Claude improved a proven bound on the Riemann hypothesis

Anthropic says an unreleased research version of Claude improved a longstanding result tied to the Riemann hypothesis. Working through Claude Code, the model raised the proven lower bound for the fraction of the Riemann zeta function's non-trivial zeros that sit on the critical line, from 41.6% to 67.2%. It did not solve the hypothesis, the century-old problem that carries a million-dollar prize. It extended what can be proven about it, building on recent work by Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh along with earlier results from Bombieri.

The write-up is specific about the process. Claude used 31 million output tokens across two sessions, coordinated about 60 subagents over roughly a day and a half, ran 2,400 shell commands and hundreds of Python scripts, and generated and tested 650 approaches before narrowing them down. It validated results numerically against known zeta zeros and searched for counterexamples. Two Anthropic mathematicians checked the work, and Claude produced a formally verifiable proof in Lean, reviewed by mathematicians Brian Conrey and Dan Goldston.

What separates this from a demo is the verification layer. The Lean proof can be checked by machine, and the review by senior mathematicians gives the result a weight that model output usually lacks.

Why it matters

If you work in mathematics or another proof-heavy field, this is a concrete workflow you can copy: run many attempts with an agent, then formalize the result in Lean so it can be checked independently. That Lean step is the part to read first before trusting any proof a model hands you.