More than 67.25% simple zeros on the line, asymptotically
Statement
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… .
Assumptions
Classical ζ; no RH assumption.
Asymptotic proportions as T → ∞; the denominator counts multiplicity.
Scope
For every ε > 0 the proportion is eventually at least k₀ − ε; dyadic (T,2T] and cumulative (0,T] versions are available.
Explanation
Proof outlines · 2
Alpöge–Furman account
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.
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.
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.
Revision & attribution
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.
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.
Revision & attribution
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.
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.
Revision & attribution
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.
History
Statement versions
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessment history · 2
Assessments retained in these publication records. Earlier statement versions retain their own source histories.