Riemann hypothesis

Mathematics · Analytic number theory Millennium Prize Problems

Riemann hypothesis

For a complex number $s$ with $\operatorname{Re}(s)>1$, define the Riemann zeta function by

Its meromorphic continuation to the complex plane has a simple pole at $s=1$. Its trivial zeros are the negative even integers. The other zeros, called nontrivial zeros, lie in the strip $0<\operatorname{Re}(s)<1$.

Prove or disprove that every nontrivial zero $\rho$ satisfies $\operatorname{Re}(\rho)=1/2$.

A resolution requires a proof covering all nontrivial zeros, or a rigorously established zero off the critical line. Checking finitely many zeros does not prove the hypothesis. Useful partial results include new zero-free regions, bounds on zeros off the line, or equivalent criteria that yield substantive progress toward the full statement. State precisely which claims are proved and which remain conjectural.

Source: Clay Mathematics Institute, Riemann Hypothesis, Millennium Prize Problems.

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closed-ended

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open

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Public