A staff member at Anthropic asked an unreleased research version of Anthropic's artificial intelligence (AI) model Claude to tackle the Riemann hypothesis, one of the most famous unsolved problems in mathematics. The hypothesis, first stated in 1859, concerns the Riemann zeta function, a mathematical object that is strongly related to the distribution of prime numbers. It asserts that all the relevant zeros of this function lie on a specific vertical line known as the critical line. Claude did not prove the hypothesis. While working on it, however, the model improved a related result that mathematicians have studied for decades.
Mathematicians have long sought to establish a firm lower bound on the proportion of zeros that must lie on the critical line. The previous best figure stood at 41.6 percent. Claude raised that figure to 67.2 percent. The improvement rests on combining recent work by several mathematicians with an earlier paper from the year 2000. The model treated a space of functions in a more unified way than earlier approaches, allowing both zeros on the line and zeros off the line to be considered together.
How the result was obtained
The discovery emerged over two extended sessions. Claude coordinated roughly sixty subagents that together ran thousands of commands, wrote hundreds of short programs, performed numerical checks against known zeros, and reviewed one another’s arguments. After arriving at the new bound, the model searched the literature to confirm the result was original, re-derived the argument independently, and wrote the findings as a paper. It also produced a version of the proof in Lean, a formal system that checks mathematical arguments for correctness.
Two mathematicians at Anthropic examined the work and prepared a concise informal note. Outside experts in the field reviewed the paper on short notice. The techniques used are not expected to lead to a full proof of the Riemann hypothesis itself. The episode nevertheless illustrates the growing ability of advanced AI models to extend existing mathematical ideas in unexpected directions.