We asked an unreleased research version of Claude to take a stab at the Riemann

Claude did not prove the Riemann hypothesis. Anthropic reports that an unreleased research version of Claude, asked to attempt it, instead improved a related partial result: the proven minimum share of the zeta function's zeros that sit where the hypothesis says they should, raised from 41.6% to 67.2%. The Riemann zeta function is tied to how prime numbers are distributed, and the hypothesis, open since 1859, says that all of its relevant zeros lie on one vertical line. Many results in mathematics assume it. Proving that a fixed fraction of the zeros lies on that line is a way to make progress without settling the whole question, and Anthropic says it does not expect Claude's techniques to lead to a proof of the hypothesis itself.

The advance came from connecting existing work, not from a new theory. A 1973 method by the mathematician Montgomery could have pushed the fraction higher, but it only worked if the hypothesis was already assumed true. Recent papers by Farzad Aryan and by Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh made parts of that method work without the assumption. Anthropic says Claude combined those results with a 2000 paper by Bombieri to beat the previous record. Claude's paper states the result as at least two thirds of the zeros being simple and on the line, rising to 0.6725 with a refined choice in the argument, against a previous unconditional record of five twelfths.

Anthropic describes a large, mostly autonomous search. Jarred Sumner, a staff member who is not a mathematician, gave the open-ended prompt and afterwards mostly sent encouragement. Claude's first session tried 650 ideas and none worked. In a second session of about a day and a half, Claude coordinated about 60 helper agents that ran thousands of numerical checks against known zeros, reviewed one another's arguments, downloaded 54 arXiv papers to check the result was not already known, and re-proved it independently. The two sessions used 31 million output tokens in total.

The checking is stronger than an AI's own say-so, but it is still largely Anthropic's. Two of Anthropic's mathematicians, Levent Alpöge and Ralph Furman, studied and validated the paper. Claude and another staff member also produced a version of the proof in Lean, a language in which a computer checks every logical step, and Anthropic says it passes the standard checking tool. Anthropic thanks Brian Conrey and Dan Goldston, two experts in the field, for examining the paper, but the post does not report their conclusions. The post was updated on August 13 with a revised paper. Until the result appears in the peer-reviewed literature, it is best read as a well-checked claim from the company that built the model.