Riemann_Problem_Description.pdf

Problems of the Millennium: the Riemann Hypothesis

E. Bombieri

I. The problem. The Riemann zeta function is the function of the complex

variable s , deﬁned in the half-plane 1

( s ) > 1 by the absolutely convergent series

ζ ( s ):= ∞

n =1

n s ,

andinthe wholecomplexplane

byanalyticcontinuation. AsshownbyRiemann,

as a meromorphic function with only a simple pole at s =1,

with residue 1, and satisﬁes the functional equation

π −s/ 2 Γ( s

2 ) ζ ( s )= π − (1 −s ) / 2 Γ( 1

−

) ζ (1

−

s ) .

(1)

In an epoch-making memoir published in 1859, Riemann [Ri] obtained an ana-

lytic formula for the number of primes up to a preassigned limit. This formula is

expressed in terms of the zeros of the zeta function, namely the solutions ρ

∈ C

the equation ζ ( ρ )=0.

Inthis paper, Riemannintroduces the function ofthe complex variable t deﬁned

ξ ( t )= 1

1) π −s/ 2 Γ( s

2 ) ζ ( s )

with s = 2 + it , andshowsthat ξ ( t ) isanevenentirefunctionof t whosezeroshave

imaginarypart between

2 s ( s

−

i/ 2and i/ 2. He further states, sketching a proof, that in

the range between 0 and T the function ξ ( t ) has about ( T/ 2 π )log( T/ 2 π )

−

T/ 2 π

zeros. Riemann then continues: “Man ﬁndet nun in der That etwa so viel reelle

Wurzeln innerhalb dieserGrenzen, und esist sehrwahrscheinlich, dassalle Wurzeln

reell sind.”, which can be translated as “Indeed, one ﬁnds between those limits

about that many real zeros, and it is very likely that all zeros are real.”

The statement that all zeros of the function ξ ( t ) are real is the Riemann hy-

pothesis.

The function ζ ( s ) has zeros at the negative even integers

−

4 ,... and one

refers to them as the trivial zeros . The other zeros are the complex numbers 2 + iα

where α is a zero of ξ ( t ). Thus, in terms of the function ζ ( s ), we can state

−

2 ,

−

2 .

In the opinion of many mathematicians the Riemann hypothesis, and its exten-

sion to general classes of L -functions, is probably today the most important open

problem in pure mathematics.

II. History and signiﬁcance of the Riemann hypothesis. For references

pertaining to the early history of zeta functions and the theory of prime numbers,

we refer to Landau [La] and Edwards [Ed].

1 We denote by ( s )and ( s ) the real and imaginary part of the complex variable s . The use

of the variable s is already in Dirichlet’s famous work of 1837 on primes in arithmetic progression.

ζ ( s ) extends to

Riemann hypothesis. The nontrivial zeros of ζ ( s ) have real part equal to

E. BOMBIERI

The connection between prime numbers and the zeta function, by means of the

celebrated Euler product

ζ ( s )=

−

p −s ) − 1

( s ) > 1,appearsfortheﬁrsttime inEuler’sbook Introductio in Analysin

Inﬁnitorum , published in 1748. Euler also studied the values of ζ ( s ) at the even

positive and the negative integers, and he divined a functional equation, equivalent

to Riemann’s functional equation, for the closely related function (

−

1) n− 1 /n s

d t

log t .

In 1837, Dirichlet proved his famous theorem of the existence of inﬁnitely many

primesinanyarithmeticprogression qn + a with q and a positivecoprimeintegers.

On May 24, 1848, Tchebychev read at the Academy of St. Petersburg his ﬁrst

memoir on the distribution of prime numbers, later published in 1850. It contains

the ﬁrst study of the function π ( x ) by analytic methods. Tchebychev begins by

taking the logarithm of the Euler product, obtaining 3

−

log(1

−

p s )+log( s

−

1)=log ( s

−

1) ζ ( s ) ,

(2)

which is his starting point.

Next, he proves the integral formula

∞

Γ( s )

x ) e −x x s− 1 d x,

ζ ( s )

−

1 =

(

1 −

(3)

−

e x

−

1) ζ ( s ) has limit 1, and also has ﬁnite derivatives

of any order, as s tends to 1 from the right. He then observes that the derivatives

of any order of the left-hand side of (2) can be written as a fraction in which the

numerator is a polynomial in the derivatives of ( s

−

1) ζ ( s ), and the denominator

1) ζ ( s ), from which it follows that the left-hand side of

(2) has ﬁnite derivatives of any order, as s tends to 1 from the right. From this,

he is able to prove that if there is an asymptotic formula for π ( x )bymeansofa

−

ﬁnite sum a k x/ (log x ) k ,uptoanorder O ( x/ (log x ) N ), then a k =( k

−

1)! for

1. This is precisely the asymptotic expansion of the function Li( x ),

thus vindicating Gauss’s intuition.

A second paper by Tchebychev gave rigorous proofs of explicit upper and lower

bounds for π ( x ), of the correct order of magnitude. Here, he introduces the count-

ing functions

ϑ ( x )=

p≤x

−

log p, ψ ( x )= ϑ ( x )+ ϑ ( √ x )+ ϑ ( √ x )+ ...

2 The integral is a principal value in the sense of Cauchy.

3 Tchebychev uses 1 + ρ in place of our

s . We write his formulas in modern notation.

validfor

(see the interesting account of Euler’s work in Hardy’s book [Hard]).

The problem of the distribution of prime numbers received attention for the ﬁrst

time with Gauss and Legendre, at the end of the eighteenth century. Gauss, in a

letter to the astronomerHencke in 1849, stated that he had found in his early years

that the number π ( x )ofprimesupto x is well approximated by the function 2

Li( x )= x

out of which he deduces that ( s

is an integral power of ( s

k =1 ,...,N

PROBLEMS OF THE MILLENNIUM: THE RIEMANN HYPOTHESIS

and proves the identity 4

ψ ( x

n )=log[ x ]! .

n≤x

From this identity, he ﬁnally obtains numerical upper and lower bounds for ψ ( x ),

ϑ ( x )and π ( x ).

Popular variants of Tchebychev’s method, based on the integrality of suitable

ratios of factorials, originate much later and cannot be ascribed to Tchebychev.

Riemann’s memoir on π ( x ) is really astonishing for the novelty of ideas intro-

duced. He ﬁrst writes ζ ( s ) using the integral formula, valid for

( s ) > 1:

ζ ( s )= 1

Γ( s )

∞

e −x

e −x x s− 1 d x,

(4)

−

and then deforms the contour of integration in the complex plane, so as to obtain

a representation valid for any s . This gives the analytic continuation and the

functionalequationof ζ ( s ). Thenhegivesa secondproofofthefunctionalequation

in the symmetric form (1), introduces the function ξ ( t ) and states some of its

properties as a function of the complex variable t .

Riemann continues by writing the logarithm of the Euler product as an integral

transform, valid for

( s ) > 1:

s log ζ ( s )= ∞

( x ) x −s− 1 d x

(5)

3 π ( √ x )+ ....

ByFourierinversion,he isabletoexpress ( x ) asacomplexintegral,andcompute

it using the calculus of residues. The residues occur at the singularities of log ζ ( s )

at s =1 and at the zeros of ζ ( s ). Finally an inversion formula expressing π ( x )in

terms of ( x ) yields Riemann’s formula.

Thiswasaremarkableachievementwhichimmediatelyattractedmuchattention.

Even if Riemann’s initial line of attack may have been inﬂuenced by Tchebychev

(we ﬁnd several explicit references to Tchebychev in Riemann’s unpublished Nach-

lass 5 ) his great contribution was to see how the distribution of prime numbers is

determined by the complex zeros of the zeta function.

At ﬁrst sight, the Riemann hypothesis appears to be only a plausible interesting

propertyofthe specialfunction ζ ( s ), andRiemann himself seemstotakethat view.

He writes: “Hiervon ware allerdings ein strenger Beweis zu wunschen; ich habe

indess die Aufsuchung desselben nach einigen ﬂuchtigen vergeblichen Versuchen

vorlauﬁg bei Seite gelassen, da er fur den nachsten Zweck meiner Untersuchung

entbehrlich schien.”, which can be translated as “Without doubt it would be desir-

able to have a rigorous proof of this proposition; however I have left this research

aside for the time being after some quick unsuccessful attempts, because it appears

to be unnecessary for the immediate goal of my study.”

( x )= π ( x )+ 1

2 π ( √ x )+ 1

4 Here [ x ] denotes the integral part of x .

5 The Nachlass consists of Riemann’s unpublished notes and is preserved in the mathematical

library of the University of Gottingen. The part regarding the zeta function was analyzed in depth

by C.L. Siegel [Sie].

where

E. BOMBIERI

On the other hand, one should not draw from this comment the conclusion that

the Riemann hypothesis was for Riemann only a casual remark of minor interest.

The validity of the Riemann hypothesis is equivalent to saying that the deviation

of the number of primes from the mean Li( x )is

π ( x )=Li( x )+ O √ x log x ;

the error term cannot be improved by much, since it is known to oscillate in both

directions to order at least Li( √ x ) logloglog x (Littlewood). In view of Riemann’s

comments at the end of his memoir about the approximation of π ( x )byLi( x ), it

is quite likely that he saw how his hypothesis was central to the question of how

good an approximation to π ( x ) one may get from his formula.

The failure of the Riemann hypothesis would create havoc in the distribution of

prime numbers. This fact alone singles out the Riemann hypothesis as the main

open question of prime number theory.

TheRiemannhypothesishasbecomeacentralproblemofpuremathematics,and

notjustbecauseofitsfundamentalconsequencesforthelawofdistributionofprime

numbers. One reason is that the Riemann zeta function is not an isolated object,

ratheristheprototypeofageneralclassoffunctions, called L -functions ,associated

with algebraic (automorphic representations) or arithmetical objects (arithmetic

varieties); we shall refer to them as global L -functions . They are Dirichlet series

with a suitableEuler product, and areexpected to satisfyan appropriatefunctional

equation and a Riemann hypothesis. The factors of the Euler product may also

be considered as some kind of zeta functions of a local nature, which also should

satisfy an appropriate Riemann hypothesis (the so-called Ramanujan property).

The most important properties of the algebraic or arithmetical objects underlying

an L -function can or should be described in terms of the location of its zeros and

poles, and values at special points.

The consequences of a Riemann hypothesis for global L -functions are important

and varied. We mention here, to indicate the variety of situations to which it can

be applied, an extremely strong eﬀective form of Tchebotarev’s density theorem

for number ﬁelds, the non-trivial representability of 0 by a non-singular cubic form

in 5 or more variables (provided it satisﬁes the appropriate necessary congruence

conditions for solubility, Hooley), and Miller’s deterministic polynomial time pri-

mality test. On the other hand, many deep results in number theory which are

consequences of a general Riemann hypothesis can be shown to hold independently

of it, thus adding considerable weight to the validity of the conjecture.

It is outside the scope of this article even to outline the deﬁnition of global L -

functions, referring instead to Iwaniec and Sarnak [IS] for a survey of the expected

properties satisﬁed by them; it suLces here to say that the study of the analytic

properties of these functions presents extraordinary diLculties.

Already the analytic continuation of L -functions as meromorphic or entire func-

tions is known only in special cases. For example, the functional equation for the

L -function of anelliptic curveover

has been established directly, ﬁrst in the semistable case in the spectacular work

andfor its twistsby Dirichletcharactersis an

easy consequence of, and is equivalent to, the existence of a parametrization of the

curve by means of modular functions for a Hecke group Γ 0 ( N ); the real diLculty

lies in establishing this modularity. No one knows how to prove this functional

equation by analytic methods. However the modularity of elliptic curves over

PROBLEMS OF THE MILLENNIUM: THE RIEMANN HYPOTHESIS

of Wiles [Wi] and Taylor and Wiles [TW] leading to the solution of Fermat’s Last

Theorem, and then in the general case in a recent preprint by Breuil, Conrad,

Diamond and Taylor.

Not all L -functions are directly associated to arithmetic or geometric objects.

The simplest example of L -functions not of arithmetic/geometric nature are those

arising from Maass waveforms for a Riemann surface X uniformized by an arith-

metic subgroup Γ of PGL(2 ,

). They are pull-backs f ( z ), to the universal cov-

( z ) > 0of X , of simultaneous eigenfunctions for the action of the

hyperbolic Laplacian and of the Hecke operators on X .

The most important case is again the group Γ 0 ( N ). In this case one can intro-

duce a notion of primitive waveform, analogous to the notion of primitive Dirichlet

character, meaning that the waveform is not induced from another waveform for a

Γ 0 ( N )with N a proper divisor of N . For a primitive waveform, the action of

the Hecke operators T n is deﬁned for every n and the L -function can be deﬁned

as λ f ( n ) n −s where λ f ( n ) is the eigenvalue of T n acting on the waveform f ( z ).

Such an L -function has an Euler product and satisﬁes a functional equation anal-

ogous to that for ζ ( s ). It is also expected that it satisﬁes a Riemann hypothesis.

Not a single example of validity or failure of a Riemann hypothesis for an L -

function is known up to this date. The Riemann hypothesis for ζ ( s ) does not seem

to be any easier than for Dirichlet L -functions (except possibly for non-trivial

real zeros), leading to the view that its solution may require attacking much more

general problems, by means of entirely new ideas.

III. Evidence for the Riemann hypothesis. Notwithstanding some skepticism

voiced in the past, based perhaps more on the number of failed attempts to a

proof rather than on solid heuristics, it is fair to say that today there is quite a

bit of evidence in its favor. We have already emphasized that the general Riemann

hypothesis is consistent with our present knowledge of number theory. There is

also speciﬁc evidence of a more direct nature, which we shall now examine.

First, strong numerical evidence.

Interestingly enough, the ﬁrst numerical computation of the ﬁrst few zeros of

the zeta function already appears in Riemann’s Nachlass. A rigorousveriﬁcation of

the Riemann hypothesis in a given range can be done numerically as follows. The

number N ( T ) of zeros of ζ ( s ) in the rectangle

with vertices at

−

iT, 2

−

iT, 2+ iT,

−

1+ iT is given by Cauchy’s integral

N ( T ) − 1= 1

2 πi

∂R −

ζ ( s )d s,

1 in the left-hand side of

this formula is due to the simple pole of ζ ( s )at s =1). The zeta function and

its derivative can be computed to arbitrary high precision using the MacLaurin

summation formula or the Riemann-Siegel formula [Sie]; the quantity N ( T )

−

which is an integer, is then computed exactly by dividing by 2 πi the numerical

evaluation of the integral, and rounding oﬀ its real part to the nearest integer (this

is only of theoretical interest and much better methods are available in practice for

computing N ( T ) exactly). On the other hand, since ξ ( t ) is continuous and real

for real t , there will be a zero of odd order between any two points at which ξ ( t )

changes sign. By judiciously choosing sample points, one can detect sign changes

−

ering space

provided T is not the imaginary part of a zero (the

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