Conjecture Fieldbook

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Riemann hypothesis

Canonical problem dossier

Every non-trivial zero of the Riemann zeta function is conjectured to have real part 1/2.

Status
Open
Field
Number theory
Verification
Reviewed dossier
Source grade
mixed
Reviewed
2026-07-27T18:04:33.834Z
Reviewer
Conjecture Fieldbook maintainers

Formal expression

ζ(s)=∑n=1∞1ns=∏p prime11−p−sζ(ρ)=0⇒Re⁡(ρ)=12\begin{aligned}\zeta(s)&=\sum_{n=1}^{\infty}\frac{1}{n^s}=\prod_{p\ \mathrm{prime}}\frac{1}{1-p^{-s}}\\[6pt]\zeta(\rho)=0&\Rightarrow \operatorname{Re}(\rho)=\frac12\end{aligned}

Why it matters

The hypothesis controls how far the distribution of prime numbers can deviate from its average behavior. Many conditional results across analytic number theory become sharper if it is true.

Context and known approaches

The zeros of the analytically continued zeta function encode error terms in prime-counting estimates. The conjecture places every non-trivial zero on one vertical line in the complex plane.

Formal boundary

If ζ(s) = 0 and s is a non-trivial zero, then Re(s) = 1/2. Trivial zeros at the negative even integers are excluded.

Known partial results

  • Infinitely many non-trivial zeros are known to lie on the critical line; the dossier timeline links the supporting reference trail.

Equivalent formulations

  • Many equivalent statements are known; treat each equivalence as a separate theorem requiring its own source.

Source trail

  1. Clay Mathematics Institute — Official Millennium problem overview and resources.
  2. Wikipedia — General overview and reference trail.
  3. Wikipedia list subsection — Placement in the accepted revision-pinned living-list snapshot.

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