The Proportion of Zeta Zeros on the Critical Line
The Riemann hypothesis asserts that every nontrivial zero of the zeta function lies on the critical line. Short of proving it, the standard measure of progress is the proportion of zeros known unconditionally to lie there: Selberg established a positive proportion, Levinson reached a third in 1974, Conrey two fifths in 1989, and the record had crept to about 41.6%. This proves at least of zeros are simple and on the line, and that at least 0.83625 of zeros are distinct.
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Analytic number theory
- Posed by
- Bernhard Riemann (1859) for the hypothesis; the proportion ladder runs from Hardy and Selberg through Levinson and Conrey
- Year posed
- —
- Years open
- —
- Solved
- 2026-08-10
- Model
- Claude (unreleased research version)
- Vendor
- Anthropic
- Collaborators
- Jarred Sumner, Levent Alpöge, Ralph Furman, Eric Easley
- Verification
- Lean-checked, statement unaudited
- Publication
- Preprint
- Significance
- 68 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
An unconditional record, not a resolution: the Riemann hypothesis is untouched, and Anthropic states it does not expect these techniques to lead to a proof of it. The improvement is large, from about 41.6% to 67.25%, and the constant is closed form.
What the AI did
Claude was asked to take a real stab at the Riemann hypothesis, with the mathematical choices left to it, and the bound improvement came out as a byproduct of failing at that. It generated and discarded roughly 650 ideas in a first session; in a second it coordinated about 60 subagents which ran some 2,400 shell commands, wrote hundreds of scripts, checked numerically against known zeros and refereed one another. Two subagents developed the key ideas, thirteen fed them, thirty tried and failed, thirteen validated, two drafted the paper. Roughly 31 million output tokens across two Claude Code sessions. The decisive step was combining the unconditional pair-correlation work of Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh with a 2000 paper of Bombieri, treating the whole function space at once with the quadratic form allowed to be non-diagonal rather than splitting it. Claude also proposed writing the result up, checked 54 arXiv papers for prior art, and recommended that a human number theorist validate it.
Verification
A sorry-free Lean 4 / Mathlib formalization accompanies the paper, and the headline statements are built from Mathlib's own riemannZeta and analyticOrderAt with the corresponding counting functions, rather than from an assumed form of the result. Curator check: the repository is real and substantial, 329 Lean files, Apache licensed to Anthropic. Held at the unaudited rung rather than promoted, for a specific reason. Building on Mathlib primitives is real statement anchoring and reduces drift, but the assembly of those primitives into the proportion claim is the paper's own, and that assembly is exactly what nobody independent has audited. The same organisation produced the proof and the formalization. On human review: two Anthropic mathematicians studied and validated the work, and Brian Conrey and Dan Goldston examined the paper on short notice. That is meaningful scrutiny but not the top verification rung, which asks for named experts with no stake who endorse the claim. The announcement reports examination rather than endorsement, and Goldston is an author of the prior work the result builds on.
Sources
Submitted by SwiftTapir695 on
Astonishing!