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, meier@mathematik.uni-wuerzburg.de,

steuding@mathematik.uni-wuerzburg.de