Recently, a staff member at Anthropic challenged Claude with a task on the Riemann hypothesis. The model attempted a full proof, but as expected, it did not succeed. During the process, however, it made significant progress on a related problem, improving the lower bound for the fraction of zeros of the Riemann zeta function that lie on the critical line.

Using prior work by Baluyot, Goldston, Suriajaya, Turnage‑Butterbaugh, and Bombieri, Claude raised the known bound from 41.6% to 67.2%. Two Anthropic mathematicians, Brian Conrey and Dan Goldston, reviewed the result and produced an informal note. The model also supplied a formally verifiable proof via Lean, which passed the standard validator.

The breakthrough came after two sessions in Claude Code, generating 31 million tokens and coordinating about 60 subagents that executed 2,400 shell commands and hundreds of Python scripts. Jarred Sumner’s encouraging prompts helped the model overcome initial skepticism. Claude’s unexpected success illustrates the speed with which AI models can extend human mathematical research, even if they do not solve the original problem.

The work demonstrates AI’s potential to push boundaries in number theory and encourages further collaboration between human experts and large language models.