Math Conjectures

Riemann Hypothesis

OPENIconicConjectureProposed 1859 · Full conjecture

Canonical statement

Let
ζ(s)=∑n=1∞n−s(ℜs>1) \zeta(s)=\sum_{n=1}^{\infty}n^{-s}\qquad(\Re s>1)
and let the same symbol denote its meromorphic continuation to C\mathbb C, whose only pole is the simple pole at s=1s=1. Every zero ρ\rho of ζ\zeta with 0<ℜρ<10<\Re\rho<1 satisfies ℜρ=12\Re\rho=\tfrac12.
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Let
\[
  \zeta(s)=\sum_{n=1}^{\infty}n^{-s}\qquad(\Re s>1)
\] and let the same symbol denote its meromorphic continuation to \(\mathbb C\), whose only pole is the simple pole at \(s=1\). Every zero \(\rho\) of \(\zeta\) with \(0<\Re\rho<1\) satisfies \(\Re\rho=\tfrac12\).
NT-AN-001:[Riemann1859][Bombieri2006RH][Conrey2003RH]
Number theoryAnalysis and functional analysis

Notes

The Riemann Hypothesis concerns the zeros of the Riemann zeta function ζ(s)\zeta(s), defined by the Dirichlet series ∑n≥1n−s\sum_{n\ge1} n^{-s} for ℜs>1\Re s>1 and continued meromorphically to the whole plane, with a single simple pole at s=1s=1. It asserts that every zero ρ\rho with 0<ℜρ<10<\Re\rho<1 lies on the critical line ℜρ=12\Re\rho=\tfrac12. Riemann proposed it, almost in passing, in his 1859 memoir on the number of primes below a given magnitude [Riemann1859].

Much is known short of the conjecture itself. The classical zero-free region confines all nontrivial zeros strictly to the interior of the critical strip, which already yields the prime number theorem, and it has long been known that infinitely many zeros do lie on the critical line; extensive computation is consistent with the conjecture. Modern surveys of the analytic and arithmetic landscape are given by Conrey [Conrey2003RH] and Bombieri [Bombieri2006RH].

The problem remains open: no argument places every nontrivial zero on the line, and a proof would have to rule out even a single zero off ℜs=12\Re s=\tfrac12.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.