Conjecture · version 3
Every nontrivial zero of the meromorphic continuation of has .
Every zero at every height; negative-even trivial zeros and the pole at 1 are excluded.
Open
Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.
Bombieri · official RH description · §I, PDF page 1 →
Observed Oct 10, 2026
In this problem · publication v12 · Selected context
Primary formulation · Open
The established equivalence joins two open formulations of the same classical question.
The asymptotic estimate has no specified effective constant here.
Open · version 3
Every nontrivial zero of the meromorphic continuation of has .
Scope: Every zero at every height; negative-even trivial zeros and the pole at 1 are excluded.
Open · version 2
π(x) = Li(x) + O(√x log x) as x → ∞.
Scope: An open target equivalent to classical RH, stronger than the prime number theorem.
Relationship statement · Proved · version 3
RH holds if and only if π(x) = Li(x) + O(√x log x) as x → ∞, where π counts primes and Li denotes the logarithmic integral.
Scope: An equivalent formulation of the same canonical question.
Relationship not independently reviewed.
Connection in Riemann hypothesis →RH gives Λ ≤ 0; the proved Λ ≥ 0 turns this into the target Λ = 0.
Λ ≤ 0.2 and a proposed smaller positive upper bound do not reach zero.
Open · version 3
Every nontrivial zero of the meromorphic continuation of has .
Scope: Every zero at every height; negative-even trivial zeros and the pole at 1 are excluded.
Open · version 2
Λ = 0 for the classical de Bruijn–Newman heat-flow family H_t and Riemann kernel defined in Rodgers–Tao §1.
Scope: A heat-flow target equivalent to classical RH; positive upper bounds leave this target open.
Relationship statement · Proved · version 3
For the classical de Bruijn–Newman constant, RH is equivalent to Λ = 0. Λ is defined by H_t(z) = ∫₀∞ exp(tu²) Φ(u) cos(zu) du, using the classical Riemann kernel Φ and normalization in the cited paper; all H_t zeros are real exactly when t ≥ Λ.
Scope: The equivalence is a theorem; whether the constant is zero remains the unresolved target.
Relationship not independently reviewed.
Connection in Riemann hypothesis →Dirichlet GRH includes classical ζ as the conductor-one case.
The direction is GRH → RH. No converse is asserted.
Open · version 2
For every primitive Dirichlet character χ, each nontrivial zero of L(s,χ) in 0 < Re(s) < 1 has real part 1/2.
Scope: The Dirichlet version of GRH, a broader family than classical ζ.
Open · version 3
Every nontrivial zero of the meromorphic continuation of has .
Scope: Every zero at every height; negative-even trivial zeros and the pole at 1 are excluded.
Relationship statement · Proved · version 2
If Dirichlet GRH holds for every primitive character, including the trivial primitive character of conductor 1, then classical RH holds.
Scope: A one-way specialization; a proof of classical RH supplies no proof for every other character.
Relationship not independently reviewed.
Connection in Riemann hypothesis →The finite-field curve theorem places Frobenius eigenvalues on the expected circle, paralleling a zero-location assertion.
Its function field, cohomology and variable differ; it is not proved classical RH.
Open · version 3
Every nontrivial zero of the meromorphic continuation of has .
Scope: Every zero at every height; negative-even trivial zeros and the pole at 1 are excluded.
Proved · version 2
For a smooth projective geometrically connected curve C/F_q of genus g, Z_C(T) = P_C(T)/((1−T)(1−qT)), with P_C(T)=∏_{j=1}^{2g}(1−α_j T) and |α_j|=√q under every complex embedding.
Scope: Weil’s RH analogue for curves over finite fields.
Relationship not independently reviewed.
Connection in Riemann hypothesis →Line membership and multiplicity are distinct questions. A zero can be on the line and repeated.
The simple-and-on-line proportion theorem addresses an intersection, not all-zero simplicity.
Open · version 3
Every nontrivial zero of the meromorphic continuation of has .
Scope: Every zero at every height; negative-even trivial zeros and the pole at 1 are excluded.
Open · version 2
Every nontrivial zero of ζ(s) has multiplicity one.
Scope: A multiplicity assertion, separate from the horizontal-position assertion of RH.
Relationship not independently reviewed.
Connection in Riemann hypothesis →Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v13
Primary formulation · Open
The established equivalence joins two open formulations of the same classical question.
The asymptotic estimate has no specified effective constant here.
Open · version 3
Every nontrivial zero of the meromorphic continuation of has .
Scope: Every zero at every height; negative-even trivial zeros and the pole at 1 are excluded.
Open · version 2
π(x) = Li(x) + O(√x log x) as x → ∞.
Scope: An open target equivalent to classical RH, stronger than the prime number theorem.
Relationship statement · Proved · version 3
RH holds if and only if π(x) = Li(x) + O(√x log x) as x → ∞, where π counts primes and Li denotes the logarithmic integral.
Scope: An equivalent formulation of the same canonical question.
Relationship not independently reviewed.
Connection in Riemann hypothesis →RH gives Λ ≤ 0; the proved Λ ≥ 0 turns this into the target Λ = 0.
Λ ≤ 0.2 and a proposed smaller positive upper bound do not reach zero.
Open · version 3
Every nontrivial zero of the meromorphic continuation of has .
Scope: Every zero at every height; negative-even trivial zeros and the pole at 1 are excluded.
Open · version 2
Λ = 0 for the classical de Bruijn–Newman heat-flow family H_t and Riemann kernel defined in Rodgers–Tao §1.
Scope: A heat-flow target equivalent to classical RH; positive upper bounds leave this target open.
Relationship statement · Proved · version 3
For the classical de Bruijn–Newman constant, RH is equivalent to Λ = 0. Λ is defined by H_t(z) = ∫₀∞ exp(tu²) Φ(u) cos(zu) du, using the classical Riemann kernel Φ and normalization in the cited paper; all H_t zeros are real exactly when t ≥ Λ.
Scope: The equivalence is a theorem; whether the constant is zero remains the unresolved target.
Relationship not independently reviewed.
Connection in Riemann hypothesis →Dirichlet GRH includes classical ζ as the conductor-one case.
The direction is GRH → RH. No converse is asserted.
Open · version 2
For every primitive Dirichlet character χ, each nontrivial zero of L(s,χ) in 0 < Re(s) < 1 has real part 1/2.
Scope: The Dirichlet version of GRH, a broader family than classical ζ.
Open · version 3
Every nontrivial zero of the meromorphic continuation of has .
Scope: Every zero at every height; negative-even trivial zeros and the pole at 1 are excluded.
Relationship statement · Proved · version 2
If Dirichlet GRH holds for every primitive character, including the trivial primitive character of conductor 1, then classical RH holds.
Scope: A one-way specialization; a proof of classical RH supplies no proof for every other character.
Relationship not independently reviewed.
Connection in Riemann hypothesis →The finite-field curve theorem places Frobenius eigenvalues on the expected circle, paralleling a zero-location assertion.
Its function field, cohomology and variable differ; it is not proved classical RH.
Open · version 3
Every nontrivial zero of the meromorphic continuation of has .
Scope: Every zero at every height; negative-even trivial zeros and the pole at 1 are excluded.
Proved · version 2
For a smooth projective geometrically connected curve C/F_q of genus g, Z_C(T) = P_C(T)/((1−T)(1−qT)), with P_C(T)=∏_{j=1}^{2g}(1−α_j T) and |α_j|=√q under every complex embedding.
Scope: Weil’s RH analogue for curves over finite fields.
Relationship not independently reviewed.
Connection in Riemann hypothesis →Line membership and multiplicity are distinct questions. A zero can be on the line and repeated.
The simple-and-on-line proportion theorem addresses an intersection, not all-zero simplicity.
Open · version 3
Every nontrivial zero of the meromorphic continuation of has .
Scope: Every zero at every height; negative-even trivial zeros and the pole at 1 are excluded.
Open · version 2
Every nontrivial zero of ζ(s) has multiplicity one.
Scope: A multiplicity assertion, separate from the horizontal-position assertion of RH.
Relationship not independently reviewed.
Connection in Riemann hypothesis →Relationship not independently reviewed. This concerns its use in this problem.
Do all nontrivial zeros of the Riemann zeta function have real part 1/2?
paraphrase · No upstream full text or code is republished by this record.
RH is equivalent to the square-root-scale error term for counting primes.
paraphrase · No upstream full text or code is republished by this record.
The finite-field curve analogue has been proved.
paraphrase · No upstream full text or code is republished by this record.
Enrico Bombieri · Published
Official Clay exposition. Historical results and approaches must not be read as current record bounds.
Source revision: official-description. Observed Oct 10, 2026.
The cited work does not report AI involvement; absence of attribution is not evidence of absence.
Official description PDF: official-description. External source evidence; no OpenMath execution or independent reproduction is attached.
The corresponding all-zeros-on-line conjecture for Dirichlet L-functions is a broader target.
paraphrase · No upstream full text or code is republished by this record.
Levent Alpöge · Ralph Furman · Preprint
August 2026 preprint. Named mathematical authors explicitly state that they verified the proof and take responsibility. Journal acceptance and independent OpenMath expert review are not established.
Source revision: arxiv-v2. Observed Oct 10, 2026.
The paper credits Claude (Anthropic) with discovery and writing; Alpöge and Furman with verification, Jarred Sumner with guiding the investigation, and Eric Easley with orchestrating Lean formalization. Company attribution is source-reported, not independently reproduced here.
Source updated Aug 19, 2026.
Lamzouri gives another proof of the same k₀ bound.
paraphrase · No upstream full text or code is republished by this record.
Simplicity of every nontrivial zeta zero is a separate conjecture.
paraphrase · No upstream full text or code is republished by this record.
Youness Lamzouri · Preprint
Author preprint at the cited revision; journal acceptance has not been established by this curation.
Source revision: arxiv-v2. Observed Oct 10, 2026.
Lamzouri gives a human-authored proof inspired by the Claude result. Appendix A reports AxiomProver (Axiom Math) autoformalization; this is a different role from discovering the original bound.
Source updated Sep 8, 2026.
After the nonnegativity theorem, RH is equivalent to Λ = 0.
paraphrase · No upstream full text or code is republished by this record.
Brad Rodgers · Terence Tao · Published
Published in Forum of Mathematics, Pi (2020). The pinned 2021 author revision records a correction to Section 8; earlier manuscript history is not silently substituted.
Source revision: arxiv-v5. Observed Oct 10, 2026.
The cited work does not report AI involvement; absence of attribution is not evidence of absence.
Source updated Jul 3, 2021.
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Open
Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.
Observed Oct 10, 2026 · Source passage →