Bound · version 2
Let N(T) count nontrivial zeros with 0 < Im ρ ≤ T, with multiplicity, and N₀ˢ(T) count simple zeros on Re ρ = 1/2. Then liminf as T → ∞ of N₀ˢ(T)/N(T) ≥ k₀ = 3/2 − cot(1/√2)/√2 = 0.6725007037… .
For every ε > 0 the proportion is eventually at least k₀ − ε; dyadic (T,2T] and cumulative (0,T] versions are available.
In Riemann hypothesis · publication v10
Simple-zero proportion estimates
The cited theorem establishes the lower asymptotic proportion k₀ of simple zeros on the critical line.
This proportion does not prove that every nontrivial zero is on the line.
Independent mathematical assessment of this argument remains separate from the statement’s attributed status.
More than 67.25% simple zeros on the line, asymptotically →Another proof of the same proportion
The source gives another proof of k₀ and further bounds for distinct zeros and related counting quantities.
The additional counting quantities have different numerators and must not be combined into one percentage.
Independent mathematical assessment of this argument remains separate from the statement’s attributed status.
More than 67.25% simple zeros on the line, asymptotically →Compare the quantity and hypotheses within each group. A finite check, conditional theorem and proposed improvement cover different scopes.
| Result | Value | Conditions |
|---|---|---|
| More than 67.25% simple zeros on the line, asymptotically → Proved Result date not recorded | ≥ k₀ = 0.6725007037… | Asymptotic liminf; denominator N(T) counts all zeros with multiplicity. |
In Riemann hypothesis · publication v13
Simple-zero proportion estimates
The cited theorem establishes the lower asymptotic proportion k₀ of simple zeros on the critical line.
This proportion does not prove that every nontrivial zero is on the line.
Proved: Scoped theorem in cited source ↗ · source dated Aug 19, 2026 · observed Oct 10, 2026. Only the exact statement, domain and revision cited by this Claim; this is not a resolution of classical RH or independent OpenMath expert assessment.
Independent mathematical assessment of this argument remains separate from the statement’s attributed status.
More than 67.25% simple zeros on the line, asymptotically →Another proof of the same proportion
The source gives another proof of k₀ and further bounds for distinct zeros and related counting quantities.
The additional counting quantities have different numerators and must not be combined into one percentage.
Proved: Scoped theorem in cited source ↗ · source dated Sep 8, 2026 · observed Oct 10, 2026. Only the exact statement, domain and revision cited by this Claim; this is not a resolution of classical RH or independent OpenMath expert assessment.
Independent mathematical assessment of this argument remains separate from the statement’s attributed status.
More than 67.25% simple zeros on the line, asymptotically →Compare the quantity and hypotheses within each group. A finite check, conditional theorem and proposed improvement cover different scopes.
| Result | Value | Conditions |
|---|---|---|
| More than 67.25% simple zeros on the line, asymptotically → Proved Selected source date: | ≥ k₀ = 0.6725007037… | Asymptotic liminf; denominator N(T) counts all zeros with multiplicity. |
Proved
Only the exact statement, domain and revision cited by this Claim; this is not a resolution of classical RH or independent OpenMath expert assessment.
Lamzouri · another proof and further proportions · Theorem 1.1; definitions of N, N₀ˢ in §1 →
Source dated Sep 8, 2026 · Observed Oct 10, 2026
Proved
Only the exact statement, domain and revision cited by this Claim; this is not a resolution of classical RH or independent OpenMath expert assessment.
Source dated Aug 19, 2026 · Observed Oct 10, 2026
In this problem · publication v10 · Selected context
Bound · Proved
Simple AND on-line zeros form a subset of on-line zeros. This lower bound does not eliminate all exceptions, or specify a universal finite cutoff.
Scope: An unconditional lower bound above 67.25% for simple zeros on the line; two cited proofs support one Claim.
Limit: This is a lower bound as height tends to infinity, not a finite-window promise or a percentage of RH solved.
Finite verification and asymptotic proportion address different regimes; their percentages and heights cannot be added.
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v13
Bound · Proved
Simple AND on-line zeros form a subset of on-line zeros. This lower bound does not eliminate all exceptions, or specify a universal finite cutoff.
Scope: An unconditional lower bound above 67.25% for simple zeros on the line; two cited proofs support one Claim.
Limit: This is a lower bound as height tends to infinity, not a finite-window promise or a percentage of RH solved.
Finite verification and asymptotic proportion address different regimes; their percentages and heights cannot be added.
Relationship not independently reviewed. This concerns its use in this problem.
An unconditional asymptotic lower bound k₀ = 3/2 − cot(1/√2)/√2 holds for simple critical-line zeros relative to total multiplicity.
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.
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.
All nontrivial zeros with positive imaginary part at most 3,000,175,332,800 lie on the critical line.
paraphrase · No upstream full text or code is republished by this record.
Dave Platt · Tim Trudgian · Published
Published in Bulletin of the London Mathematical Society, DOI 10.1112/blms.12460. Statements here use the pinned author manuscript, Theorem 1 and Corollary 2.
Source revision: arxiv-v1. Observed Oct 10, 2026.
The cited work does not report AI involvement; absence of attribution is not evidence of absence.
Source updated Apr 21, 2020.
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.
Proved
Only the exact statement, domain and revision cited by this Claim; this is not a resolution of classical RH or independent OpenMath expert assessment.
Source dated Sep 8, 2026 · Observed Oct 10, 2026 · Source passage →
Proved
Only the exact statement, domain and revision cited by this Claim; this is not a resolution of classical RH or independent OpenMath expert assessment.
Source dated Aug 19, 2026 · Observed Oct 10, 2026 · Source passage →