Anthropic gave an unreleased research version of Claude an unreasonable challenge: “take a real stab at the Riemann hypothesis.” It didn’t crack the 1859 problem — but as a byproduct, it improved a longstanding lower bound. The proven fraction of zeta zeros on the critical line rose from 41.6% to 67.2%.
The mechanics matter as much as the number:
- Two sessions in Claude Code, 31 million output tokens
- First 650 ideas failed; a second push ran ~60 subagents
- 2,400 shell commands and hundreds of Python scripts
- Subagents cross-refereed one another’s work and ran thousands of numerical checks against known zeros
Claude built on Montgomery’s pair-correlation techniques (made RH-free by Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh) plus a 2000 Bombieri paper. Its key step: treating the whole quadratic-form space — positive- and negative-definite subspaces together, non-diagonal forms allowed.
The result was validated end to end:
- Anthropic’s in-house mathematicians (Levent Alpöge, Ralph Furman)
- External experts Brian Conrey and Dan Goldston, examined on short notice
- A Lean formalization (github.com/anthropics/zeta-23-lean) passing the standard comparator
Claude even volunteered to write the paper and recommended human verification. Anthropic is clear the techniques won’t prove the full hypothesis — but a model producing a verified, novel mathematical advance as the byproduct of a failed attempt, with the loop closed by Lean and human experts, is the clearest capability signal yet for AI-driven research.