Bound · version 2
For fixed 3/4 ≤ σ < 1, let N(σ,T) count zeta zeros with Re ρ ≥ σ and |Im ρ| ≤ T, with multiplicity. Then N(σ,T) ≤ T^(15(1−σ)/(3+5σ)+o(1)) as T → ∞; combining classical estimates also gives T^(30(1−σ)/13+o(1)).
This selection states the theorem on a conservative subrange of its domain. The o(1) tends to zero with T, for fixed σ.
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.
Guth–Maynard · zero-density estimates · Theorem 1.2 and equation (1.4), p. 2 →
Source dated Apr 7, 2026 · Observed Oct 10, 2026
In this problem · publication v7 · Selected context
Bound · Proved
A zero-density estimate constrains how numerous zeros in a region can be. RH requires their absence at every Re ρ > 1/2.
Scope: The Guth–Maynard estimate improves quantitative bounds on possible zeros away from the critical line.
Limit: An upper bound on the number of zeros is not proof that there are none; the selected estimate leaves the strip just right of 1/2 untreated.
This counts zeros in a horizontal-position range, whereas the proportion bound counts simple zeros on the line.
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v13
Bound · Proved
A zero-density estimate constrains how numerous zeros in a region can be. RH requires their absence at every Re ρ > 1/2.
Scope: The Guth–Maynard estimate improves quantitative bounds on possible zeros away from the critical line.
Limit: An upper bound on the number of zeros is not proof that there are none; the selected estimate leaves the strip just right of 1/2 untreated.
This counts zeros in a horizontal-position range, whereas the proportion bound counts simple zeros on the line.
Relationship not independently reviewed. This concerns its use in this problem.
A new bound limits the number of zeros to the right of a fixed vertical line.
paraphrase · No upstream full text or code is republished by this record.
Larry Guth · James Maynard · Published
Published online 1 March 2026 in Annals of Mathematics 203, pp. 623–675, DOI 10.4007/annals.2026.203.2.6. The author manuscript was revised April 2026; the pinned theorem statement is cited here.
Source revision: arxiv-v2. Observed Oct 10, 2026.
The cited work does not report AI involvement; absence of attribution is not evidence of absence.
Source updated Apr 7, 2026.
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.
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 Apr 7, 2026 · Observed Oct 10, 2026 · Source passage →