The Riemann Hypothesis

In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part ⁠1/2⁠. Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by Bernhard Riemann, after whom it is named. According to a 2026 survey, there is overwhelming numerical evidence for the hypothesis, but no proof is known.

The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, make up Hilbert's eighth problem in David Hilbert's list of twenty-three unsolved problems; it is also one of the Millennium Prize Problems of the Clay Mathematics Institute, which offers US$1 million for a solution to any of them. The name is also used for some closely related analogues, some of which have been proved, such as the Riemann hypothesis for curves over finite fields, which was proved by André Weil.

The Riemann zeta function \(\zeta\) is a function whose argument may be any complex number other than 1, and whose values are also complex. It has zeros at the negative even integers; that is, \(\zeta(s)=0\) when \(s\) is one of \(-2,-4,-6,\dots\) These are called its trivial zeros. The zeta function is also zero for other values of \(s\), which are called nontrivial zeros. The Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that:

Thus, the hypothesis states that all the nontrivial zeros lie on the critical line, consisting of the complex numbers \(\tfrac{1}{2}+it\) where \(t\) is a real number and \(i\) is the imaginary unit.

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 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

\(\zeta(s) = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}}= \frac{1}{1-2^{-s}}\cdot\frac{1}{1-3^{-s}}\cdot\frac{1}{1-5^{-s}}\cdot\frac{1}{1-7^{-s}} \cdots\)

where the infinite product extends over all prime numbers \(p\).

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 \(s\). Because the zeta function is meromorphic, all choices of how to perform this analytic continuation will lead to the same result, by the identity theorem. A first step in this continuation observes that the series for the zeta function and the Dirichlet eta function satisfy the relation

\(\left(1-\frac{2}{2^s}\right)\zeta(s) = \eta(s) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^s} = \frac{1}{1^s} - \frac{1}{2^s} + \frac{1}{3^s} - \cdots,\)

within the region of convergence for both series. But the eta function 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, the zeta function can be redefined as \(\eta(s)/(1-2/2^s)\), extending it from \(\operatorname{Re}(s)>1\) to the larger domain \(\operatorname{Re}(s)>0\), except for the points where \(1-2/2^s\) is zero. These are the points \(s = 1 + 2\pi in/\log 2\), where \(n\) can be any nonzero integer; the zeta function can be extended to these values too by taking limits (see the article on the Dirichlet eta function), giving a finite value for all values of \(s\) with positive real part except the simple pole at \(s = 1\).

In the strip \(0<\operatorname{Re}(s)<1\) this extension of the zeta function satisfies the functional equation

\(\zeta(s) = 2^s\pi^{s-1}\ \sin\left(\frac{\pi s}{2}\right)\ \Gamma(1-s)\ \zeta(1-s).\)

One may then define \(\zeta(s)\) for all remaining nonzero complex numbers \(s\) (\(\operatorname{Re}(s)\leq 0\) and \(s\neq 0\)) by applying this equation outside the strip, and letting \(\zeta(s)\) equal the right side of the equation whenever \(s\) has non-positive real part (and \(s\neq 0\)).

Condensed: the full section is in Wikipedia.

, Riemann's statement of the Riemann hypothesis, from (Riemann 1859). (He was discussing a variant of the zeta function, modified in a way that the real line be mapped to the critical line.)

, Jacques Hadamard, The Mathematician's Mind, VIII. Paradoxical Cases of Intuition

Riemann's original motivation for studying the zeta function and its zeros was their occurrence in his explicit formula for the number of primes \(\pi(x)\) less than or equal to 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

\(\Pi(x) = \pi(x) + \frac{\pi(x^{1/2})}{2} +\frac{\pi(x^{1/3})}{3} + \frac{\pi(x^{1/4})}{4} + \frac{\pi(x^{1/5})}{5} +\frac{\pi(x^{1/6})}{6} +\cdots\)

which counts the primes and prime powers up to \(x\), counting a prime power \(p^n\) as \(1/n\). The number of primes can be recovered from this function by using the Möbius inversion formula:

\(\begin{aligned} \pi(x) &= \sum_{n=1}^\infty \frac{\mu(n)} n \Pi(x^{1/n}) \\ &= \Pi(x) -\frac{1}{2}\Pi(x^{1/2}) - \frac{1}{3}\Pi(x^{1/3}) - \frac{1}{5}\Pi(x^{1/5}) + \frac{1}{6} \Pi(x^{1/6}) -\cdots, \end{aligned}\)

where \(\mu\) is the Möbius function. Riemann's formula is then

\(\Pi_0(x) = \operatorname{li}(x) - \sum_\rho \operatorname{li}(x^\rho) -\log 2 + \int_x^\infty\frac{dt}{t(t^2-1) \log t}\),

where the sum is over the nontrivial zeros of the zeta function and where \(\Pi_0\) is a slightly modified version of \(\Pi\) 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 \(\rho\) in order of the absolute value of their imaginary part. The function \(\operatorname{li}\) occurring in the first term is the (unoffset) logarithmic integral function given by the Cauchy principal value of the divergent integral

\(\operatorname{li}(x) = \int_0^x \frac{dt}{\log t}.\)

, (Burton 2006, p. 376)

Condensed: the full section is in Wikipedia.

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

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 \(\beta\) is the upper bound of the real parts of the zeros, then \(\pi(x) - \operatorname{li}(x) = O\!\left( x^\beta \log x \right)\), where \(\pi(x)\) is the prime-counting function and \(\operatorname{li}(x)\) is the logarithmic integral function. It is already known that \(1/2\leq\beta\leq 1\).

Helge von Koch proved that the Riemann hypothesis implies the "best possible" bound for the error of the prime number theorem. A precise version of von Koch's result, due to Schoenfeld (1976), says that the Riemann hypothesis implies

\(|\pi(x) - \operatorname{li}(x)| < \frac{1}{8\pi} \sqrt{x} \log(x)\)

for all \(x\geq 2657\). Schoenfeld (1976) also showed that the Riemann hypothesis implies

\(|\psi(x) - x| < \frac{1}{8\pi} \sqrt{x} \log^2 x\)

for all \(x\geq 73.2\), where \(\psi(x)\) is Chebyshev's second function.

Adrian Dudek proved that the Riemann hypothesis implies that for \(x \geq 2\), there is a prime \(p\) satisfying

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

The constant \(4/\pi\) may be reduced to \(1+\varepsilon\) provided that \(x\) is taken to be sufficiently large. This is an explicit version of a theorem of Cramér.

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

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

for every positive ε is equivalent to the Riemann hypothesis (J. E. Littlewood, 1912; see for instance: paragraph 14.25 in Titchmarsh (1986)). 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, Edward Charles Titchmarsh, Helmut Maier and Hugh Montgomery, and Kannan Soundararajan. Soundararajan's result is that, conditional on the Riemann hypothesis,

\(M(x) = O\left(x^{1/2}\exp\left((\log x)^{1/2}(\log \log x)^{14}\right)\right).\)

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

\(|M(x)| \le \sqrt x.\)

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 \(o(n^{1/2+\epsilon})\) for all \(\epsilon>0\) (see incidence algebra).

\(\sigma(n) = \sum_{d\mid n} d\)

\sigma(n) < e^\gamma n \log \log n\)

\sigma(n) < H_n + \log(H_n)e^{H_n}\)

\frac{n}{\varphi (n)}

\(\sum_{i=1}^m|F_n(i) - \tfrac{i}{m}| = O\left(n^{\frac{1}{2}+\epsilon}\right)\)

\(m = \sum_{i=1}^n\varphi(i)\)

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

Condensed: the full section is in Wikipedia.

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,

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

as t → \(\infty\).

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

\(e^\gamma\le \limsup_{t\rightarrow +\infty}\frac{|\zeta(1+it)|}{\log\log t}\le 2e^\gamma\)

\(\frac{6}{\pi^2}e^\gamma\le \limsup_{t\rightarrow +\infty}\frac{1/|\zeta(1+it)|}{\log\log t}\le \frac{12}{\pi^2}e^\gamma\)

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

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)), which, while larger than the average gap, is far smaller than the bound implied by the Riemann hypothesis. Numerical evidence supports Cramér's conjecture.

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

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

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

\(f(x) = \sum_{\nu=1}^nc_\nu\rho \left(\frac{\theta_\nu}{x} \right)\)

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

\(\sum_{\nu=1}^nc_\nu\theta_\nu=0,\)

is dense in the Hilbert space L(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(0,1). This Nyman-Beurling criterion was strengthened by Baez-Duarte to the case where \(\theta_\nu \in \{1/k\}_{k\geq 1}\).

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

\(\int_0^\infty\frac{z^{-\sigma-1}\varphi(z)}{{e^{x/z}}+1}\,dz=0\)

\(H(\lambda, z):=\int_{0}^{\infty} e^{\lambda u^{2}} \Phi(u) \cos (z u)\, d u\),

\(\Phi(u) = \sum_{n=1}^{\infty} (2\pi^2n^4e^{9u} - 3 \pi n^2 e^{5u} ) e^{-\pi n^2 e^{4u}}\),

Condensed: the full section is in Wikipedia.

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).

Condensed: the full section is in Wikipedia.

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

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.

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 most common 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. Since Dedekind zeta function for abelian extension of the rationals can be expressed as product of Dirichlet L-functions and only possible pole is in 1 for Riemann zeta function (thus no pole can cancel nontrivial zero), this version of Riemann hypothesis implies generalized Riemann hypothesis.

The Riemann hypothesis can also be extended to the L-functions of Hecke characters of number fields. Since Dirichlet L-functions are Hecke L-functions for finite characters, then this hypothesis directly implies generalized Riemann hypothesis. Dedekind zeta functions can be expressed as product of Hecke L-functions and only possible pole of Hecke L-function is at 1, then this version of Riemann Hypothesis implies also version for Dedekind zeta functions.

There are two approaches for extension of Riemann hypothesis that seem to be the most general. The grand Riemann hypothesis extends it to all Automorphic L-functions, such as Mellin transforms of Hecke eigenforms. The Riemann hypothesis for Selberg class extends it rather for functions satisfying some properties (at least conjecturaly satisfied by most functions usually called zeta functions or L-functions) than for functions defined by direct formula. Though it is expected that Selberg class should be equal to class of automorphic L-functions, and thus this approaches should be equivalent, this is important open problem itself and part of Langlands program.

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 \(\sqrt{q}\) 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.

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. 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 Pierre Deligne, but remains entirely unknown in characteristic zero.