Proportion of zeta zeros on the critical line raised to 67.25%
An unconditional proof that at least 67.25% of the nontrivial zeros of the Riemann zeta function are simple and lie on the critical line, up from a previous record of about 41.6%, with a machine-checked Lean formalization.
- Lab
- Anthropic
- Model
- Claude (unreleased research version)
- Field
- Mathematics
- Date
- 2026-08-10
- Human collaborators
- Jarred Sumner, Levent Alpöge, Ralph Furman, Eric Easley
Sources
Original work
Announcement
Independent commentary
What was found
The Riemann hypothesis asserts every nontrivial zero 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 roughly 41.6%. This work proves at least 3/2 - (1/sqrt 2)cot(1/sqrt 2) = 67.25...% of zeros are simple and on the line, and that at least 0.83625 of zeros are distinct. The decisive step combined 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. The process is itself the record worth noting: the model was asked to attempt the Riemann hypothesis outright, with the mathematical choices left to it, and the bound improvement fell out 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, across roughly 31 million output tokens in two Claude Code sessions. The model also proposed writing the result up, checked 54 arXiv papers for prior art, and recommended a human number theorist validate it.
Novelty check
The proportion ladder is a documented literature running from Hardy (1914) and Selberg (1942) through Levinson's one third (1974) and Conrey's two fifths (1989), with subsequent refinements reaching about 41.6%; the paper states the prior record and cites it. The model itself searched 54 arXiv papers for prior art before write-up. No earlier unconditional bound above 42% appears in the literature. The result is a new bound, not a retrieval of a known one.
Caveats and known objections
Not peer-reviewed. The Riemann hypothesis itself is untouched, and Anthropic states it does not expect these techniques to lead to a proof of it. The Lean formalization is sorry-free and anchors its statements on Mathlib's own riemannZeta and analyticOrderAt with the corresponding counting functions rather than an assumed form of the result, which is real statement anchoring; 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 both the proof and its formalization. Two Anthropic mathematicians studied and validated the work, which is author-side review, not independent replication. Autonomy graded ai-led rather than autonomous: humans posed the problem, supplied the environment and validated the output.
Nobody outside the lab has checked this yet.
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Entry history (1 event)
- AddedEntered the registry graded Formally verified and AI-led.
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Graded formally verified for verification and ai-led for autonomy. What these mean.
Cite this entry
whataifound.org. (2026). Proportion of zeta zeros on the critical line raised to 67.25%. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-08-10-zeta-zeros-critical-lineBibTeX
@misc{whataifound-anthropic-2026-line,
title = {Proportion of zeta zeros on the critical line raised to 67.25%},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Anthropic. Verification: Formally verified. Autonomy: AI-led.},
url = {https://whataifound.org/finding/2026-08-10-zeta-zeros-critical-line}
}