Bound · version 2
With N_d(T) counting distinct nontrivial zeros and N(T) counting total multiplicity in (0,T], liminf N_d(T)/N(T) ≥ (1+k₀)/2 = 0.8362503518…; k₀ = 3/2 − cot(1/√2)/√2.
A multiplicity bound, regardless of whether the distinct zeros lie on the line.
Compare the quantity and hypotheses within each group. A finite check, conditional theorem and proposed improvement cover different scopes.
| Result | Value | Conditions |
|---|---|---|
| At least 83.625% distinct zeros, asymptotically → Proved Selected source date: | ≥ 0.8362503518… | Asymptotic liminf; numerator counts distinct zeros, denominator counts 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.
Alpöge–Furman · simple zeros on the line · Theorem A(ii), Montgomery–Taylor refinement →
Source dated Aug 19, 2026 · Observed Oct 10, 2026
In this problem · publication v6 · Selected context
Supporting result · Proved
RH permits multiple zeros. This bound is additional knowledge about multiplicity, not an 83.625% on-line guarantee.
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v13
Supporting result · Proved
RH permits multiple zeros. This bound is additional knowledge about multiplicity, not an 83.625% on-line guarantee.
Relationship not independently reviewed. This concerns its use in this problem.
The proportion of distinct zeros relative to total multiplicity has liminf at least (1+k₀)/2.
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.
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 Aug 19, 2026 · Observed Oct 10, 2026 · Source passage →