Open problemChecked July 24, 2026

Riemann Hypothesis

The best-known open question about the distribution of prime numbers.

Number theory Complex analysis Prime numbers

Mathematical statement

Every nontrivial zero $\rho$ of the Riemann zeta function satisfies $\operatorname{Re}(\rho)=\tfrac12$ .

Statement source: Therefore editorial statement

Statement terms: Source-specific

Editorial wording by Therefore. No statement reuse license is granted.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

_root_.RiemannHypothesis

Canonical source

Complete library proposition, no proof artifact attachedLean 4

def RiemannHypothesis : Prop :=  ∀ (s : ℂ) (_ : riemannZeta s = 0)    (_ : ¬∃ n : ℕ, s = -2 * (n + 1)) (_ : s ≠ 1),    s.re = 1 / 2

Statement source: Mathlib
Lean version: v4.27.0
Placeholder: Absent; this artifact only defines the proposition
Source evidence: Pinned source index
Fidelity review: Community formulation

Pinned Lean formulation 2

Millennium.clay_prize_riemann_hypothesis

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem clay_prize_riemann_hypothesis :    ClayRiemannHypothesis := by  sorry

Statement source: Lean Millennium Prize Problems
Lean version: v4.31.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Clay Mathematics Institute problem pageofficial Riemann hypothesis overviewbackground