Two thirds of zeta zeros on the line

On 10 August 2026 Anthropic announced that an unreleased research version of Claude had proved unconditionally that at least two thirds of the nontrivial zeros of the Riemann zeta function are simple and lie on the critical line, against an earlier record of 5/12 (41.6%); with a refined window the paper gives 0.6725. The proof was checked by two Anthropic mathematicians and formalised in Lean; it does not prove the Riemann hypothesis.

State of the claim. Claimed: Anthropic’s post of 10 August (updated 13 August with a revised paper; the 10 August version of the paper was not read) and arXiv 2608.13637 by Levent Alpöge and Ralph Furman (v1 13 August, v2 19 August). Checked by people: the paper says the listed authors "verified the proof and take responsibility for its content"; Anthropic says Brian Conrey and Dan Goldston "generously examined the paper on short notice" (their own statement was not found); Youness Lamzouri’s arXiv 2609.02882 (2 September) gives "a new, conceptually simpler" proof of the 67.25% bound by his own route. Formalised in Lean: Appendix A of the paper — Lean 4.33.0-rc2, Mathlib 51e6992efd06, repository tag v1.0 of github.com/anthropics/zeta-23-lean (GitHub now redirects it to anthropics/formal-math); #print axioms returns only propext, Classical.choice and Quot.sound, with no sorry; the formalisation was orchestrated by Eric Easley. The pass read the paper and the repository’s metadata but did not build the proof. No dispute was found in what was read. Numbers. Theorem A in v1: at least 2/3 of the nontrivial zeros (counted with multiplicity) are simple and lie on the critical line, at least 5/6 are distinct; with the Montgomery–Taylor window 0.6725 and 0.8362. The previous unconditional records were 5/12 and 0.6603. Anthropic’s post says 67.2% and describes it as a share of zeros that "satisfy the Riemann hypothesis"; the paper makes this precise as simple zeros on the line. The model’s role. By Anthropic, a staff member who is not a mathematician (Jarred Sumner) asked Claude to "take a real stab" at the Riemann hypothesis; after 650 failed ideas the model coordinated about 60 subagents for a day and a half; 31 million output tokens; it recommended that a human number theorist validate the result. The paper says the argument "was discovered and written by Claude". What the record does not claim: that the Riemann hypothesis is proved. A later preprint reports a further improvement (arXiv 2609.33043; its author says one constant is unverified); it is not entered.