζ(5), ζ(7), ζ(9), and ζ(11) is irra-
tional.
Riemann and the zeta-function
Riemann was the first to consider ζ(s)
as a function of a complex variable. If
the real part of s is greater than 1, the
series defining ζ(s) converges. Since Eu-
ler considerd ζ(s) as a real function, he
could not avoid the barrier at s = 1.¶ In
the complex domain in contrast, Riemann
could circumvent the singularity of ζ(s) at
s = 1 by analytic continuation. Besides,
Riemann found the functional equation
π− s2 Γ( s
2 )ζ(s) = π− 12(1−s)Γ(1
2 (1−s))ζ(1−s),
showing a point symmetry with respect to
s = 12 . He gave two different proofs of
this identity, one is by contour integration,
one relies on the functional equation of the
theta-function, resp. Poisson’s summa-
tion formula, and marks the beginning of
Hecke’s theory of modular forms and asso-
ciated Dirichlet series. Besides Riemann’s
functional equation we have ζ(s) = ζ(s),
and so it suffices to study the zeta-function
in the upper half-plane.
¶Although Euler was summing ζ(s) as a diver-
gent series for negative values of s.
THE RIEMANN HYPOTHESIS 5
Now we shall investigate the zeros of
the zeta-function. There are no zeros in
the half-plane Re s > 1 of absolute con-
vergence. This follows immediately from
the Euler product representation (2). The
Gamma-function Γ(z) has simple poles at
z = −m, m ∈ N0, and is analytic else-
where. Now, putting s = −2n for any pos-
itive integer n in the functional equation,
the right-hand side is finite and different
from zero. It thus follows that ζ(s) = 0
for s = −2n; these zeros are called trivial
zeros and it follows from (2) that ζ(s) has
no other zeros in the half-plane Re s < 0.
Hence, all other zeros have to lie inside the
so-called critical strip 0 ≤ Re s ≤ 1; these
zeros are said to be nontrivial. Riemann
conjectured that there are infinitely many
nontrivial zeros; more precisely, if N(T )
counts the nontrivial zeros with imaginary
part in between 0 and T , then
(3) N(T ) ∼ T
2πlog
T
2πe.
Although Riemann had an idea how to
prove this asymptotic formula, it took
more than thirty years before a rigorous
proof was given by von Mangoldt [38].
For the horizontal distribution of the zeros
Riemann claimed that they are all located
on the critical line Re s = 12 although after
some attempts to prove this statement he
put it aside for the time being. This open
statement is now known as the Riemann
hypothesis.
Hardy [26] was the first to show that
there are infinitely many zeros on the crit-
ical line. His reasoning is based on the
function
(4) Z(t) := π− it2
Γ(14 + i t
2 )
|Γ(14 + i t
2 )|ζ(12 + it).
This function takes real values for real t
and vansihes if and only if t is the ordinate
of a nontrivial zero of the zeta-function.
-15 -10 -5 5
-40
-20
20
40
Figure 3: Here one can see the qualitative
properties of the zeta-function. The colour red
relates to big values of |ζ(s)|, whereas small
values are coloured in blue; yellow is in be-
tween. One can see the first nontrivial zeros,
the pole at s = 1, and some of the trivial zeros.
Using Hardy’s Z-function it is easy to lo-
calize zeta zeros by the mean-value theo-
rem from real analysis. Hardy showed that
∫ 2T
T
|Z(t)| dt >
∫ 2T
T
Z(t) dt
for any sufficiently large T , which yields
his result. Refining ideas of Selberg, Levin-
son, and others, Conrey [12] proved that
more than forty percent of the zeros lie on
6 PETER MEIER & JORN STEUDING
-2 -1 1 2 3
-2
-1
1
2
-2 -1 1 2 3
-2
-1
1
2
Figure 4: This picture shows the values
ζ( 1
2+ it) (upper picture) and ζ( 5
8+ it) (lower
picture) each with 0 ≤ t ≤ 50. On the upper
picture one can localize some zeros, however,
there are no zeros on the bottom picture - con-
sistent with Riemann’s hypothesis.
the critical line and are simple. The un-
derlying mollyfier method is too difficult
to be explained here.
Primes vs. zeros
Why is it so important to know the lo-
cation of the zeros inside the critical strip?
By the work of Riemann it turned out
that there is a close connection between
the zeros of the zeta-function and the
prime numbers. Actually, Riemann found
a formula how to compute the number of
primes below a given magnitude in terms
of the zeros.
Recall the Euler product representation
for the zeta-function. In the previous sec-
tion we deduced important information on
the non-vanishing of ζ(s). The core of
Riemann’s idea is an alternative product
representation for the zeta-function where
20 40 60 80 100
-4
-2
2
4
5020 5040 5060 5080 5100
-6
-4
-2
2
4
6
Figure 5: The values of Z(t) for 0 ≤ t ≤ 100
(upper picture) and for 5000 ≤ t ≤ 5100
(lower picture). The number of zeros in-
creases with t → ∞ according to (3).
each nontrivial zero corresponds to a linear
factor. For polynomials such a factoriza-
tion follows immediately from the funda-
mental theorem of algebra. However, for a
function like ζ(s) with infinitely many ze-
ros such a representation is anything but
trivial. Riemann conjectured
12s(s − 1)π− s
2 Γ( s2 )ζ(s)(5)
= exp(A + Bs)∏
ρ
(1 − sρ) exp( s
ρ ),
where A and B are certain constants and
the product is taken over all nontrivial ze-
ros ρ. The expression on the left is half
of the functional equation multiplied with12s(s − 1) in order to get rid of the singu-
larities at s = 1 and the trivial zeros. This
formula was established by Hadamard [24]
in 1893; his general theory for zeros of
entire functions forms now an important
part of complex analysis. Comparing the
Hadamard product formula with the Euler
product (2) verifies another conjecture of
THE RIEMANN HYPOTHESIS 7
Riemann – the so-called explicit formula:
π(x) +
∞∑
n=2
1nπ(x
1n )
= li(x) −∑
ρ=β+iγ
γ>0
(
li(xρ) + li(x1−ρ))
+
∫ ∞
x
du
u(u2 − 1) log(u)− log(2),
being valid for every x ≥ 2 not equal to
a power of a prime (otherwise one has to
add 12k on the left if x = pk). Here π(x)
Out[19]=
20 40 60 80 100
5
10
15
20
25
Out[20]=
10 000 20 000 30 000 40 000 50 000
1000
2000
3000
4000
5000
Figure 6: The prime counting function π(x)
has jumps of 1 at every prime. From a dis-
tance, π(x) looks pretty smooth.
counts the numbers of primes p ≤ x and
li(x) is the logarithmic integral
li(x) =
∫ x
0
du
log(u),
which is of magnitude xlog(x) in first ap-
proximation. The explicit formula shows
an astonishing duality between two rather
different objects, primes and zeros. By
counting primes, Gauss had conjectured
that π(x) is asymptotically equal to li(x).
The explicit formula provides a much more
accurate approximation (see Figure 6 and
7). Note that there is no error term in-
volved – the formula is exact! From this
fact it already follows that there are infin-
itely many nontrivial zeros since the left
hand side jumps at prime powers.
1.´107 2.´107 3.´107 4.´107 5.´107
-600
-400
-200
200
400
600
Figure 7: The difference between Gauss’ ap-
proximation and π(x) is painted red. Although
only the first 10 summands of the infinite se-
ries and only 300 non-trivial zeros have been
taken into account, the explicit formula pro-
vides an excellent approximation of π(x). The
difference to π(x), painted blue, is for all x less
than or equal to 50 million never greater than
200.
The ideas and conjectures of Riemann
stimulated research in this direction. The
open conjectures from his path-breaking
paper have pushed the development of
complex analysis forwards. These ef-
forts were crowned by the proof of Gauss’
conjecture for the asymptotic behaviour
of the prime counting function, the so-
called prime number theorem, found by
Hadamard [25] and de la Vallee-Poussin
[55] (independently) in 1896:
(6) π(x) ∼ li(x).
Actually, they both prove an asymptotic
formula with a reminder term. The an-
alytic proof of the prime number theo-
rem follows from contour integration. The
main term arises from the pole of the zeta-
function at s = 1 (as in Euler’s analytic
proof of the infinitude of primes). For this
aim one integrates the logarithmic deriva-
tive ζ′/ζ(s) and each zero in the contour
8 PETER MEIER & JORN STEUDING
yields a residue. Hence, the error term in
the prime number theorem depends on the
location of the zeros of ζ(s). The details
are rather technical and we refer to [16].‖
The present best estimate is due to Vino-
gradov and Korobov (independently) who
proved in 1958
π(x) − li (x)
= O
(
x exp
(
−C(log x)
35
(log log x)15
))
.
In 1900 von Koch showed
ζ(s) 6= 0 for Re s > θ
⇐⇒π(x) − li(x) ≪ xθ+ǫ
(for any positive ǫ). If the Riemann hy-
pothesis is true, then the prime numbers
are distributed as uniformly as possible,
or, in the language of probability theory,
then the error term in the prime num-
ber theorem behaves like an unbiased ran-
dom walk. An error term less than x12
is impossible by the existence of infinitely
many zeros on the critical line. Little-
wood [36] proved that the error term os-
cillates in both directions to order at least
x12 (log x)−1 log log log x. However, there is
no θ < 1 known such that ζ(s) is non-
vanishing for Re s > θ.
There is also an elementary proof of
the prime number theorem, which might
be very surprising since the prime num-
ber theorem without reminder term (6) is
equivalent to the non-vanishing of ζ(s) on
the line Re s = 1. Here the attribute ’ele-
mentary’ indicates that only number the-
oretical means are used in the proof, no
analysis. Such a proof was found by Erdos
[20] and Selberg [49] in 1949 with slightly
‖It should be noted that Riemann worked with
log ζ(s) and not with ζ′/ζ(s). For historical de-
tails and further reading we refer to [41].
different arguments. For the priority dis-
pute between Erdos and Selberg we refer
to the paper [21] of Goldfeld.
Lets discuss a consequence of the prime
number theorem. For the nth prime num-
ber pn we have π(pn) = n. Therefore,
it follows from (6) that pn ∼ n logn as
n → ∞. Hence, the quantity
pn+1 − pn
log n
is on average equal to 1. The result (1)
of Goldston, Pintz & Yildirim shows that
small gaps between primes do exist. In
the other direction it was already in 1931
shown by Westzynthius [58] that
(7) lim supn→∞
pn+1 − pn
log pn= +∞
holds. For the difference
dn := pn+1 − pn
between consecutive primes Baker & Har-
man [5] obtained the estimate
dn ≪ p0.534n .
The Riemann hypothesis would imply
dn ≪ p12n log pn; however, this still does not
lead to the existence of a prime in between
two consecutive squares.
Attempts to prove Riemann’s
hypothesis
The Riemann hypothesis is one of the
most important problems in mathemat-
ics. There are hundreds of articles which
investigate its consequences. Why do
most mathematicians believe in the Rie-
mann hypothesis? Obviously, the regu-
larity in the distribution of zeros, and
thus in prime number distribution as well,
proposed by the Riemann hypothesis, is
the most aestethic of all possible scenar-
ios. In 1932, Siegel [51] published an
account of Riemann’s work on the zeta-
function found in Riemann’s private pa-
pers in the archive of the university library
THE RIEMANN HYPOTHESIS 9
in Gottingen. It became evident that be-
hind Riemann’s speculation there was ex-
tensive analysis and computation. Rie-
mann himself computed some zeros, the
one with smallest positive imaginary part
being
ρ =1
2+ i 14.34725 . . . .
Many computations were done to find a
counterexample to the Riemann hypoth-
esis. In 1986, Van de Lune, te Riele &
Winter [37] localized the first 1 500 000 001
zeros, all lying without exception on the
critical line; moreover they all are simple.∗
There are quite many equivalent for-
mulations of Riemann’s hypothesis known.
One of the easiest might be Riesz’s crite-
rion [44]: Riemann’s hypothesis is true if
and only if
∞∑
k=1
(−1)k+1xk
(k − 1)!ζ(2k)≪ x
14+ǫ.
Another completely elementary equivalent
for the truth of the Riemann hypothesis is
the system of inequalities
∑
d|n
d ≤ Hn+exp(Hn) log Hn for n ∈ N,
where Hn :=∑n
j=11j . This criterion was
found by Lagarias [32], building on former
work of Robin.
Weil [57] gave a far-reaching refinement
of the explicit formula: let h be an even
function which is holomorphic in the strip
|Im t| ≤ 12 + δ and satisfies h(t) = O((1 +
|t|)−2−δ) for some δ > 0, and let
g(u) =1
2π
∫ ∞
−∞
h(r) exp(−iur) dr,
∗In the meantime, the project ZetaGrid has
extended this verification of the Riemann hy-
pothesis to the first 100 billion zeros; see
http://www.zetagrid.net/.
then∑
γ
h(γ) = 2h( i2 ) − g(0) log π
+ 12π
∫ ∞
−∞
h(r)Γ′
Γ(14 + ir
2 ) dr
−2∞∑
n=1
Λ(n)√n
g(log n),
where Λ(n) is equal to log p if n = pk for
some prime p and a positive integer k, and
zero otherwise. Here a zero is written as
ρ = 12 + iγ with γ ∈ C; hence the Rie-
mann hypothesis is the assertion that all γ
are real. Based on this duality Weil gave
a criterion for the truth of the Riemann
hypothesis. Here we state the following
version due to Bombieri [9]: the Riemann
hypothesis is true if and only if∑
ρ
g(ρ)g(1 − ρ) > 0
for every complex-valued g(x) ∈ C∞0 (0,∞)
which is not identically 0, where
g(s) =
∫ ∞
0
g(x)xs−1 dx.
An old idea to solve the Riemann hy-
pothesis is due to Hilbert and Polya who
asked for a self-adjoint linear operator on
an appropriate Hilbert space having an
eigenvalue spectrum equal to the set of ze-
ros of the function ξ(t) := ξ(12 + it) de-
fined by the expression (5). Clearly, the
self-adjointness would force the zeros t to
be real, resp. the nontrivial zeros of ζ(s)
to lie on the vertical line Re s = 12 .
Alain Connes [11] found an approach
via noncommutative geometry in order to
reduce the problem to the existence of a
trace formula in a certain noncommuta-
tive space. This idea is related to an old
observation Selberg [50] discovered for an-
other type of zeta-function, so-called Sel-
berg zeta-functions associated to compact
Riemann surfaces. Selberg’s trace for-
mula relates geometrical information to
10 PETER MEIER & JORN STEUDING
the eigenvalue spectrum of the hyperbolic
Laplacian. As an immediate consequence,
Selberg zeta-functions satisfy the analogue
of Riemann’s hypothesis. However, these
approaches have not led to anything with
respect to the Riemann zeta-function.
We should mention that there are ana-
logues of the Riemann hypothesis for
other zeta- and L-functions† encoding
arithmetic information about multiplica-
tive structures, e.g., for Dirichlet L-
functions to Dirichlet characters (group
homomorphisms on the group of prime
residue classes), Dedekind and Hecke zeta-
functions built from prime ideals in alge-
braic number fields, or L-functions to cusp
forms. It is widely believed that a solu-
tion of the Riemann hypothesis should not
only work for the Riemann zeta-function
but, with slight modifications, should lead
to a better understanding of other zeta-
and L-functions as well. A nice overview
over all arithmetically relevant L-functions
provides the monograph [10]. By study-
ing generalizations of the Riemann zeta-
function it seems to be evident that an
Euler product representation is crucial for
a zero-distribution as predicted by Rie-
mann’s hypothesis (although the Euler
product (2) itself does not converge inside
the critical strip). Davenport & Heilbronn
[17] proved that Epstein zeta-functions to
binary quadratic forms have an infinitude
of zeros in the half-plane of absolute con-
vergence if the class number is greater than
one.
There is an analogue of the Riemann
hypothesis for curves and abelian varieties.
The concept of a zeta-function associated
with a nonsingular projective curve over a
finite field was introduced by Emil Artin,
†The name L-function refers to a Dirichlet se-
ries with multiplicative coefficients satisfying a
Riemann-type functional equation; there seems to
be no exact definition in the literature.
Hasse, and F.K. Schmidt in the first half
of the twentieth century. In 1934 Hasse
succeeded to prove the analogue of the
Riemann hypothesis for the (local) zeta-
function to elliptic curves. This generating
function shares indeed some patterns with
the Riemann zeta-function, for instance a
functional equation, however, it is essen-
tially a rational function. Important ex-
tensions of Hasse’s proof to zeta-functions
of general algebraic varieties over finite
fields were obtained by Weil and Deligne
in the 1940s and the 1970s, respectively.
Random Matrices
In 1973 Montgomery [39] came up with
an interesting conjecture how the nontriv-
ial zeros should be distributed on the crit-
ical line. Assuming the Riemann hypothe-
sis, he conjectured that the number of non-
trivial zeros 12 +iγ, 1
2 +iγ′ of ζ(s) satisfying
the inequalities
0 < α ≤ log T
2π(γ − γ′) ≤ β
is asymptotically equal to
N(T )
∫ β
α
(
1 −(
sin πu
πu
)2)
du
as T → ∞. This so-called pair correlation
conjecture plays a complementary role to
the Riemann hypothesis: vertical vs. hor-
izontal distribution of the nontrivial ze-
ros. There are plenty of important conse-
quences of this far reaching conjecture; for
instance, the pair correlation conjecture
implies that almost all zeros of the zeta-
function are simple. As noticed by the
physicist Dyson, the predicted pair corre-
lation matches to the one of the eigenan-
gles of certain random matrix ensembles;
here the corresponding asymptotics is a
theorem in random matrix theory, not only
a conjecture. By computations of Odlyzko
[42] it turned out that the pair correlation
and the nearest neighbour spacing for the
THE RIEMANN HYPOTHESIS 11
zeros of ζ(s) were amazingly close to those
for the Gaussian Unitary Ensemble. The
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 8: The upper figure depicts
Odlyzko’s pair correlation picture for 2 × 108
zeros of ζ(s) near the 1023rd zero. The lower
figure shows the difference between the his-
togram in the first graph and 1 − ( sin πt
πt)2. In
the interval displayed, the two agree to within
about 0.002 . . . .
so-called Montgomery-Odlyzko law claims
that these distributions are, statistically,
the same (see [13, 47] for details). Un-
fortunately, no one knows how to relate
these two rather different fields, the zeros
of ζ(s) on one side, and the eigenvalues of
random matrices on the other side. There
is more evidence for this hypothetical link,
however, the only proved cases of pair cor-
relation asymptotics are those of Katz &
Sarnak [30] for certain local zeta-functions.
Lets explain some details. Random ma-
trices were introduced to describe the en-
ergy levels in many particle systems in
mathematical physics. The unitary group
U(N) is the group of all N × N matri-
ces U with complex entries which satisfy
the condition UUt
= idN , where Ut
de-
notes the transpose of the complex con-
jugate of U and idN is the N ×N identity
matrix. Any U ∈ U(N) has eigenvalues
of the form exp(iθj) with real eigenangles
θj ∈ (−π, π] for 1 ≤ j ≤ N . Since U(N)
is a Lie-group, there exists a uniquely de-
termined, translation invariant probability
measure on U(N), namely the Haar mea-
sure which we denote by m. The Circular
Unitary Ensemble is the group U(N) with
its respective Haar measure. The charac-
teristic polynomial ZN (θ, U) of a unitary
matrix U ∈ U(N) is defined by
ZN (θ; U) = det (idN − U exp(−iθ))
=N∏
j=1
(1 − exp(i(θj − θ))) .
Keating & Snaith [31] showed that char-
acteristic polynomials of the Circular Uni-
tary Ensemble have a similar value distri-
bution as the Riemann zeta-function on
the critical line. They proved
limN→∞
m
{
U ∈ U(N) :logZN (θ; U)√
12 log N
∈ R}
=1
2π
∫∫
R
exp(
− 12 (x2 + y2)
)
dxdy,
where R is any rectangle in the complex
plane with edges parallel to the real and
the imaginary axis. For the zeta-function
there is an old (unpublished) result of Sel-
berg showing the same Gaussian normal
distribution:
limT→∞
1
Tµ
{
t ∈ [T, 2T ] :log ζ(1
2 + it)√
12 log log T
∈ R}
=1
2π
∫∫
R
exp(
− 12 (x2 + y2)
)
dxdy,
where the measure µ on the left-hand
side is the Lebesgue measure. This and
other analogies led Keating & Snaith to
12 PETER MEIER & JORN STEUDING
the idea that characteristic polynomials
of large random matrices can be used to
model the analytic behaviour of the Rie-
mann zeta-function on the critical line.
By the Riemann-von Mangoldt formula (3)
the average spacing of consecutive ordi-
nates γ ≍ T of zeros of ζ(12 + it) is 2π
log T .
Comparing with the average spacing of the
eigenangles θj of ZN (θ; U) on the unit cir-
cle, it makes sense to scale according to
N ∼ log T2π . Next we describe how the
random matrix model is used to make pre-
dictions for the zeta-function.
In zeta-function theory there is a long
standing conjecture that for k ≥ 0, there
exists a contant C(k) such that
(8)
1
T
∫ T
0
|ζ(12 + it)|2kdt ∼ C(k)(log T )k2
,
as T → ∞. These asymptotics are only
known in the trivial case k = 0, and the
cases k = 1 and k = 2 by classical re-
sults of Hardy & Littlewood [27] and Ing-
ham [28], respectively. Very little is known
for higher moments. By the work of Bal-
asubramanian & Ramachandra [6] a lower
bound of the expected size holds for any
arbitrary positive integer k
1
T
∫ T
0
|ζ(12 + it)|2kdt ≫ (log T )k2
,
where the implicit constant depends on k.
Recently, Soundararajan [52] has shown
under assumption of the Riemann hypoth-
esis that
1
T
∫ T
0
|ζ(12 + it)|2kdt ≪ (log T )k2+ǫ
for any positive real k and any positive ǫ;
here the implicit constant depends on k
and ǫ. On the contrary, Conrey & Gonek
[15] and Keating & Snaith [31] stated a
conjecture for the constant C(k) appearing
in (8); remarkably, their heuristics differ
one from another (see also the survey [13]).
To state this conjecture define
a(k) =∏
p
(
1 − 1
p2
)k2
×
×∞∑
m=0
(
Γ(m + k)
m!Γ(k)
)21
pm.
Here one has to take an appropriate limit
if k is an integer less than or equal to zero.
It is not difficult to compute a(1) = 1 and
a(2) = 6π2 ; however, further values are not
explicitly known. Then the constant C(k)
in (8) is conjectured to be given by
C(k) = a(k)G(k + 1)2
G(2k + 1),
where G(z) is Barnes’ double Gamma-
function, defined by
G(z + 1) = Γ(z)G(z), G(1) = 1.
The approach of Conrey & Gonek [15] is
of combinatorial nature. On the contrary,
Keating & Snaith [31] used the random
matrix analogue. In fact, they proved, for
fixed k > − 12 ,
EN1
2π
∫ 2π
0
|ZN (θ; U)|2kdθ
∼ G(k + 1)2
G(2k + 1)Nk2
,
where EN stands for the expectation with
respect to the corresponding Haar measure
on U(N). The factor on the right-hand
side was found to coincide with some data
from the Conrey & Gonek-approach. How-
ever, the standard random matrix model
cannot detect the arithmetic factor a(k)
since prime numbers do not occur in this
model. Consequently, the arithmetic infor-
mation a(k), appearing in the heuristics of
Conrey & Ghosh, has to be inserted in an
ad hoc way.
In the meantime quite many conjectures
were formulated based on the random ma-
trix model. The hope is that these random
matrix conjectures lead to a better un-
derstanding of the zeta-function and thus
THE RIEMANN HYPOTHESIS 13
maybe to a solution of the Riemann hy-
pothesis. Here we mention an applica-
tion of the random matrix model to an old
question about the spacing of the zeros.
Denote by γn the positive ordinates of the
nontrivial zeros of the zeta-function in as-
cending order, then, by (3), the quantity
(9) (γn+1 − γn)1
2πlog
γn
2π
is equal to 1 on average. Similarly to the
case of primes, formulae (1) and (7), we
may ask for deviations. Steuding & Steud-
ing [54] have proved that the limit superior
of the expression in (9), λ say, equals in-
finity provided that the Riemann hypoth-
esis is true as well as the moment con-
jecture (8) and a discrete analogue. The
only unconditional bound is due to Selberg
[48] who showed λ > 1; under assump-
tion of the Riemann hypothesis Mueller
[40] obtained λ > 1.9. The case of small
gaps between consecutive zeros seems to
be much harder (even under assumption
of deep conjectures), similar to the case of
their dual analogue, gaps between consec-
utive primes. The best result in this direc-
tion was found by Conrey et al. [14] where
it was shown under assumption of the Rie-
mann hypothesis that the limit inferior of
(9) is less than 0.5172.
Universality
Another approach to Riemann’s hy-
pothesis relies on a remarkable approx-
imation property of the Riemann zeta-
function. In 1975, Voronin [56] proved
that, roughly speaking, the zeta-function
can approximate any non-vanishing ana-
lytic function uniformly. More precisely:
Given 0 < r < 14 and a continuous non-
vanishing function f(s) on the disk |s| ≤ r,
which is analytic in the interior of the disk,
then for any positive ǫ, there exists a pos-
itive real number τ such that
(10) max|s|≤r
|ζ(s + 34 + iτ) − f(s)| < ǫ;
moreover, there exist quite many such ap-
proximating shifts τ : the set of τ ∈ [0, T ]
for which the preceeding inequality holds,
has positive lower density as T → ∞ (with
respect to the Lebesgue measure). Since
the class of target functions is extremely
large and their approximation can be re-
alized by a single function — ζ(s) —, this
theorem is called the universality theorem.
In particular, it follows that
• the set of values ζ(σ + it) on ver-
tical lines in the open right half
of the critical strip is dense in C
(however, this is unknown for the
critical line);
• the zeta-function does not satisfy
any algebraic differential equation
(hypertranscendence).
There are further applications of univer-
sality to zeta-function theory, and even in
theoretical physics. Bitar, Khuri & Ren [7]
applied Voronin’s theorem to Feynman’s
path integral in quantum physics. They
obtained a formula for the partition func-
tion as a discrete sum over paths with each
path labeled by an integer and given by
a zeta-function evaluated at a fixed set of
points in the critical strip. These points
are the image of the space-time lattice re-
sulting from a linear mapping.
It is impossible to approximate func-
tions f(s) in the sense of Voronin’s the-
orem if they have a zero inside the disk.
This follows from Rouche’s theorem. If an
approximation would be possible, i.e. (10)
holds, the function ζ(s+ 34 +iτ) would have
a zero too, and by the positive lower den-
sity of the set of approximating τ , the zeta-
function ζ(s) would have at least constant
times T many zeros with imaginary part
14 PETER MEIER & JORN STEUDING
in between 0 and T . However, this con-
tradicts classical density theorems which
state for the number N(α, T ) of nontrivial
zeros ρ = β+iγ with β > α and 0 < γ ≤ T
that
(11) limT→∞
N(α, T )
T= 0.
The latter formula indicates that zero is
indeed a very special value in the value-
distribution of the zeta-function. In fact,
it can be shown that a corresponding for-
mula is false if the zero-counting function
is replaced by the counting function of
any other complex number different from
zero. However, given any complex number
c, almost all of the roots of the equation
ζ(s) = c are clustered around the critical
line as was shown by Levinson [35].
This negative property, that ζ(s) can-
not approximate functions with zeros, has
a direct application to the Riemann hy-
pothesis. For this purpose we reformu-
late Voronin’s theorem slightly. Let K be
a compact subset of the right open half
of the critical strip such that its comple-
ment is connected. Suppose that f(s) is
a continuous non-vanishing function on Kwhich is analytic in the interior of K and
let ǫ be an arbitrary but fixed positive real
number. Then the Lebesgue measure of
τ ∈ [0, T ] with
maxs∈K
|ζ(s + iτ) − f(s)| < ǫ
has positive lower density as T → ∞.
This generalization of Voronin’s universal-
ity theorem is realized by Mergelyan’s cele-
brated approximation theorem (for the de-
tails see [53]). Now assume that ζ(s) is
non-vanishing on K, then by the univer-
sality property (10)
maxs∈K
|ζ(s + iτ) − ζ(s)| < ǫ.
On the contrary, if the latter inequality
holds for a set of τ with positive lower den-
sity, any hypothetical zero ρ = β + iγ in K
of ζ(s) would lead (by Rouche’s theorem)
to another zero of the zeta-function inside
K+ iτ := {s + iτ : s ∈ K}; since any such
zero would lie to the right of the critical
line and since there would be about con-
stant times T many such zeros, we obtain
a contradiction to classical density theo-
rems, e.g. (11). Hence we have shown that
the Riemann hypothesis is true if and only
if for any compact subset K of the strip12 < σ < 1 with connected complement and
for any ǫ > 0 the measure of the set of all
τ ∈ [0, T ] satisfying
maxs∈K
|ζ(s + iτ) − ζ(s)| < ǫ
has positive lower density with respect to
the Lebesgue measure. This equivalent
was first discovered by Harald Bohr‡ [8]
in 1922. For lack of Voronin’s universality
theorem Bohr was only able to prove this
equivalent for Dirichlet L-series but not for
ζ(s). In the case of Dirichlet L-series one
can use the almost periodicity of Dirchlet
series as substitute for the universal ap-
proximation property (different from the
ζ-defining series, Dirichlet L-series con-
verge inside the critical strip). Actually
Bohr has invented the concept of almost
periodic functions in order to prove the
Riemann hypothesis. In the 1980s Bagchi
[4] extended Bohr’s theorem to the zeta-
function by use of Voronin’s universality
theorem.
What if not?
One may also consider the possibility of
the Riemann hypothesis being false. Here
we refer to the interesting paper of Ivic [29]
for some reasons to doubt the Riemann
hypothesis. In particular, Lehmer’s phe-
nomenon should be mentioned here. The
function Z(t) (given by (4)) has a nega-
tive local maximum at t = 2.4757 . . . (see
‡the brother of the Nobel laureate in physics
Niels Bohr
THE RIEMANN HYPOTHESIS 15
Figure 5), and this is the only known neg-
ative local maximum in the range t ≥ 0;
a positive local minimum is not known.
The occurrence of a negative local maxi-
mum, besides the one at t = 2.4757 . . ., or
a positive local minimum of Z(t), would
disprove Riemann’s hypothesis. This fol-
lows from the fact that if the Riemann’s
hypothesis is true, the graph of the loga-
rithmic derivative Z′
Z (t) is monotonically
decreasing between the zeros of Z(t) for
t ≥ 1000. The proof of this proposition is
not difficult and can be found in the mono-
graph [19] of Edwards. The Riemann-
Siegel formula (discovered by Riemann,
rediscovered by Siegel [51] while study-
ing Riemann’s unpublished papers as al-
ready mentioned above) provides a very
good approximation of the zeta-function
on the critical line. In terms of Hardy’s
Z-function,
Z(t) = 2∑
n≤√
t/(2π)
cos(ϑ(t) − t log n)
n12
+
+O(
t−14
)
,
valid for t ≥ 1. The Riemann-Siegel
formula is the basis of all high preci-
sion computations of the zeta-function on
the critical line. Lehmer [33] detected
that the zeta-function occasionally has two
very close zeros on the critical line; for
instance the zeros at t = 7005.0629 . . .
and t = 7005.1006 . . .. So the graph of
Z(t) sometimes barely crosses the t-axis
(see Figure 4). In view of our observa-
tion relating the graph of Z′
Z (t) with Rie-
mann’s hypothesis from the previous sec-
tion, Z(t) has exactly one critical point be-
tween successive zeros for sufficiently large
t. Hence, Lehmer’s observation, in the lit-
erature called Lehmer’s phenomenon, is a
near-counterexample to the Riemann hy-
pothesis.
7005.06 7005.07 7005.08 7005.09 7005.10 7005.11
-0.006
-0.004
-0.002
0.002
0.004
Figure 9: Lehmer’s phenomenon.
We conclude with a quotation from Pe-
ter Sarnak (cf. [46]):
”If [the Riemann Hypoth-
esis is] not true, then
the world is a very dif-
ferent place. The whole
structure of integers and
prime numbers would be
very different to what we
could imagine. In a way,
it would be more inter-
esting if it were false,
but it would be a disas-
ter because we’ve built so
much round assuming its
truth.”
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Peter Meier & Jorn Steuding, De-
partment of Mathematics, Wurzburg
University, Am Hubland, 97 074 Wurzburg,
Germany, [email protected],