Riemann hypothesis

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Riemann hypothesis
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{{short description|Conjecture in mathematics linked to the distribution of prime numbers}}{{For|the musical term|Riemannian theory}}(File:RiemannCriticalLine.svg|thumb|The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(s) = 1/2. The first non-trivial zeros can be seen at Im(s) = ±14.135, ±21.022 and ±25.011.){{Millennium Problems}}In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part {{sfrac|2}}. Many consider it to be the most important unsolved problem in pure mathematics {{harv|Bombieri|2000}}. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by {{harvs|txt|first=Bernhard|last= Riemann|year=1859|author-link=Bernhard Riemann}}, after whom it is named. The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, comprise Hilbert's eighth problem in David Hilbert's list of 23 unsolved problems; it is also one of the Clay Mathematics Institute's Millennium Prize Problems. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields.The Riemann zeta function ζ(s) is a function whose argument s may be any complex number other than 1, and whose values are also complex. It has zeros at the negative even integers; that is, ζ(s) = 0 when s is one of −2, −4, −6, .... These are called its trivial zeros. However, the negative even integers are not the only values for which the zeta function is zero. The other ones are called non-trivial zeros. The Riemann hypothesis is concerned with the locations of these non-trivial zeros, and states that:{{bquote|The real part of every non-trivial zero of the Riemann zeta function is {{sfrac|2}}.}}Thus, if the hypothesis is correct, all the non-trivial zeros lie on the critical line consisting of the complex numbers {{nowrap|{{sfrac|2}} + i{{hsp}}t,}} where t is a real number and i is the imaginary unit.There are several nontechnical books on the Riemann hypothesis, such as {{harvtxt|Derbyshire|2003}}, {{harvtxt|Rockmore|2005}}, {{harvs|last=Sabbagh|year=2003a|year2=2003b}},{{harvtxt|du Sautoy|2003}}. The books {{harvtxt|Edwards|1974}}, {{harvtxt|Patterson|1988}}, {{harvtxt|Borwein|Choi|Rooney|Weirathmueller|2008}}, {{harvtxt|Mazur|Stein|2015}} and {{harvtxt|Broughan|2017}} give mathematical introductions, while{{harvtxt|Titchmarsh|1986}}, {{harvtxt|Ivić|1985}} and {{harvtxt|Karatsuba|Voronin|1992}} are advanced monographs.

Riemann zeta function

The Riemann zeta function is defined for complex s with real part greater than 1 by the absolutely convergent infinite series
zeta(s) = sum_{n=1}^infty frac{1}{n^s} = frac{1}{1^s} + frac{1}{2^s} + frac{1}{3^s} + cdots
Leonhard Euler already considered this series in the 1730s for real values of s, in conjunction with his solution to the Basel problem. He also proved that it equals the Euler product
zeta(s) = prod_{p text{ prime}} frac{1}{1-p^{-s}}= frac{1}{1-2^{-s}}cdotfrac{1}{1-3^{-s}}cdotfrac{1}{1-5^{-s}}cdotfrac{1}{1-7^{-s}} cdot frac{1}{1-11^{-s}} cdots
where the infinite product extends over all prime numbers p.Leonhard Euler. Variae observationes circa series infinitas. Commentarii academiae scientiarum Petropolitanae 9, 1744, pp. 160–188, Theorems 7 and 8. In Theorem 7 Euler proves the formula in the special case s=1, and in Theorem 8 he proves it more generally. In the first corollary to his Theorem 7 he notes that zeta(1)=loginfty, and makes use of this latter result in his Theorem 19, in order to show that the sum of the inverses of the prime numbers is logloginfty.The Riemann hypothesis discusses zeros outside the region of convergence of this series and Euler product. To make sense of the hypothesis, it is necessary to analytically continue the function to obtain a form that is valid for all complex s. This is permissible because the zeta function is meromorphic, so its analytic continuation is guaranteed to be unique and functional forms equivalent over their domains. One begins by showing that the zeta function and the Dirichlet eta function satisfy the relation
left(1-frac{2}{2^s}right)zeta(s) = eta(s) = sum_{n=1}^infty frac{(-1)^{n+1}}{n^s} = frac{1}{1^s} - frac{1}{2^s} + frac{1}{3^s} - cdots.
But the series on the right converges not just when the real part of s is greater than one, but more generally whenever s has positive real part. Thus, this alternative series extends the zeta function from {{nowrap|Re(s) > 1}} to the larger domain {{nowrap|Re(s) > 0}}, excluding the zeros s = 1 + 2pi in/ln(2) of 1-2/2^s (see Dirichlet eta function). The zeta function can be extended to these values too by taking limits, giving a finite value for all values of s with positive real part except for the simple pole at s = 1.In the strip {{nowrap|0 < Re(s) < 1}} (and everywhere else) the zeta function satisfies the functional equation
zeta(s) = 2^spi^{s-1} sinleft(frac{pi s}{2}right) Gamma(1-s) zeta(1-s).
One may then define ζ(s) for all remaining nonzero complex numbers s by applying this equation outside the strip, and letting ζ(s) equal the right-hand side of the equation whenever s has non-positive real part.If s is a negative even integer then ζ(s) = 0 because the factor sin(Ï€s/2) vanishes; these are the trivial zeros of the zeta function. (If s is a positive even integer this argument does not apply because the zeros of the sine function are cancelled by the poles of the gamma function as it takes negative integer arguments.) The value ζ(0) = âˆ’1/2 is not determined by the functional equation, but is the limiting value of ζ(s) as s approaches zero. The functional equation also implies that the zeta function has no zeros with negative real part other than the trivial zeros, so all non-trivial zeros lie in the critical strip where s has real part between 0 and 1.


"("…it is very probable that all roots are real. Of course one would wish for a rigorous proof here; I have for the time being, after some fleeting vain attempts, provisionally put aside the search for this, as it appears dispensable for the immediate objective of my investigation.")|source=Riemann's statement of the Riemann hypothesis, from {{harv|Riemann|1859}}. (He was discussing a version of the zeta function, modified so that its roots (zeros) are real rather than on the critical line.)}}Riemann's original motivation for studying the zeta function and its zeros was their occurrence in his explicit formula for the number of primes π(x) less than a given number x, which he published in his 1859 paper "On the Number of Primes Less Than a Given Magnitude". His formula was given in terms of the related function
begin{align}Pi(x) = {} & pi(x) +tfrac{1}{2}pi(x^{frac{1}{2}}) +tfrac{1}{3} pi(x^{frac{1}{3}}) +tfrac{1}{4}pi(x^{frac{1}{4}})
& {} +tfrac{1}{5}pi(x^{frac{1}{5}}) +tfrac{1}{6}pi(x^{frac{1}{6}}) +cdotsend{align}which counts the primes and prime powers up to x, counting a prime power pn as {{frac|1|n}} of a prime. The number of primes can be recovered from this function by using the Möbius inversion formula,
begin{align} pi(x) &= sum_{n=1}^{infty}frac{mu(n)}{n}Pi(x^{frac{1}{n}}) &= Pi(x) -frac{1}{2}Pi(x^{frac{1}{2}}) -frac{1}{3}Pi(x^{frac{1}{3}}) -frac{1}{5}Pi(x^{frac{1}{5}}) + frac{1}{6}Pi(x^{frac{1}{6}}) -cdots, end{align}
where μ is the Möbius function. Riemann's formula is then
begin{align}Pi_0(x) &= operatorname{Li}(x) - sum_rho operatorname{Li}(x^rho) -log(2) + int_x^inftyfrac{dt}{t(t^2-1)log(t)}end{align}
where the sum is over the nontrivial zeros of the zeta function and where Π0 is a slightly modified version of Π that replaces its value at its points of discontinuity by the average of its upper and lower limits:
Pi_0(x) = lim_{varepsilon to 0}frac{Pi(x-varepsilon) + Pi(x+varepsilon)}2.
The summation in Riemann's formula is not absolutely convergent, but may be evaluated by taking the zeros ρ in order of the absolute value of their imaginary part. The function Li occurring in the first term is the (unoffset) logarithmic integral function given by the Cauchy principal value of the divergent integral
operatorname{Li}(x) = int_0^xfrac{dt}{log(t)}.
The terms Li(xρ) involving the zeros of the zeta function need some care in their definition as Li has branch points at 0 and 1, and are defined (for x > 1) by analytic continuation in the complex variable ρ in the region Re(ρ) > 0, i.e. they should be considered as {{nowrap|Ei(ρ ln x)}}. The other terms also correspond to zeros: the dominant term Li(x) comes from the pole at s = 1, considered as a zero of multiplicity −1, and the remaining small terms come from the trivial zeros. For some graphs of the sums of the first few terms of this series see {{harvtxt|Riesel|Göhl|1970}} or {{harvtxt|Zagier|1977}}.This formula says that the zeros of the Riemann zeta function control the oscillations of primes around their "expected" positions. Riemann knew that the non-trivial zeros of the zeta function were symmetrically distributed about the line {{nowrap|s {{=}} 1/2 + it,}} and he knew that all of its non-trivial zeros must lie in the range {{nowrap|0 ≤ Re(s) ≤ 1.}} He checked that a few of the zeros lay on the critical line with real part 1/2 and suggested that they all do; this is the Riemann hypothesis.}}{{clear}}


The practical uses of the Riemann hypothesis include many propositions known true under the Riemann hypothesis, and some that can be shown to be equivalent to the Riemann hypothesis.

Distribution of prime numbers

Riemann's explicit formula for the number of primes less than a given number in terms of a sum over the zeros of the Riemann zeta function says that the magnitude of the oscillations of primes around their expected position is controlled by the real parts of the zeros of the zeta function. In particular the error term in the prime number theorem is closely related to the position of the zeros. For example, if β is the upper bound of the real parts of the zeros, then {{harv|Ingham|1932}}{{rp|Theorem 30, p.83}}, {{harv|Montgomery|Vaughan|2007}}{{rp|p. 430}}
pi(x) - operatorname{li}(x) = O left( x^{beta} log x right) .
It is already known that 1/2 ≤ β ≤ 1 {{harv|Ingham|1932}}.{{rp|p. 82}}Von Koch (1901) proved that the Riemann hypothesis implies the "best possible" bound for the error of the prime number theorem. A precise version of Koch's result, due to {{harvtxt|Schoenfeld|1976}}, says that the Riemann hypothesis implies
|pi(x) - operatorname{li}(x)| < frac{1}{8pi} sqrt{x} log(x), qquad text{for all } x ge 2657,
where π(x) is the prime-counting function, and log(x) is the natural logarithm of x.{{harvtxt|Schoenfeld|1976}} also showed that the Riemann hypothesis implies
|psi(x) - x| < frac{1}{8pi} sqrt{x} log^2(x), qquad text{for all } x ge 73.2,
where ψ(x) is Chebyshev's second function.{{harvtxt|Dudek|2014}} proved that the Riemann hypothesis implies there is a prime p satisfyingx - frac{4}{pi} sqrt{x} log x < p leq xfor all x geq 2. This is an explicit version of a theorem of Cramér.

Growth of arithmetic functions

The Riemann hypothesis implies strong bounds on the growth of many other arithmetic functions, in addition to the primes counting function above.One example involves the Möbius function μ. The statement that the equation
frac{1}{zeta(s)} = sum_{n=1}^infty frac{mu(n)}{n^s}
is valid for every s with real part greater than 1/2, with the sum on the right hand side converging, is equivalent to the Riemann hypothesis. From this we can also conclude that if the Mertens function is defined by
M(x) = sum_{n le x} mu(n)
then the claim that
M(x) = Oleft(x^{frac{1}{2}+varepsilon}right)
for every positive ε is equivalent to the Riemann hypothesis (J.E. Littlewood, 1912; see for instance: paragraph 14.25 in {{harvtxt|Titchmarsh|1986}}). (For the meaning of these symbols, see Big O notation.) The determinant of the order n Redheffer matrix is equal to M(n), so the Riemann hypothesis can also be stated as a condition on the growth of these determinants. The Riemann hypothesis puts a rather tight bound on the growth of M, since {{harvtxt|Odlyzko|te Riele|1985}} disproved the slightly stronger Mertens conjecture
|M(x)| le sqrt x.
The Riemann hypothesis is equivalent to many other conjectures about the rate of growth of other arithmetic functions aside from μ(n). A typical example is Robin's theorem {{harv|Robin|1984}}, which states that if σ(n) is the divisor function, given by
sigma(n) = sum_{dmid n} d
sigma(n) < e^gamma n log log n
for all n > 5040 if and only if the Riemann hypothesis is true, where γ is the Euler–Mascheroni constant.Another example was found by Jérôme Franel, and extended by Landau (see {{harvtxt|Franel|Landau|1924}}). The Riemann hypothesis is equivalent to several statements showing that the terms of the Farey sequence are fairly regular. One such equivalence is as follows: if Fn is the Farey sequence of order n, beginning with 1/n and up to 1/1, then the claim that for all ε > 0
sum_{i=1}^m|F_n(i) - tfrac{i}{m}| = O(n^{frac{1}{2}+epsilon})
is equivalent to the Riemann hypothesis. Here
m = sum_{i=1}^nphi(i)
is the number of terms in the Farey sequence of order n.For an example from group theory, if g(n) is Landau's function given by the maximal order of elements of the symmetric group Sn of degree n, then {{harvtxt|Massias|Nicolas|Robin|1988}} showed that the Riemann hypothesis is equivalent to the bound
log g(n) < sqrt{operatorname{Li}^{-1}(n)}
for all sufficiently large n.

Lindelöf hypothesis and growth of the zeta function

The Riemann hypothesis has various weaker consequences as well; one is the Lindelöf hypothesis on the rate of growth of the zeta function on the critical line, which says that, for any ε > 0,
zetaleft(frac{1}{2} + itright) = O(t^varepsilon),
as t → ∞.The Riemann hypothesis also implies quite sharp bounds for the growth rate of the zeta function in other regions of the critical strip. For example, it implies that
e^gammale limsup_{trightarrow +infty}frac{|zeta(1+it)|}{loglog t}le 2e^gamma frac{6}{pi^2}e^gammale limsup_{trightarrow +infty}frac{1/|zeta(1+it)|}{loglog t}le frac{12}{pi^2}e^gamma
so the growth rate of ζ(1+it) and its inverse would be known up to a factor of 2 {{harv|Titchmarsh|1986}}.

Large prime gap conjecture

The prime number theorem implies that on average, the gap between the prime p and its successor is log p. However, some gaps between primes may be much larger than the average. Cramér proved that, assuming the Riemann hypothesis, every gap is O({{radic|p}} log p). This is a case in which even the best bound that can be proved using the Riemann hypothesis is far weaker than what seems true: Cramér's conjecture implies that every gap is O((log p)2), which, while larger than the average gap, is far smaller than the bound implied by the Riemann hypothesis. Numerical evidence supports Cramér's conjecture {{harv|Nicely|1999}}.

Analytic criteria equivalent to the Riemann hypothesis

Many statements equivalent to the Riemann hypothesis have been found, though so far none of them have led to much progress in proving (or disproving) it. Some typical examples are as follows. (Others involve the divisor function σ(n).)The Riesz criterion was given by {{harvtxt|Riesz|1916}}, to the effect that the bound
-sum_{k=1}^infty frac{(-x)^k}{(k-1)! zeta(2k)}= Oleft(x^{frac{1}{4}+epsilon}right)
holds for all ε > 0 if and only if the Riemann hypothesis holds.{{harvtxt|Nyman|1950}} proved that the Riemann hypothesis is true if and only if the space of functions of the form
f(x) = sum_{nu=1}^nc_nurho left(frac{theta_nu}{x} right)
where ρ(z) is the fractional part of z, {{nowrap|0 ≤ θν ≤ 1}}, and
is dense in the Hilbert space L2(0,1) of square-integrable functions on the unit interval. {{harvtxt|Beurling|1955}} extended this by showing that the zeta function has no zeros with real part greater than 1/p if and only if this function space is dense in Lp(0,1){{harvtxt|Salem|1953}} showed that the Riemann hypothesis is true if and only if the integral equation
has no non-trivial bounded solutions phi for 1/22}(-1)^{(p+1)/2}x^p=+infty,
which says that in some sense primes 3 mod 4 are more common than primes 1 mod 4.
  • In 1923 Hardy and Littlewood showed that the generalized Riemann hypothesis implies a weak form of the Goldbach conjecture for odd numbers: that every sufficiently large odd number is the sum of three primes, though in 1937 Vinogradov gave an unconditional proof. In 1997 Deshouillers, Effinger, te Riele, and Zinoviev showed that the generalized Riemann hypothesis implies that every odd number greater than 5 is the sum of three primes.
  • In 1934, Chowla showed that the generalized Riemann hypothesis implies that the first prime in the arithmetic progression a mod m is at most Km2log(m)2 for some fixed constant K.
  • In 1967, Hooley showed that the generalized Riemann hypothesis implies Artin's conjecture on primitive roots.
  • In 1973, Weinberger showed that the generalized Riemann hypothesis implies that Euler's list of idoneal numbers is complete.
  • {{harvtxt|Weinberger|1973}} showed that the generalized Riemann hypothesis for the zeta functions of all algebraic number fields implies that any number field with class number 1 is either Euclidean or an imaginary quadratic number field of discriminant −19, −43, −67, or −163.
  • In 1976, G. Miller showed that the generalized Riemann hypothesis implies that one can test if a number is prime in polynomial time via the Miller test. In 2002, Manindra Agrawal, Neeraj Kayal and Nitin Saxena proved this result unconditionally using the AKS primality test.
  • {{harvtxt|Odlyzko|1990}} discussed how the generalized Riemann hypothesis can be used to give sharper estimates for discriminants and class numbers of number fields.
  • {{harvtxt|Ono|Soundararajan|1997}} showed that the generalized Riemann hypothesis implies that Ramanujan's integral quadratic form {{nowrap|x2 + y2 + 10z2}} represents all integers that it represents locally, with exactly 18 exceptions.

Excluded middle

Some consequences of the RH are also consequences of its negation, and are thus theorems. In their discussion of the Hecke, Deuring, Mordell, Heilbronn theorem, {{harv|Ireland|Rosen|1990|p=359}} sayThe method of proof here is truly amazing. If the generalized Riemann hypothesis is true, then the theorem is true. If the generalized Riemann hypothesis is false, then the theorem is true. Thus, the theorem is true!!     (punctuation in original)Care should be taken to understand what is meant by saying the generalized Riemann hypothesis is false: one should specify exactly which class of Dirichlet series has a counterexample.

Littlewood's theorem

This concerns the sign of the error in the prime number theorem.It has been computed that π(x) < Li(x) for all x ≤ 1023,{{Citation needed|date=November 2016}} and no value of x is known for which π(x) > Li(x). See this table.In 1914 Littlewood proved that there are arbitrarily large values of x for which
pi(x)>operatorname{Li}(x) +frac13frac{sqrt x}{log x}logloglog x,
and that there are also arbitrarily large values of x for which
pi(x) Cfrac{sqrt{|D|}}{log |D|}.
Theorem (Deuring; 1933). If the RH is false then h(D) > 1 if |D| is sufficiently large.Theorem (Mordell; 1934). If the RH is false then h(D) → ∞ as D → −∞.Theorem (Heilbronn; 1934). If the generalized RH is false for the L-function of some imaginary quadratic Dirichlet character then h(D) → ∞ as D → −∞.(In the work of Hecke and Heilbronn, the only L-functions that occur are those attached to imaginary quadratic characters, and it is only for those L-functions that GRH is true or GRH is false is intended; a failure of GRH for the L-function of a cubic Dirichlet character would, strictly speaking, mean GRH is false, but that was not the kind of failure of GRH that Heilbronn had in mind, so his assumption was more restricted than simply GRH is false.)In 1935, Carl Siegel later strengthened the result without using RH or GRH in any way.

Growth of Euler's totient

In 1983 J. L. Nicolas proved {{harv|Ribenboim|1996|p=320}} that
varphi(n) < e^{-gamma}frac {n} {log log n}
for infinitely many n, where φ(n) is Euler's totient function and γ is Euler's constant.Ribenboim remarks that:The method of proof is interesting, in that the inequality is shown first under the assumption that the Riemann hypothesis is true, secondly under the contrary assumption.

Generalizations and analogs

Dirichlet L-series and other number fields

The Riemann hypothesis can be generalized by replacing the Riemann zeta function by the formally similar, but much more general, global L-functions. In this broader setting, one expects the non-trivial zeros of the global L-functions to have real part 1/2. It is these conjectures, rather than the classical Riemann hypothesis only for the single Riemann zeta function, which account for the true importance of the Riemann hypothesis in mathematics.The generalized Riemann hypothesis extends the Riemann hypothesis to all Dirichlet L-functions. In particular it implies the conjecture that Siegel zeros (zeros of L-functions between 1/2 and 1) do not exist.The extended Riemann hypothesis extends the Riemann hypothesis to all Dedekind zeta functions of algebraic number fields. The extended Riemann hypothesis for abelian extension of the rationals is equivalent to the generalized Riemann hypothesis. The Riemann hypothesis can also be extended to the L-functions of Hecke characters of number fields.The grand Riemann hypothesis extends it to all automorphic zeta functions, such as Mellin transforms of Hecke eigenforms.

Function fields and zeta functions of varieties over finite fields

{{harvtxt|Artin|1924}} introduced global zeta functions of (quadratic) function fields and conjectured an analogue of the Riemann hypothesis for them, which has been proved by Hasse in the genus 1 case and by {{harvtxt|Weil|1948}} in general. For instance, the fact that the Gauss sum, of the quadratic character of a finite field of size q (with q odd), has absolute value sqrt{q} is actually an instance of the Riemann hypothesis in the function field setting. This led {{harvtxt|Weil|1949}} to conjecture a similar statement for all algebraic varieties; the resulting Weil conjectures were proved by {{harvs|txt|first=Pierre|last=Deligne|authorlink=Pierre Deligne|year1=1974|year2=1980}}.

Arithmetic zeta functions of arithmetic schemes and their L-factors

Arithmetic zeta functions generalise the Riemann and Dedekind zeta functions as well as the zeta functions of varieties over finite fields to every arithmetic scheme or a scheme of finite type over integers. The arithmetic zeta function of a regular connected equidimensional arithmetic scheme of Kronecker dimension n can be factorized into the product of appropriately defined L-factors and an auxiliary factor {{harvs|txt|first=Jean-Pierre|last=Serre|authorlink=J.-P. Serre|year1=1969–1970}}. Assuming a functional equation and meromorphic continuation, the generalized Riemann hypothesis for the L-factor states that its zeros inside the critical strip Re(s)in (0,n) lie on the central line. Correspondingly, the generalized Riemann hypothesis for the arithmetic zeta function of a regular connected equidimensional arithmetic scheme states that its zeros inside the critical strip lie on vertical lines Re(s)=1/2,3/2,dots,n-1/2 and its poles inside the critical strip lie on vertical lines Re(s)=1, 2, dots,n-1. This is known for schemes in positive characteristic and follows from {{harvs|txt|first=Pierre|last=Deligne|authorlink=Pierre Deligne|year1=1974|year2=1980}}, but remains entirely unknown in characteristic zero.

Selberg zeta functions

{{harvtxt|Selberg|1956}} introduced the Selberg zeta function of a Riemann surface. These are similar to the Riemann zeta function: they have a functional equation, and an infinite product similar to the Euler product but taken over closed geodesics rather than primes. The Selberg trace formula is the analogue for these functions of the explicit formulas in prime number theory. Selberg proved that the Selberg zeta functions satisfy the analogue of the Riemann hypothesis, with the imaginary parts of their zeros related to the eigenvalues of the Laplacian operator of the Riemann surface.

Ihara zeta functions

The Ihara zeta function of a finite graph is an analogue of the Selberg zeta function, which was first introduced by Yasutaka Ihara in the context of discrete subgroups of the two-by-two p-adic special linear group. A regular finite graph is a Ramanujan graph, a mathematical model of efficient communication networks, if and only if its Ihara zeta function satisfies the analogue of the Riemann hypothesis as was pointed out by T. Sunada.

Montgomery's pair correlation conjecture

{{harvtxt|Montgomery|1973}} suggested the pair correlation conjecture that the correlation functions of the (suitably normalized) zeros of the zeta function should be the same as those of the eigenvalues of a random hermitian matrix. {{harvtxt|Odlyzko|1987}} showed that this is supported by large scale numerical calculations of these correlation functions.Montgomery showed that (assuming the Riemann hypothesis) at least 2/3 of all zeros are simple, and a related conjecture is that all zeros of the zeta function are simple (or more generally have no non-trivial integer linear relations between their imaginary parts). Dedekind zeta functions of algebraic number fields, which generalize the Riemann zeta function, often do have multiple complex zeros {{harv|Radziejewski|2007}}. This is because the Dedekind zeta functions factorize as a product of powers of Artin L-functions, so zeros of Artin L-functions sometimes give rise to multiple zeros of Dedekind zeta functions. Other examples of zeta functions with multiple zeros are the L-functions of some elliptic curves: these can have multiple zeros at the real point of their critical line; the Birch-Swinnerton-Dyer conjecture predicts that the multiplicity of this zero is the rank of the elliptic curve.

Other zeta functions

There are many other examples of zeta functions with analogues of the Riemann hypothesis, some of which have been proved. Goss zeta functions of function fields have a Riemann hypothesis, proved by {{harvtxt|Sheats|1998}}. The main conjecture of Iwasawa theory, proved by Barry Mazur and Andrew Wiles for cyclotomic fields, and Wiles for totally real fields, identifies the zeros of a p-adic L-function with the eigenvalues of an operator, so can be thought of as an analogue of the Hilbert–Pólya conjecture for p-adic L-functions {{harv|Wiles|2000}}.

Attempted proofs

Several mathematicians have addressed the Riemann hypothesis, but none of their attempts have yet been accepted as a correct solution. {{harvtxt|Watkins|2007}} lists some incorrect solutions, and more are frequently announced.

Operator theory

Hilbert and Pólya suggested that one way to derive the Riemann hypothesis would be to find a self-adjoint operator, from the existence of which the statement on the real parts of the zeros of ζ(s) would follow when one applies the criterion on real eigenvalues. Some support for this idea comes from several analogues of the Riemann zeta functions whose zeros correspond to eigenvalues of some operator: the zeros of a zeta function of a variety over a finite field correspond to eigenvalues of a Frobenius element on an étale cohomology group, the zeros of a Selberg zeta function are eigenvalues of a Laplacian operator of a Riemann surface, and the zeros of a p-adic zeta function correspond to eigenvectors of a Galois action on ideal class groups.{{harvtxt|Odlyzko|1987}} showed that the distribution of the zeros of the Riemann zeta function shares some statistical properties with the eigenvalues of random matrices drawn from the Gaussian unitary ensemble. This gives some support to the Hilbert–Pólya conjecture.In 1999, Michael Berry and Jonathan Keating conjectured that there is some unknown quantization hat H of the classical Hamiltonian H = xp so that
zeta (1/2+ihat H) = 0
and even more strongly, that the Riemann zeros coincide with the spectrum of the operator 1/2 + i hat H. This is in contrast to canonical quantization, which leads to the Heisenberg uncertainty principle [x,p]=1/2 and the natural numbers as spectrum of the quantum harmonic oscillator. The crucial point is that the Hamiltonian should be a self-adjoint operator so that the quantization would be a realization of the Hilbert–Pólya program. In a connection with this quantum mechanical problem Berry and Connes had proposed that the inverse of the potential of the Hamiltonian is connected to the half-derivative of the function
N(s)= frac{1}{pi}operatorname{Arg}xi(1/2+isqrt s)
then, in Berry–Connes approach
V^{-1}(x) = sqrt{4pi} frac{d^{1/2}N(x)}{dx^{1/2}}
{{harv|Connes|1999}}. This yields a Hamiltonian whose eigenvalues are the square of the imaginary part of the Riemann zeros, and also that the functional determinant of this Hamiltonian operator is just the Riemann Xi function. In fact the Riemann Xi function would be proportional to the functional determinant (Hadamard product)
as proved by Connes and others, in this approach
The analogy with the Riemann hypothesis over finite fields suggests that the Hilbert space containing eigenvectors corresponding to the zeros might be some sort of first cohomology group of the spectrum Spec (Z) of the integers. {{harvtxt|Deninger|1998}} described some of the attempts to find such a cohomology theory {{harv|Leichtnam|2005}}.{{harvtxt|Zagier|1981}} constructed a natural space of invariant functions on the upper half plane that has eigenvalues under the Laplacian operator that correspond to zeros of the Riemann zeta function—and remarked that in the unlikely event that one could show the existence of a suitable positive definite inner product on this space, the Riemann hypothesis would follow. {{harvtxt|Cartier|1982}} discussed a related example, where due to a bizarre bug a computer program listed zeros of the Riemann zeta function as eigenvalues of the same Laplacian operator.{{harvtxt|Schumayer|Hutchinson|2011}} surveyed some of the attempts to construct a suitable physical model related to the Riemann zeta function.

Lee–Yang theorem

The Lee–Yang theorem states that the zeros of certain partition functions in statistical mechanics all lie on a "critical line" with their real part equals to 0, and this has led to some speculation about a relationship with the Riemann hypothesis {{harv|Knauf|1999}}.

Turán's result

{{harvs|authorlink=Pál Turán|first=Pál |last=Turán|year=1948|txt}} showed that if the functions
sum_{n=1}^N n^{-s}
have no zeros when the real part of s is greater than one then
T(x) = sum_{nle x}frac{lambda(n)}{n}ge 0text{ for } x > 0,
where λ(n) is the Liouville function given by (−1)r if n has r prime factors. He showed that this in turn would imply that the Riemann hypothesis is true. However {{harvtxt|Haselgrove|1958}} proved that T(x) is negative for infinitely many x (and also disproved the closely related Pólya conjecture), and {{harvtxt|Borwein|Ferguson|Mossinghoff|2008}} showed that the smallest such x is {{gaps|72|185|376|951|205}}. {{harvtxt|Spira|1968}} showed by numerical calculation that the finite Dirichlet series above for N=19 has a zero with real part greater than 1. Turán also showed that a somewhat weaker assumption, the nonexistence of zeros with real part greater than 1+N−1/2+ε for large N in the finite Dirichlet series above, would also imply the Riemann hypothesis, but {{harvtxt|Montgomery|1983}} showed that for all sufficiently large N these series have zeros with real part greater than {{nowrap|1 + (log log N)/(4 log N)}}. Therefore, Turán's result is vacuously true and cannot be used to help prove the Riemann hypothesis.

Noncommutative geometry

{{harvs|last=Connes|authorlink=Alain Connes|year1=1999|year2=2000|txt}} has described a relationship between the Riemann hypothesis and noncommutative geometry, and shows that a suitable analog of the Selberg trace formula for the action of the idèle class group on the adèle class space would imply the Riemann hypothesis. Some of these ideas are elaborated in {{harvtxt|Lapidus|2008}}.

Hilbert spaces of entire functions

{{harvs|txt||first =Louis|last=de Branges |year=1992 |authorlink=Louis de Branges}} showed that the Riemann hypothesis would follow from a positivity condition on a certain Hilbert space of entire functions.However {{harvtxt|Conrey|Li|2000}} showed that the necessary positivity conditions are not satisfied. Despite this obstacle, de Branges has continued to work on an attempted proof of the Riemann hypothesis along the same lines, but this has not been widely accepted by other mathematicians {{harv|Sarnak|2005}}.


The Riemann hypothesis implies that the zeros of the zeta function form a quasicrystal, meaning a distribution with discrete support whose Fourier transform also has discrete support.{{harvtxt|Dyson|2009}} suggested trying to prove the Riemann hypothesis by classifying, or at least studying, 1-dimensional quasicrystals.

Arithmetic zeta functions of models of elliptic curves over number fields

When one goes from geometric dimension one, e.g. an algebraic number field, to geometric dimension two, e.g. a regular model of an elliptic curve over a number field, the two-dimensional part of the generalized Riemann hypothesis for the arithmetic zeta function of the model deals with the poles of the zeta function. In dimension one the study of the zeta integral in Tate's thesis does not lead to new important information on the Riemann hypothesis. Contrary to this, in dimension two work of Ivan Fesenko on two-dimensional generalisation of Tate's thesis includes an integral representation of a zeta integral closely related to the zeta function. In this new situation, not possible in dimension one, the poles of the zeta function can be studied via the zeta integral and associated adele groups. Related conjecture of {{harvs|last=Fesenko|authorlink=Ivan Fesenko|year1=2010|txt}} on the positivity of the fourth derivative of a boundary function associated to the zeta integral essentially implies the pole part of the generalized Riemann hypothesis. {{harvs|last=Suzuki|year1=2011|txt}} proved that the latter, together with some technical assumptions, implies Fesenko's conjecture.

Multiple zeta functions

Deligne's proof of the Riemann hypothesis over finite fields used the zeta functions of product varieties, whose zeros and poles correspond to sums of zeros and poles of the original zeta function, in order to bound the real parts of the zeros of the original zeta function. By analogy, {{harvtxt|Kurokawa|1992}} introduced multiple zeta functions whose zeros and poles correspond to sums of zeros and poles of the Riemann zeta function. To make the series converge he restricted to sums of zeros or poles all with non-negative imaginary part. So far, the known bounds on the zeros and poles of the multiple zeta functions are not strong enough to give useful estimates for the zeros of the Riemann zeta function.

Location of the zeros

Number of zeros

The functional equation combined with the argument principle implies that the number of zeros of the zeta function with imaginary part between 0 and T is given by
N(T)=frac{1}{pi}mathop{mathrm{Arg}}(xi(s)) = frac{1}{pi}mathop{mathrm{Arg}}(Gamma(tfrac{s}{2})pi^{-frac{s}{2}}zeta(s)s(s-1)/2)
for s=1/2+iT, where the argument is defined by varying it continuously along the line with Im(s)=T, starting with argument 0 at ∞+iT. This is the sum of a large but well understood term
frac{1}{pi}mathop{mathrm{Arg}}(Gamma(tfrac{s}{2})pi^{-s/2}s(s-1)/2) = frac{T}{2pi}logfrac{T}{2pi}-frac{T}{2pi} +7/8+O(1/T)
and a small but rather mysterious term
S(T) = frac{1}{pi}mathop{mathrm{Arg}}(zeta(1/2+iT)) =O(log(T)).
So the density of zeros with imaginary part near T is about log(T)/2π, and the function S describes the small deviations from this. The function S(t) jumps by 1 at each zero of the zeta function, and for {{nowrap|t ≥ 8}} it decreases monotonically between zeros with derivative close to −log t.Karatsuba (1996) proved that every interval (T, T+H] for H ge T^{frac{27}{82}+varepsilon} contains at least
H(ln T)^{frac{1}{3}}e^{-csqrt{lnln T}}
points where the function S(t) changes sign.{{harvtxt|Selberg|1946}} showed that the average moments of even powers of S are given by
int_0^T|S(t)|^{2k}dt = frac{(2k)!}{k!(2pi)^{2k}}T(log log T)^k + O(T(log log T)^{k-1/2}).
This suggests that S(T)/(log log T)1/2 resembles a Gaussian random variable with mean 0 and variance 2Ï€2 ({{harvtxt|Ghosh|1983}} proved this fact).In particular |S(T)| is usually somewhere around (log log T)1/2, but occasionally much larger. The exact order of growth of S(T) is not known. There has been no unconditional improvement to Riemann's original bound S(T)=O(log T), though the Riemann hypothesis implies the slightly smaller bound S(T)=O(log T/log log T) {{harv|Titchmarsh|1986}}. The true order of magnitude may be somewhat less than this, as random functions with the same distribution as S(T) tend to have growth of order about log(T)1/2. In the other direction it cannot be too small: {{harvtxt|Selberg|1946}} showed that {{nowrap|S(T) ≠ o((log T)1/3/(log log T)7/3)}}, and assuming the Riemann hypothesis Montgomery showed that {{nowrap|S(T) ≠ o((log T)1/2/(log log T)1/2)}}.Numerical calculations confirm that S grows very slowly: |S(T)| < 1 for {{nowrap|T < 280}}, |S(T)| < 2 for T < {{gaps|6|800|000}}, and the largest value of |S(T)| found so far is not much larger than 3 {{harv|Odlyzko|2002}}.Riemann's estimate S(T) = O(log T) implies that the gaps between zeros are bounded, and Littlewood improved this slightly, showing that the gaps between their imaginary parts tends to 0.

Theorem of Hadamard and de la Vallée-Poussin

{{harvtxt|Hadamard|1896}} and {{harvtxt|de la Vallée-Poussin|1896}} independently proved that no zeros could lie on the line Re(s) = 1. Together with the functional equation and the fact that there are no zeros with real part greater than 1, this showed that all non-trivial zeros must lie in the interior of the critical strip {{nowrap|0 < Re(s) < 1}}. This was a key step in their first proofs of the prime number theorem.Both the original proofs that the zeta function has no zeros with real part 1 are similar, and depend on showing that if ζ(1+it) vanishes, then ζ(1+2it) is singular, which is not possible. One way of doing this is by using the inequality
|zeta(sigma)^3zeta(sigma+it)^4zeta(sigma+2it)|ge 1
for σ > 1, t real, and looking at the limit as σ → 1. This inequality follows by taking the real part of the log of the Euler product to see that
|zeta(sigma+it)| = expResum_{p^n}frac{p^{-n(sigma+it)}}{n}=expsum_{p^n}frac{p^{-nsigma}cos(tlog p^n)}{n},
where the sum is over all prime powers pn, so that
|zeta(sigma)^3zeta(sigma+it)^4zeta(sigma+2it)| = expsum_{p^n}p^{-nsigma}frac{3+4cos(tlog p^n)+cos(2tlog p^n)}{n}
which is at least 1 because all the terms in the sum are positive, due to the inequality
3+4cos(theta)+cos(2theta) = 2 (1+cos(theta))^2ge0.

Zero-free regions

De la Vallée-Poussin (1899–1900) proved that if {{nowrap|σ + i{{hsp}}t}} is a zero of the Riemann zeta function, then {{nowrap|1 − σ ≥ {{sfrac|C|log(t)}}}} for some positive constant C. In other words, zeros cannot be too close to the line {{nowrap|σ {{=}} 1:}} there is a zero-free region close to this line. This zero-free region has been enlarged by several authors using methods such as Vinogradov's mean-value theorem. {{harvtxt|Ford|2002}} gave a version with explicit numerical constants: {{nowrap|ζ(σ + i{{hsp}}t ) ≠ 0}} whenever {{nowrap begin}}|t | ≥ 3{{nowrap end}} and
sigmage 1-frac{1}{57.54(log{|t|})^{2/3}(log{log{|t|}})^{1/3}}.

Zeros on the critical line

{{harvtxt|Hardy|1914}} and {{harvtxt|Hardy|Littlewood|1921}} showed there are infinitely many zeros on the critical line, by considering moments of certain functions related to the zeta function. {{harvtxt|Selberg|1942}} proved that at least a (small) positive proportion of zeros lie on the line. {{harvtxt|Levinson|1974}} improved this to one-third of the zeros by relating the zeros of the zeta function to those of its derivative, and {{harvtxt|Conrey|1989}} improved this further to two-fifths.Most zeros lie close to the critical line. More precisely, {{harvtxt|Bohr|Landau|1914}} showed that for any positive ε, all but an infinitely small proportion of zeros lie within a distance ε of the critical line. {{harvtxt|Ivić|1985}} gives several more precise versions of this result, called zero density estimates, which bound the number of zeros in regions with imaginary part at most T and real part at least 1/2+ε.

Hardy–Littlewood conjectures

In 1914 Godfrey Harold Hardy proved that zetaleft(tfrac{1}{2}+itright) has infinitely many real zeros.Let N(T) be the total number of real zeros, N_0(T) be the total number of zeros of odd order of the function
lying on the interval (0, T].The next two conjectures of Hardy and John Edensor Littlewood on the distance between real zeros of zetaleft(tfrac{1}{2}+itright) and on the density of zeros of zetaleft(tfrac{1}{2}+itright) on intervals (T, T+H] for sufficiently large T > 0, H = T^{a + varepsilon} and with as less as possible value of a > 0, where ε > 0 is an arbitrarily small number, open two new directions in the investigation of the Riemann zeta function:1. for any ε > 0 there exists T_0 = T_0(varepsilon) > 0 such that for T geq T_0 and H=T^{0.25+varepsilon} the interval (T,T+H] contains a zero of odd order of the function zetabigl(tfrac{1}{2}+itbigr).2. for any ε > 0 there exist T_0 = T_0(varepsilon) > 0 and c = c(ε) > 0, such that for T geq T_0 and H=T^{0.5+varepsilon} the inequality N_0(T+H)-N_0(T) geq cH is true.

Selberg's zeta function conjecture

{{harvs|first=Atle|last=Selberg|authorlink=Atle Selberg||year=1942|txt}} investigated the problem of Hardy–Littlewood 2 and proved that for any ε > 0 there exists such T_0 = T_0(varepsilon) > 0 and c = c(ε) > 0, such that for T geq T_0 and H=T^{0.5+varepsilon} the inequality N(T+H)-N(T) geq cHlog T is true. Selberg conjectured that this could be tightened to H=T^{0.5}. {{harvs|first=A. A. |last=Karatsuba |year=1984a |year2=1984b |year3=1985 |txt |authorlink=Anatolii Alexeevitch Karatsuba}} proved that for a fixed ε satisfying the condition 0 < ε < 0.001, a sufficiently large T and H = T^{a+varepsilon}, a = tfrac{27}{82} = tfrac{1}{3} -tfrac{1}{246}, the interval (T, T+H) contains at least cHln(T) real zeros of the Riemann zeta function zetaleft(tfrac{1}{2}+itright) and therefore confirmed the Selberg conjecture. The estimates of Selberg and Karatsuba can not be improved in respect of the order of growth as T → ∞.{{harvtxt|Karatsuba|1992}} proved that an analog of the Selberg conjecture holds for almost all intervals (T, T+H], H = T^{varepsilon}, where ε is an arbitrarily small fixed positive number. The Karatsuba method permits to investigate zeros of the Riemann zeta-function on "supershort" intervals of the critical line, that is, on the intervals (T, T+H], the length H of which grows slower than any, even arbitrarily small degree T. In particular, he proved that for any given numbers ε, varepsilon_1 satisfying the conditions 0

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