Anthropic staff set Claude an unreasonable challenge: take a serious attempt at the Riemann hypothesis, a famously unsolved problem from 1859 with a million-dollar bounty. Claude didn’t solve the hypothesis, but during the attempt it unexpectedly improved a related result. An unreleased research version of Claude increased the proven lower bound for the fraction of zeros of the Riemann zeta function that lie on the critical line from 41.6% to 67.2%.

The result draws on a series of recent papers by Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh, which allowed Montgomery’s 1973 techniques to be applied without assuming the Riemann hypothesis itself, combined with a 2000 paper by Bombieri. Claude constructed a suitable space of functions with a quadratic form induced by Weil, separated into positive-definite and negative-definite subspaces corresponding to zeros on and off the line. By writing an inequality on the rank of the quadratic form in terms of first- and second-moment information, it derived the improved bound.

Two mathematicians at Anthropic studied and validated Claude‘s paper and produced a concise informal note for experts. Claude also produced a formally verifiable proof in Lean. External experts Brian Conrey and Dan Goldston generously examined the paper on short notice.

The work was carried out over two sessions in Claude Code using a total of 31 million output tokens. A non-mathematician staff member prompted Claude to “take a real stab” at the hypothesis, then left the mathematical choices to the model. Claude initially generated and tried 650 ideas that didn’t work, then spent a day and a half coordinating about 60 subagents that ran 2,400 shell commands, wrote hundreds of Python scripts, performed thousands of numerical checks against known zeta zeros, and refereed one another’s work. The human input was mostly encouragement.

Claude tested its own work by having subagents review proofs, search for counterexamples, download 54 arXiv papers to check the result wasn’t already known, and independently re-prove the finding from scratch. Claude volunteered to write up its findings as a paper and recommended human validation.

The authors note they do not expect the techniques used will lead to proving the Riemann hypothesis itself. The result serves as an example of progress in AI mathematical capabilities, and an unintended byproduct of the original request.