An unreleased research version of Anthropic's Claude has improved a longstanding lower bound in research related to the Riemann hypothesis — one of the most famous unsolved problems in mathematics — the company announced on August 10. Claude increased the proven lower bound on the fraction of zeros of the Riemann zeta function that lie on the critical line from 41.6% to 67.2%.
A 31-Million-Token Effort
Anthropic staff member Jarred Sumner prompted Claude to 'take a real stab' at the hypothesis itself. After an initial run that generated and discarded 650 ideas, a second session deployed about 60 Claude subagents over a day and a half, running 2,400 shell commands and writing hundreds of Python scripts. The model downloaded 54 papers from arXiv to check whether its finding already existed, had subagents referee one another's work, and independently re-proved the result from scratch.
Formal Verification and Human Review
Two mathematicians at Anthropic, Levent Alpöge and Ralph Furman, studied and validated Claude's paper, while Claude worked with staff member Eric Easley to produce a Lean formalization that passes the standard validation tool. Experts Brian Conrey and Dan Goldston — prominent researchers in the area — examined the work on short notice, Anthropic said.
Anthropic cautioned that the techniques are not expected to prove the Riemann hypothesis itself, but called the result the latest example of the speed of progress in AI models' mathematical capabilities.
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