On August 10, 2026 Anthropic reported that an unreleased research version of Claude improved a longstanding lower bound for the fraction of zeros of the Riemann zeta function that satisfy the Riemann hypothesis, raising it from 41.6 percent to 67.2 percent. The result was an unintended byproduct. Jarred Sumner, an Anthropic staff member who is not a mathematician, had simply prompted the model to “take a real stab” at the Riemann hypothesis itself. It did not solve the hypothesis, and Anthropic says it does not expect the techniques used to lead to a proof of it.
The mathematical route was to combine recent work by Aryan and by Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh, which lets Montgomery’s 1973 techniques operate without assuming the hypothesis, with a 2000 paper by Bombieri. Anthropic describes the model forming a space of functions with a quadratic form induced by Weil, with positive-definite and negative-definite subspaces arising from zeros on and off the critical line, then bounding the rank of that quadratic form using first- and second-moment information. Anthropic characterizes the key step as the willingness to treat the entire space at once, with positive- and negative-definiteness handled together and the quadratic form allowed to be non-diagonal.
The methodology figures are the part practitioners should read closely. The model found the bound across two sessions in Claude Code using 31 million output tokens in total. It first generated and tried 650 ideas, none of which worked. Prompted to try again, it spent a day and a half coordinating about 60 Claude subagents, which between them ran 2,400 shell commands and wrote hundreds of Python scripts, ran thousands of numerical checks against known zeta zeros, and refereed one another’s work. The human’s contribution during that stretch was mostly encouragement, variants of “keep going” and “believe in yourself,” which Anthropic says helped the model past its own initial skepticism.
Verification was layered. Subagents reviewed the proofs, searched for counterexamples, downloaded 54 arXiv papers to confirm the result was not already known, and independently re-proved it from scratch. The model volunteered to write the finding up as a paper and recommended that a human number theorist validate it. Two Anthropic mathematicians, Levent Alpoge and Ralph Furman, studied and validated the work and produced a short note stating the proof concisely; Brian Conrey and Dan Goldston, outside experts in the area, also examined the paper. Separately the model worked with Anthropic staff to produce a Lean formalization that passes the standard comparator validation tool.
The signal here is not that AI is closing in on the Riemann hypothesis, because Anthropic explicitly says it is not. The signal is the shape of the workflow: a non-expert prompt, a long-horizon multi-agent search costing tens of millions of tokens, machine-generated formal verification, and human experts brought in at the end to confirm rather than to direct. That is a template any research-heavy organization can copy, and it puts a price on a category of literature-synthesis work that previously required a specialist.