OP-0001OpenNumber theory$1,000,000 prize

The Riemann hypothesis

Do all non-trivial zeros of the Riemann zeta function lie on the critical line?

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Problem statement

Let $\zeta(s)$ denote the analytic continuation of $\sum_{n\ge 1} n^{-s}$ to $\mathbb{C}\setminus\{1\}$ . The function has trivial zeros at the negative even integers. The hypothesis asserts that every other zero $\rho$ satisfies

\operatorname{Re}(\rho) = \tfrac{1}{2}.

Equivalently, for every $\varepsilon > 0$ ,

\pi(x) = \operatorname{li}(x) + O\!\left(x^{1/2+\varepsilon}\right),

where $\pi(x)$ counts primes up to $x$ and $\operatorname{li}$ is the logarithmic integral.

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Background

Riemann introduced the hypothesis in his 1859 memoir on the distribution of primes, where it appears almost in passing as a remark he could not prove and did not need for his main results. It is the eighth of Hilbert's problems and the first of Smale's, and a Clay Millennium Prize problem.

Why it matters

Hundreds of published theorems are stated conditionally on the Riemann hypothesis or its generalisations. It controls the error term in the prime number theorem, and through the generalised hypothesis it governs the distribution of primes in arithmetic progressions, class numbers, and the running time of several number-theoretic algorithms.

Known partial results

The zeros are known to lie in the critical strip $0 < \operatorname{Re}(s) < 1$ , and a positive proportion (at least $41\%$ , by work of Conrey, Bui–Conrey–Young and successors) lie exactly on the critical line.
Over $10^{13}$ zeros have been verified numerically to lie on the line.
The function-field analogue was proved by Weil, and the higher-dimensional analogue by Deligne — which shows the statement is not merely a numerical accident, while offering no route to the classical case.

References

[1]B. Riemann, Über die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsberichte der Berliner Akademie, 1859.
[2]Clay Mathematics Institute, official problem description link
[3]Wikipedia contributors — Riemann hypothesis, mathematics problem list (inspected 2026-09-09) link

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