Distribution of prime numbers

Riemann's explicit formula for the number of primes less than a given number states that, in terms of a sum over the zeros of the Riemann zeta function, 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^[4] ${\displaystyle \pi (x)-\operatorname {li} (x)=O\left(x^{\beta }\log x\right).}$ It is already known that 1/2 ≤ β ≤ 1.^[5]

Corrections to an estimate of the prime-counting function using zeros of the zeta function. The magnitude of the correction term is determined by the real part of the zero being added in the correction.

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 Schoenfeld (1976), says that the Riemann hypothesis implies

${\displaystyle |\pi (x)-\operatorname {li} (x)|<{\frac {1}{8\pi }}{\sqrt {x}}\log(x),\qquad {\text{for all }}x\geq 2657,}$

where ${\displaystyle \pi (x)}$ is the prime-counting function, ${\displaystyle \operatorname {li} (x)}$ is the logarithmic integral function, and ${\displaystyle \log(x)}$ is the natural logarithm of x.

Schoenfeld (1976) also showed that the Riemann hypothesis implies

${\displaystyle |\psi (x)-x|<{\frac {1}{8\pi }}{\sqrt {x}}\log ^{2}x,\qquad {\text{for all }}x\geq 73.2,}$

where ${\displaystyle \psi (x)}$ is Chebyshev's second function.

Dudek (2014) proved that the Riemann hypothesis implies that for all ${\displaystyle x\geq 2}$ there is a prime ${\displaystyle p}$ satisfying

${\displaystyle x-{\frac {4}{\pi }}{\sqrt {x}}\log x<p\leq x}$.

The constant 4/π may be reduced to (1 + ε) provided that x is taken to be sufficiently large. 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

${\displaystyle {\frac {1}{\zeta (s)}}=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}}$

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

${\displaystyle M(x)=\sum _{n\leq x}\mu (n)}$

then the claim that

${\displaystyle M(x)=O\left(x^{{\frac {1}{2}}+\varepsilon }\right)}$

for every positive ε is equivalent to the Riemann hypothesis (J. E. Littlewood, 1912; see for instance: paragraph 14.25 in 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. Littlewood's result has been improved several times since then, by Edmund Landau,^[6] Edward Charles Titchmarsh,^[7] Helmut Maier and Hugh Montgomery,^[8] and Kannan Soundararajan.^[9] Soundararajan's result is that

${\displaystyle M(x)=O(x^{1/2}\exp {\big (}(\log x)^{1/2}(\log \log x)^{14}{\big )}.}$

The Riemann hypothesis puts a rather tight bound on the growth of M, since Odlyzko & te Riele (1985) disproved the slightly stronger Mertens conjecture

${\displaystyle |M(x)|\leq {\sqrt {x}}.}$

Another closely related result is due to Björner (2011), that the Riemann hypothesis is equivalent to the statement that the Euler characteristic of the simplicial complex determined by the lattice of integers under divisibility is ${\displaystyle o(n^{1/2+\epsilon })}$ for all ${\displaystyle \epsilon >0}$ (see incidence algebra).

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,^[10] which states that if σ(n) is the sigma function, given by

${\displaystyle \sigma (n)=\sum _{d\mid n}d}$

then

${\displaystyle \sigma (n)<e^{\gamma }n\log \log n}$

for all n > 5040 if and only if the Riemann hypothesis is true, where γ is the Euler–Mascheroni constant.

A related bound was given by Jeffrey Lagarias in 2002, who proved that the Riemann hypothesis is equivalent to the statement that:

${\displaystyle \sigma (n)<H_{n}+\log(H_{n})e^{H_{n}}}$

for every natural number n > 1, where ${\displaystyle H_{n}}$ is the nth harmonic number.^[11]

The Riemann hypothesis is also true if and only if the inequality

${\displaystyle {\frac {n}{\varphi (n)}}<e^{\gamma }\log \log n+{\frac {e^{\gamma }(4+\gamma -\log 4\pi )}{\sqrt {\log n}}}}$

is true for all n ≥ 120569# where φ(n) is Euler's totient function and 120569# is the product of the first 120569 primes.^[12]

Another example was found by Jérôme Franel, and extended by Landau (see 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 F_n is the Farey sequence of order n, beginning with 1/n and up to 1/1, then the claim that for all ε > 0

${\displaystyle \sum _{i=1}^{m}|F_{n}(i)-{\tfrac {i}{m}}|=O\left(n^{{\frac {1}{2}}+\epsilon }\right)}$

is equivalent to the Riemann hypothesis. Here

${\displaystyle m=\sum _{i=1}^{n}\phi (i)}$

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 S_n of degree n, then Massias, Nicolas & Robin (1988) showed that the Riemann hypothesis is equivalent to the bound

${\displaystyle \log g(n)<{\sqrt {\operatorname {Li} ^{-1}(n)}}}$

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,

${\displaystyle \zeta \left({\frac {1}{2}}+it\right)=O(t^{\varepsilon }),}$

as ${\displaystyle t\to \infty }$.

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

${\displaystyle e^{\gamma }\leq \limsup _{t\rightarrow +\infty }{\frac {|\zeta (1+it)|}{\log \log t}}\leq 2e^{\gamma }}$

${\displaystyle {\frac {6}{\pi ^{2}}}e^{\gamma }\leq \limsup _{t\rightarrow +\infty }{\frac {1/|\zeta (1+it)|}{\log \log t}}\leq {\frac {12}{\pi ^{2}}}e^{\gamma }}$

so the growth rate of ζ(1+it) and its inverse would be known up to a factor of 2.^[13]

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(√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.^[14]

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 Riesz (1916), to the effect that the bound

${\displaystyle -\sum _{k=1}^{\infty }{\frac {(-x)^{k}}{(k-1)!\zeta (2k)}}=O\left(x^{{\frac {1}{4}}+\epsilon }\right)}$

holds for all ε > 0 if and only if the Riemann hypothesis holds. See also the Hardy–Littlewood criterion.

Nyman (1950) proved that the Riemann hypothesis is true if and only if the space of functions of the form

${\displaystyle f(x)=\sum _{\nu =1}^{n}c_{\nu }\rho \left({\frac {\theta _{\nu }}{x}}\right)}$

where ρ(z) is the fractional part of z, 0 ≤ θ_ν ≤ 1, and

${\displaystyle \sum _{\nu =1}^{n}c_{\nu }\theta _{\nu }=0}$,

is dense in the Hilbert space L²(0,1) of square-integrable functions on the unit interval. 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 L^p(0,1). This Nyman-Beurling criterion was strengthened by Baez-Duarte^[15] to the case where ${\displaystyle \theta _{\nu }\in \{1/k\}_{k\geq 1}}$.

Salem (1953) showed that the Riemann hypothesis is true if and only if the integral equation

${\displaystyle \int _{0}^{\infty }{\frac {z^{-\sigma -1}\phi (z)}{{e^{x/z}}+1}}\,dz=0}$

has no non-trivial bounded solutions ${\displaystyle \phi }$ for ${\displaystyle 1/2<\sigma <1}$.

Weil's criterion is the statement that the positivity of a certain function is equivalent to the Riemann hypothesis. Related is Li's criterion, a statement that the positivity of a certain sequence of numbers is equivalent to the Riemann hypothesis.

Speiser (1934) proved that the Riemann hypothesis is equivalent to the statement that ${\displaystyle \zeta '(s)}$, the derivative of ${\displaystyle \zeta (s)}$, has no zeros in the strip

${\displaystyle 0<\Re (s)<{\frac {1}{2}}.}$

That ${\displaystyle \zeta (s)}$ has only simple zeros on the critical line is equivalent to its derivative having no zeros on the critical line.

The Farey sequence provides two equivalences, due to Jerome Franel and Edmund Landau in 1924.

The de Bruijn–Newman constant denoted by Λ and named after Nicolaas Govert de Bruijn and Charles M. Newman, is defined as the unique real number such that the function

${\displaystyle H(\lambda ,z):=\int _{0}^{\infty }e^{\lambda u^{2}}\Phi (u)\cos(zu)\,du}$,

that is parametrised by a real parameter λ, has a complex variable z and is defined using a super-exponentially decaying function

${\displaystyle \Phi (u)=\sum _{n=1}^{\infty }(2\pi ^{2}n^{4}e^{9u}-3\pi n^{2}e^{5u})e^{-\pi n^{2}e^{4u}}}$.

has only real zeros if and only if λ ≥ Λ. Since the Riemann hypothesis is equivalent to the claim that all the zeroes of H(0, z) are real, the Riemann hypothesis is equivalent to the conjecture that ${\displaystyle \Lambda \leq 0}$. Brad Rodgers and Terence Tao discovered the equivalence is actually ${\displaystyle \Lambda =0}$ by proving zero to be the lower bound of the constant.^[16] Proving zero is also the upper bound would therefore prove the Riemann hypothesis. As of April 2020 the upper bound is ${\displaystyle \Lambda \leq 0.2}$.^[17]

Consequences of the generalized Riemann hypothesis

Several applications use the generalized Riemann hypothesis for Dirichlet L-series or zeta functions of number fields rather than just the Riemann hypothesis. Many basic properties of the Riemann zeta function can easily be generalized to all Dirichlet L-series, so it is plausible that a method that proves the Riemann hypothesis for the Riemann zeta function would also work for the generalized Riemann hypothesis for Dirichlet L-functions. Several results first proved using the generalized Riemann hypothesis were later given unconditional proofs without using it, though these were usually much harder. Many of the consequences on the following list are taken from Conrad (2010).

• In 1913, Grönwall showed that the generalized Riemann hypothesis implies that Gauss's list of imaginary quadratic fields with class number 1 is complete, though Baker, Stark and Heegner later gave unconditional proofs of this without using the generalized Riemann hypothesis.
• In 1917, Hardy and Littlewood showed that the generalized Riemann hypothesis implies a conjecture of Chebyshev that
  $${\displaystyle \lim _{x\to 1^{-}}\sum _{p>2}(-1)^{(p+1)/2}x^{p}=+\infty ,}$$

  
  which says that primes 3 mod 4 are more common than primes 1 mod 4 in some sense. (For related results, see Prime number theorem § Prime number race.)
• 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 2013 Harald Helfgott proved the ternary Goldbach conjecture without the GRH dependence, subject to some extensive calculations completed with the help of David J. Platt.
• 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.
• 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.
• Odlyzko (1990) discussed how the generalized Riemann hypothesis can be used to give sharper estimates for discriminants and class numbers of number fields.
• Ono & Soundararajan (1997) showed that the generalized Riemann hypothesis implies that Ramanujan's integral quadratic form x² + y² + 10z² represents all integers that it represents locally, with exactly 18 exceptions.
• In 2021, Alexander (Alex) Dunn and Maksym Radziwill proved Patterson's conjecture under the assumption of the GRH.^[18]

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, Ireland & Rosen (1990, p. 359) say

The 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.

Gauss's class number conjecture

This is the conjecture (first stated in article 303 of Gauss's Disquisitiones Arithmeticae) that there are only finitely many imaginary quadratic fields with a given class number. One way to prove it would be to show that as the discriminant D → −∞ the class number h(D) → ∞.

The following sequence of theorems involving the Riemann hypothesis is described in Ireland & Rosen 1990, pp. 358–361:

Theorem (Hecke; 1918) — Let D < 0 be the discriminant of an imaginary quadratic number field K. Assume the generalized Riemann hypothesis for L-functions of all imaginary quadratic Dirichlet characters. Then there is an absolute constant C such that

$${\displaystyle h(D)>C{\frac {\sqrt {|D|}}{\log |D|}}.}$$

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 that

$${\displaystyle \varphi (n)<e^{-\gamma }{\frac {n}{\log \log n}}}$$

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."^[19]

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

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 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 ${\displaystyle {\sqrt {q}}}$ is actually an instance of the Riemann hypothesis in the function field setting. This led Weil (1949) to conjecture a similar statement for all algebraic varieties; the resulting Weil conjectures were proved by Pierre Deligne (1974, 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 Jean-Pierre Serre (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 ${\displaystyle \Re (s)\in (0,n)}$ 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 ${\displaystyle \Re (s)=1/2,3/2,\dots ,n-1/2}$ and its poles inside the critical strip lie on vertical lines ${\displaystyle \Re (s)=1,2,\dots ,n-1}$. This is known for schemes in positive characteristic and follows from Pierre Deligne (1974, 1980), but remains entirely unknown in characteristic zero.

Selberg zeta functions

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

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. 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.^[20] 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 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.^[21]

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.

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 ${\displaystyle {\hat {H}}}$ of the classical Hamiltonian H = xp so that

$${\displaystyle \zeta (1/2+i{\hat {H}})=0}$$

and even more strongly, that the Riemann zeros coincide with the spectrum of the operator ${\displaystyle 1/2+i{\hat {H}}}$. This is in contrast to canonical quantization, which leads to the Heisenberg uncertainty principle ${\displaystyle \sigma _{x}\sigma _{p}\geq {\frac {\hbar }{2}}}$ 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

$${\displaystyle N(s)={\frac {1}{\pi }}\operatorname {Arg} \xi (1/2+i{\sqrt {s}})}$$

then, in Berry–Connes approach^[22]

$${\displaystyle V^{-1}(x)={\sqrt {4\pi }}{\frac {d^{1/2}N(x)}{dx^{1/2}}}.}$$

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)

$${\displaystyle \det(H+1/4+s(s-1))}$$

as proved by Connes and others, in this approach

$${\displaystyle {\frac {\xi (s)}{\xi (0)}}={\frac {\det(H+s(s-1)+1/4)}{\det(H+1/4)}}.}$$

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. Deninger (1998) described some of the attempts to find such a cohomology theory.^[23]

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. 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.

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 equal to 0, and this has led to some speculation about a relationship with the Riemann hypothesis.^[24]

Turán's result

Pál Turán (1948) showed that if the functions

$${\displaystyle \sum _{n=1}^{N}n^{-s}}$$

have no zeros when the real part of s is greater than one then

$${\displaystyle T(x)=\sum _{n\leq x}{\frac {\lambda (n)}{n}}\geq 0{\text{ for }}x>0,}$$

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. But Haselgrove (1958) proved that T(x) is negative for infinitely many x (and also disproved the closely related Pólya conjecture), and Borwein, Ferguson & Mossinghoff (2008) showed that the smallest such x is 72185376951205. 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 Montgomery (1983) showed that for all sufficiently large N these series have zeros with real part greater than 1 + (log log N)/(4 log N). Therefore, Turán's result is vacuously true and cannot help prove the Riemann hypothesis.

Noncommutative geometry

Connes (1999, 2000) has described a relationship between the Riemann hypothesis and noncommutative geometry, and showed 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 Lapidus (2008).

Hilbert spaces of entire functions

Louis de Branges (1992) showed that the Riemann hypothesis would follow from a positivity condition on a certain Hilbert space of entire functions. However 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.^[25]

Quasicrystals

The Riemann hypothesis implies that the zeros of the zeta function form a quasicrystal, a distribution with discrete support whose Fourier transform also has discrete support. 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 Fesenko (2010) 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. Suzuki (2011) 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, 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.

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

${\displaystyle N(T)={\frac {1}{\pi }}\mathop {\mathrm {Arg} } (\xi (s))={\frac {1}{\pi }}\mathop {\mathrm {Arg} } (\Gamma ({\tfrac {s}{2}})\pi ^{-{\frac {s}{2}}}\zeta (s)s(s-1)/2)}$

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

${\displaystyle {\frac {1}{\pi }}\mathop {\mathrm {Arg} } (\Gamma ({\tfrac {s}{2}})\pi ^{-s/2}s(s-1)/2)={\frac {T}{2\pi }}\log {\frac {T}{2\pi }}-{\frac {T}{2\pi }}+7/8+O(1/T)}$

and a small but rather mysterious term

${\displaystyle S(T)={\frac {1}{\pi }}\mathop {\mathrm {Arg} } (\zeta (1/2+iT))=O(\log T).}$

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 t ≥ 8 it decreases monotonically between zeros with derivative close to −log t.

Trudgian (2014) proved that, if ${\displaystyle T>e}$, then

${\displaystyle |N(T)-{\frac {T}{2\pi }}\log {\frac {T}{2\pi e}}|\leq 0.112\log T+0.278\log \log T+3.385+{\frac {0.2}{T}}}$.

Karatsuba (1996) proved that every interval (T, T+H] for ${\displaystyle H\geq T^{{\frac {27}{82}}+\varepsilon }}$ contains at least

${\displaystyle H(\log T)^{\frac {1}{3}}e^{-c{\sqrt {\log \log T}}}}$

points where the function S(t) changes sign.

Selberg (1946) showed that the average moments of even powers of S are given by

${\displaystyle \int _{0}^{T}|S(t)|^{2k}dt={\frac {(2k)!}{k!(2\pi )^{2k}}}T(\log \log T)^{k}+O(T(\log \log T)^{k-1/2}).}$

This suggests that S(T)/(log log T)^1/2 resembles a Gaussian random variable with mean 0 and variance 2π² (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).^[13] 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: Selberg (1946) showed that S(T) ≠ o((log T)^1/3/(log log T)^7/3), and assuming the Riemann hypothesis Montgomery showed that S(T) ≠ o((log T)^1/2/(log log T)^1/2).

Numerical calculations confirm that S grows very slowly: |S(T)| < 1 for T < 280, |S(T)| < 2 for T < 6800000, and the largest value of |S(T)| found so far is not much larger than 3.^[26]

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 tend to 0.

Theorem of Hadamard and de la Vallée-Poussin

Hadamard (1896) and 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 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

${\displaystyle |\zeta (\sigma )^{3}\zeta (\sigma +it)^{4}\zeta (\sigma +2it)|\geq 1}$

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

${\displaystyle |\zeta (\sigma +it)|=\exp \Re \sum _{p^{n}}{\frac {p^{-n(\sigma +it)}}{n}}=\exp \sum _{p^{n}}{\frac {p^{-n\sigma }\cos(t\log p^{n})}{n}},}$

where the sum is over all prime powers pⁿ, so that

${\displaystyle |\zeta (\sigma )^{3}\zeta (\sigma +it)^{4}\zeta (\sigma +2it)|=\exp \sum _{p^{n}}p^{-n\sigma }{\frac {3+4\cos(t\log p^{n})+\cos(2t\log p^{n})}{n}}}$

which is at least 1 because all the terms in the sum are positive, due to the inequality

${\displaystyle 3+4\cos(\theta )+\cos(2\theta )=2(1+\cos(\theta ))^{2}\geq 0.}$

Zero-free regions

The most extensive computer search by Platt and Trudgian^[17] for counter examples of the Riemann hypothesis has verified it for ${\displaystyle |t|\leq 3.0001753328\cdot 10^{12}}$. Beyond that zero-free regions are known as inequalities concerning σ + i t, which can be zeroes. The oldest version is from De la Vallée-Poussin (1899–1900), who proved there is a region without zeroes that satisfies 1 − σ ≥ C/log(t) for some positive constant C. In other words, zeros cannot be too close to the line σ = 1: there is a zero-free region close to this line. This has been enlarged by several authors using methods such as Vinogradov's mean-value theorem.

The most recent paper^[27] by Mossinghoff, Trudgian and Yang is from December 2022 and provides four zero-free regions that improved the previous results of Kevin Ford from 2002, Mossinghoff and Trudgian themselves from 2015 and Pace Nielsen's slight improvement of Ford from October 2022:

${\displaystyle \sigma \geq 1-{\frac {1}{5.558691\log |t|}}}$ whenever ${\displaystyle |t|\geq 2}$,

${\displaystyle \sigma \geq 1-{\frac {1}{55.241(\log {|t|})^{2/3}(\log {\log {|t|}})^{1/3}}}}$ whenever ${\displaystyle |t|\geq 3}$ (largest known region in the bound ${\displaystyle 3.0001753328\cdot 10^{12}\leq |t|\leq \exp(64.1)\approx 6.89\cdot 10^{27}}$),

${\displaystyle \sigma \geq 1-{\frac {0.04962-{\frac {0.0196}{1.15+\log 3+{\frac {1}{6}}\log t+\log \log t}}}{0.685+\log 3+{\frac {1}{6}}\log t+1.155\cdot \log \log t}}}$ whenever ${\displaystyle |t|\geq 1.88\cdot 10^{14}}$ (largest known region in the bound ${\displaystyle \exp(64.1)\leq |t|\leq \exp(1000)\approx 1.97\cdot 10^{434}}$) and

${\displaystyle \sigma \geq 1-{\frac {0.05035}{{\frac {27}{164}}(\log {|t|})+7.096}}+{\frac {0.0349}{({\frac {27}{164}}(\log {|t|})+7.096)^{2}}}}$ whenever ${\displaystyle |t|\geq \exp(1000)}$ (largest known region in its own bound)

The paper also presents an improvement to the second zero-free region, whose bounds are unknown on account of ${\displaystyle |t|}$ being merely assumed to be "sufficiently large" to fulfill the requirements of the paper's proof. This region is

${\displaystyle \sigma \geq 1-{\frac {1}{48.1588(\log {|t|})^{2/3}(\log {\log {|t|}})^{1/3}}}}$.

Hardy–Littlewood conjectures

In 1914 Godfrey Harold Hardy proved that ${\displaystyle \zeta \left({\tfrac {1}{2}}+it\right)}$ has infinitely many real zeros.

The next two conjectures of Hardy and John Edensor Littlewood on the distance between real zeros of ${\displaystyle \zeta \left({\tfrac {1}{2}}+it\right)}$ and on the density of zeros of ${\displaystyle \zeta \left({\tfrac {1}{2}}+it\right)}$ on the interval ${\displaystyle (T,T+H]}$ for sufficiently large ${\displaystyle T>0}$, and ${\displaystyle H=T^{a+\varepsilon }}$ and with as small as possible value of ${\displaystyle a>0}$, where ${\displaystyle \varepsilon >0}$ is an arbitrarily small number, open two new directions in the investigation of the Riemann zeta function:

1. For any ${\displaystyle \varepsilon >0}$ there exists a lower bound ${\displaystyle T_{0}=T_{0}(\varepsilon )>0}$ such that for ${\displaystyle T\geq T_{0}}$ and ${\displaystyle H=T^{{\tfrac {1}{4}}+\varepsilon }}$ the interval ${\displaystyle (T,T+H]}$ contains a zero of odd order of the function ${\displaystyle \zeta {\bigl (}{\tfrac {1}{2}}+it{\bigr )}}$.

Let ${\displaystyle N(T)}$ be the total number of real zeros, and ${\displaystyle N_{0}(T)}$ be the total number of zeros of odd order of the function ${\displaystyle ~\zeta \left({\tfrac {1}{2}}+it\right)~}$ lying on the interval ${\displaystyle (0,T]~}$.

2. For any ${\displaystyle \varepsilon >0}$ there exists ${\displaystyle T_{0}=T_{0}(\varepsilon )>0}$ and some ${\displaystyle c=c(\varepsilon )>0}$, such that for ${\displaystyle T\geq T_{0}}$ and ${\displaystyle H=T^{{\tfrac {1}{2}}+\varepsilon }}$ the inequality ${\displaystyle N_{0}(T+H)-N_{0}(T)\geq cH}$ is true.

Selberg's zeta function conjecture

Atle Selberg (1942) investigated the problem of Hardy–Littlewood 2 and proved that for any ε > 0 there exists such ${\displaystyle T_{0}=T_{0}(\varepsilon )>0}$ and c = c(ε) > 0, such that for ${\displaystyle T\geq T_{0}}$ and ${\displaystyle H=T^{0.5+\varepsilon }}$ the inequality ${\displaystyle N(T+H)-N(T)\geq cH\log T}$ is true. Selberg conjectured that this could be tightened to ${\displaystyle H=T^{0.5}}$. A. A. Karatsuba (1984a, 1984b, 1985) proved that for a fixed ε satisfying the condition 0 < ε < 0.001, a sufficiently large T and ${\displaystyle H=T^{a+\varepsilon }}$, ${\displaystyle a={\tfrac {27}{82}}={\tfrac {1}{3}}-{\tfrac {1}{246}}}$, the interval (T, T+H) contains at least cH log(T) real zeros of the Riemann zeta function ${\displaystyle \zeta \left({\tfrac {1}{2}}+it\right)}$ 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 → ∞.

Karatsuba (1992) proved that an analog of the Selberg conjecture holds for almost all intervals (T, T+H], ${\displaystyle H=T^{\varepsilon }}$, 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, ${\displaystyle \varepsilon _{1}}$ satisfying the conditions ${\displaystyle 0<\varepsilon ,\varepsilon _{1}<1}$ almost all intervals (T, T+H] for ${\displaystyle H\geq \exp {\{(\log T)^{\varepsilon }\}}}$ contain at least ${\displaystyle H(\log T)^{1-\varepsilon _{1}}}$ zeros of the function ${\displaystyle \zeta \left({\tfrac {1}{2}}+it\right)}$. This estimate is quite close to the one that follows from the Riemann hypothesis.

Numerical calculations

The function

${\displaystyle \pi ^{-{\frac {s}{2}}}\Gamma ({\tfrac {s}{2}})\zeta (s)}$

has the same zeros as the zeta function in the critical strip, and is real on the critical line because of the functional equation, so one can prove the existence of zeros exactly on the real line between two points by checking numerically that the function has opposite signs at these points. Usually one writes

${\displaystyle \zeta ({\tfrac {1}{2}}+it)=Z(t)e^{-i\theta (t)}}$

where Hardy's Z function and the Riemann–Siegel theta function θ are uniquely defined by this and the condition that they are smooth real functions with θ(0)=0. By finding many intervals where the function Z changes sign one can show that there are many zeros on the critical line. To verify the Riemann hypothesis up to a given imaginary part T of the zeros, one also has to check that there are no further zeros off the line in this region. This can be done by calculating the total number of zeros in the region using Turing's method and checking that it is the same as the number of zeros found on the line. This allows one to verify the Riemann hypothesis computationally up to any desired value of T (provided all the zeros of the zeta function in this region are simple and on the critical line).

These calculations can also be used to estimate ${\displaystyle \pi (x)}$ for finite ranges of ${\displaystyle x}$. For example, using the latest result from 2020 (zeros up to height ${\displaystyle 3\times 10^{12}}$), it has been shown that

${\displaystyle |\pi (x)-\operatorname {li} (x)|<{\frac {1}{8\pi }}{\sqrt {x}}\log(x),\qquad {\text{for }}2657\leq x\leq 1.101\times 10^{26}.}$

In general, this inequality holds if

${\displaystyle x\geq 2657}$ and ${\displaystyle {\frac {9.06}{\log {\log {x}}}}{\sqrt {\frac {x}{\log {x}}}}\leq T,}$

where ${\displaystyle T}$ is the largest known value such that the Riemann hypothesis is true for all zeros ${\displaystyle \rho }$ with ${\displaystyle \Im {\left(\rho \right)}\in \left(0,T\right]}$.^[29]

Some calculations of zeros of the zeta function are listed below, where the "height" of a zero is the magnitude of its imaginary part, and the height of the nth zero is denoted by γ_n. So far all zeros that have been checked are on the critical line and are simple. (A multiple zero would cause problems for the zero finding algorithms, which depend on finding sign changes between zeros.) For tables of the zeros, see Haselgrove & Miller (1960) or Odlyzko.

Gram points

A Gram point is a point on the critical line 1/2 + it where the zeta function is real and non-zero. Using the expression for the zeta function on the critical line, ζ(1/2 + it) = Z(t)e ^− iθ(t), where Hardy's function, Z, is real for real t, and θ is the Riemann–Siegel theta function, we see that zeta is real when sin(θ(t)) = 0. This implies that θ(t) is an integer multiple of π, which allows for the location of Gram points to be calculated fairly easily by inverting the formula for θ. They are usually numbered as g_n for n = 0, 1, ..., where g_n is the unique solution of θ(t) = nπ.

Gram observed that there was often exactly one zero of the zeta function between any two Gram points; Hutchinson called this observation Gram's law. There are several other closely related statements that are also sometimes called Gram's law: for example, (−1)ⁿZ(g_n) is usually positive, or Z(t) usually has opposite sign at consecutive Gram points. The imaginary parts γ_n of the first few zeros (in blue) and the first few Gram points g_n are given in the following table

This is a polar plot of the first 20 non-trivial Riemann zeta function zeros (including Gram points) along the critical line ${\displaystyle \zeta (1/2+it)}$ for real values of ${\displaystyle t}$ running from 0 to 50. The consecutively labeled zeros have 50 red plot points between each, with zeros identified by concentric magenta rings scaled to show the relative distance between their values of t. Gram's law states that the curve usually crosses the real axis once between zeros.

The first failure of Gram's law occurs at the 127th zero and the Gram point g₁₂₆, which are in the "wrong" order.

A Gram point t is called good if the zeta function is positive at 1/2 + it. The indices of the "bad" Gram points where Z has the "wrong" sign are 126, 134, 195, 211, ... (sequence A114856 in the OEIS). A Gram block is an interval bounded by two good Gram points such that all the Gram points between them are bad. A refinement of Gram's law called Rosser's rule due to Rosser, Yohe & Schoenfeld (1969) says that Gram blocks often have the expected number of zeros in them (the same as the number of Gram intervals), even though some of the individual Gram intervals in the block may not have exactly one zero in them. For example, the interval bounded by g₁₂₅ and g₁₂₇ is a Gram block containing a unique bad Gram point g₁₂₆, and contains the expected number 2 of zeros although neither of its two Gram intervals contains a unique zero. Rosser et al. checked that there were no exceptions to Rosser's rule in the first 3 million zeros, although there are infinitely many exceptions to Rosser's rule over the entire zeta function.

Gram's rule and Rosser's rule both say that in some sense zeros do not stray too far from their expected positions. The distance of a zero from its expected position is controlled by the function S defined above, which grows extremely slowly: its average value is of the order of (log log T)^1/2, which only reaches 2 for T around 10²⁴. This means that both rules hold most of the time for small T but eventually break down often. Indeed, Trudgian (2011) showed that both Gram's law and Rosser's rule fail in a positive proportion of cases. To be specific, it is expected that in about 66% one zero is enclosed by two successive Gram points, but in 17% no zero and in 17% two zeros are in such a Gram-interval on the long run Hanga (2020).

Popular expositions

• Sabbagh, Karl (2003a), The greatest unsolved problem in mathematics, Farrar, Straus and Giroux, New York, ISBN 978-0-374-25007-2, MR 1979664
• Sabbagh, Karl (2003b), Dr. Riemann's zeros, Atlantic Books, London, ISBN 978-1-843-54101-1
• du Sautoy, Marcus (2003), The music of the primes, HarperCollins Publishers, ISBN 978-0-06-621070-4, MR 2060134
• Rockmore, Dan (2005), Stalking the Riemann hypothesis, Pantheon Books, ISBN 978-0-375-42136-5, MR 2269393
• Derbyshire, John (2003), Prime Obsession, Joseph Henry Press, Washington, DC, ISBN 978-0-309-08549-6, MR 1968857
• Watkins, Matthew (2015), Mystery of the Prime Numbers, Liberalis Books, ISBN 978-1782797814, MR 0000000
• Frenkel, Edward (2014), The Riemann Hypothesis Numberphile, Mar 11, 2014 (video)

Derbyshire 2003, Rockmore 2005, both Sabbagh 2003, Sautoy 2003, and Watkins 2015 are non-technical. Edwards 1974, Patterson 1988, Borwein/Choi/Rooney/Weirathmueller 2008, Mazur/Stein 2015, and Broughan 2017 give mathematical introductions, while Titchmarsh 1986, Ivić 1985, and Karatsuba/Voronin 1992 are advanced monographs.