Anthropic announced today (the 11th) that an undisclosed research version of Claude has made important progress in tackling the Riemann Hypothesis.
In a test lasting several days and completed autonomously by the model, Claude raised the long-standing lower bound for the proportion of zeros of the Riemann ζ function on the critical line from 41.6% to 67.2%.
Claude conducted the research in Claude Code, generating 31 million output tokens across two tests. After 650 failed attempts, the model deployed 60 sub-agents to continue the work, executing a total of 2,400 Shell commands and writing hundreds of Python scripts.
Anthropic then asked its in-house mathematicians Levente Arbog and Ralph Furman to verify the results, and had them reviewed by external number theory experts Brian Conrey and Dan Goldston. Employee Eric Isely also helped formalize the proof for verification in Lean.
The Riemann Hypothesis was first proposed in 1859. The Clay Mathematics Institute offers a $1 million prize for solving the problem. (Note: At the current exchange rate, this is approximately 6.755 million yuan.)
Claude’s work combined existing research frameworks from mathematicians including Baluyot, Goldston, Suriajaya, Turnage-Butterbaugh, and Bombieri.
Anthropic also explained the specific method.
Claude constructed a suitable function space and introduced a quadratic form induced by Weil within it. The zeros on the critical line form a positive-definite subspace, while the zeros off the critical line form a negative-definite subspace. Claude then used first- and second-moment information to directly establish an inequality for the rank of the quadratic form. (Computing the latter second-moment information through a dual perspective over the primes, or by controlling the Hilbert transform, is not unusual in analytic number theory.)
The truly critical step, in a certain sense, was to handle the entire space uniformly while incorporating both the positive- and negative-definite parts and allowing the quadratic form to be non-diagonal. Building on important prior research, Claude reached the conclusion described above.
Anthropic also published a paper written by Claude, along with informal expert commentary, detailed process records, and a proof that can be formally verified through Lean. However, Anthropic did not disclose whether it plans to make the aforementioned multi-agent capabilities available to the public in the near future.
References
Claude’s paper
Claude’s formalization
Anthropic’s informal memorandum
Claude’s explanation of its results
Claude’s detailed process record