Claim from Claude (paper author)

${lim inf}_{T \to \infty} \frac{N_{0}^{*} (T, 2 T)}{N (T, 2 T)} \geq \frac{3}{2} - \frac{1}{\sqrt{2}} \cot (\frac{1}{\sqrt{2}}) = 0.67250 \dots$

Quoted from the source · Claude (paper author) · 2026-08-10, p.21

Author involvement

Compiled from Anthropic's public announcement, paper, technical note, and repository. Anthropic and the paper's author did not participate in or confirm this Significance record.

Record note · Significance · 2026-08-11

Summary · for readers

In one sentence: Claude's paper says at least about 67.25% of the nontrivial zeros of the Riemann zeta function are on the critical line, in a precise asymptotic sense. It does not claim to settle the Riemann hypothesis.

Reader summary · Significance · 2026-08-11

Start here · for reviewers

Main deduction

Start with Theorem D and trace the paper's passage from the finite Weil form calculation to the asymptotic lower bound.

Theorem D, p. 21; Sections 4–5 · Check the statement correspondence and the final constant before reading the full paper. · Significance’s interpretation · Significance editor · 2026-08-17

Risk points

The correspondence between the paper's Theorem D and the published Lean/comparator statements is an important independent reading target.
Theorem D and Lean repository README · Significance’s interpretation · Significance editor · 2026-08-17
The analytic argument's tail control and use of the unconditional prime-side estimate are the most substantial paper-reading targets.
Proposition 4.4 and the proof of Theorem D · Significance’s interpretation · Significance editor · 2026-08-17

Useful background

Basic familiarity with the Riemann zeta function, pair correlation, and Weil's explicit or Hermitian-form viewpoint.
Paper Sections 1–3 and the cited background results · Significance’s interpretation · Significance editor · 2026-08-17

Needs checking

Compare the informal Theorem D with the exact formal definitions and declaration used by the Lean artifact. — A successful build does not by itself establish statement correspondence.
Paper p. 21; comparator/Challenge.lean and Solution.lean · Significance’s interpretation · Significance editor · 2026-08-17
Suggest another focused check

Review activity

Independent reruns: 0
Math assessments: 0
Written reviews: 0
Open checks: 3

What Significance checked: Significance pinned and inspected Anthropic's paper, concise note, and Lean repository. The paper and note were downloaded and hashed; the repository was pinned to its public v1.0 commit.

Still open: Significance has not independently rebuilt the Lean project, run comparator, matched every formal definition to the paper, or assessed the analytic argument. Anthropic reports human examination and successful formal checks; those reports are not independent Significance receipts.

Careful wording

Anthropic published Claude's claimed unconditional 67.25% lower bound with a Lean artifact; Significance has pinned the sources but has not yet independently rebuilt or assessed them.

Wording · Significance · 2026-08-11

Copy for sharing

Copy and paste this summary.

Anthropic published Claude's claimed unconditional 67.25% lower bound with a
Lean artifact; Significance has pinned the sources but has not yet
independently rebuilt or assessed them.

Checked: Significance pinned and inspected Anthropic's paper, concise note,
and Lean repository. The paper and note were downloaded and hashed; the
repository was pinned to its public v1.0 commit.

Not checked: Significance has not independently rebuilt the Lean project,
run comparator, matched every formal definition to the paper, or assessed
the analytic argument. Anthropic reports human examination and successful
formal checks; those reports are not independent Significance receipts.

As of 2026-08-11T00:48:19Z (freshness: current)
Full record: https://hjyuh.github.io/significance/2026-anthropic-zeta-two-thirds/
Significance records evidence. It does not judge the mathematics.

Scope: N(T, 2T) counts nontrivial zeros of the Riemann zeta function in the height window with multiplicity; N₀*(T, 2T) counts distinct such zeros on the critical line. The displayed result is an asymptotic lower bound, not a statement about every finite height window. Significance’s interpretation · Significance editor · 2026-08-10, p.1
Excluded: They have no bearing on the Riemann hypothesis in either direction. The argument produces lower bounds only: it certifies that at least two thirds of the zeros are on the line, and says nothing about the remaining third. Quoted from the source · Claude (paper author) · 2026-08-10, p.5
Source: Claude, More Than Two Thirds of the Zeros of the Riemann Zeta Function Lie on the Critical Line · official-release-2026-08-10 · retrieved 2026-08-11T00:30:00Z
sha256 6792988e6cd0…

Companion: Anthropic's concise informal note for experts · retrieved 2026-08-11T00:30:00Z
sha256 45e0330ad379…

Evidence — 1 entry

Formal code published by the source
ev-anthropic-lean-artifact

Published by the source. Any independent rerun appears as a separate entry.

Anthropic published a Lean 4 companion repository at tag v1.0. Its README maps paper Theorem D to comparator declarations including montgomery_taylor_on_critical_line and reports Lean v4.33.0-rc2, Mathlib commit 51e6992efd06…, no hypotheses on the headline theorem, the three standard Lean axioms, and successful comparator runs. Significance inspected those files at the pinned commit but has not attached its own execution receipt.

Open link @ 3635e74826a4… · Significance’s interpretation · Significance editor · 2026-08-11 · View source

Plain-language explanation

The Riemann hypothesis predicts that every nontrivial zero of the zeta function lies on one vertical line. This paper does not establish that. It gives a lower-bound certificate: as higher and higher windows are examined, the claimed share of distinct zeros on that line is at least about 67.25%, after zeros are counted with multiplicity in the denominator. The other roughly one third are not claimed to lie off the line; this method simply does not account for them.

Explanation · Significance

Mathematical context

The paper replaces the positivity normally supplied by RH in Montgomery's pair-correlation argument with finite-dimensional linear algebra on a compression of Weil's Hermitian form. Theorem D optimizes the test window to obtain the Montgomery–Taylor constant 0.67250… . This is the paper's stated mechanism, not an assessment of its steps.

Importance

Drawing on extensive prior research by mathematicians over the past decades, it has increased this bound from 41.6% to 67.2%. Quoted from the source · Anthropic

Ralph Furman and Levent Alpöge studied the result in detail after the session, placed it in the context of the existing literature, checked the argument independently, and have taken responsibility for its communication. Quoted from the source · Claude (paper author)

Limits

This record does not issue a whole-paper correctness judgement.
Open invitations identify work not yet represented by a completed evidence entry.
No independent reproduction or mathematical assessment is represented unless an evidence entry above says otherwise.

Formal code published by the source