A substantial step on the Riemann hypothesis, with a precise limit

The Riemann hypothesis says that all the relevant zeros of the zeta function lie on a particular vertical line. Mathematicians had established that at least 41.6% do. Anthropic reports that Claude improved that guaranteed fraction to 67.2% by connecting existing research in a new way.

This took two sessions and 31 million output tokens. An initial sweep of 650 ideas failed; a second attempt coordinated roughly 60 agents, including agents checking arguments and looking for counterexamples.

Two Anthropic mathematicians validated the work, and outside experts Brian Conrey and Dan Goldston examined the paper. The team also produced a Lean formalization that passes its validation tool.

The company does not expect this particular approach to prove the full hypothesis. A substantial partial result remains a substantial partial result: the gain is new mathematics built on earlier human work, accompanied by unusually inspectable evidence.