Unreleased Claude model makes progress on a problem linked to the Riemann hypothesis

An unreleased research version of Claude has made progress on a problem related to the Riemann hypothesis, one of the most famous unsolved problems in mathematics, Anthropic announced.

An Anthropic employee gave the model an ambitious task to “take a real stab” at proving the hypothesis itself, which has remained unsolved since 1859 and is one of the Millennium Prize Problems carrying a $1 million reward. Claude did not manage to prove the hypothesis, but its work unexpectedly produced progress on a related problem.

Claude improved a longstanding lower bound on the fraction of zeros of the Riemann zeta function that satisfy the Riemann hypothesis. Drawing on decades of previous mathematical research, the model increased the bound from 41.6% to 67.2%. The result was independently checked by two mathematicians at Anthropic as well as external experts in the field, Brian Conrey and Dan Goldston.

The work took two Claude Code sessions and 31 million output tokens. The model initially generated and tested 650 ideas, none of which worked. Claude then spent nearly two days coordinating around 60 subagents, which collectively ran 2,400 terminal commands and wrote hundreds of Python scripts to test results against known zeros of the zeta function. The Anthropic employee’s involvement was largely limited to encouragement such as “keep going” and “believe in yourself.”

Anthropic does not expect the methods Claude used to lead to a proof of the Riemann hypothesis itself, but says the result demonstrates how quickly AI models’ mathematical capabilities are advancing.

Durov’s Code has previously covered similar cases in which AI models unexpectedly made progress on open mathematical problems. OpenAI’s Astra system recently independently disproved a 1946 Erdős conjecture and solved ten other unsolved problems.