Anthropic says Claude nearly doubled a key Riemann hypothesis bound

An unreleased research version of Anthropic's Claude AI has produced the largest single improvement to a key mathematical bound related to the Riemann hypothesis, raising the proven lower bound for the proportion of nontrivial zeros of the Riemann zeta function on the critical line from 41.6% to 67.2%.

Anthropic announced the result on August 10, describing it as an unintended byproduct of an attempt to solve the Riemann hypothesis itself — a problem unsolved since 1859 that carries a $1 million bounty from the Clay Mathematics Institute.anthropic

How It Happened

The experiment began when Anthropic staff member Jarred Sumner, a non-mathematician, prompted Claude to "take a real stab" at the Riemann hypothesis and left all mathematical decisions to the model. Claude failed at that task but stumbled onto something else.anthropic

Over two sessions in Claude Code, the model generated 31 million output tokens. An initial attempt produced roughly 650 failed ideas. Claude then coordinated approximately 60 subagents over a day and a half, executing 2,400 shell commands and writing hundreds of Python scripts. The subagents ran thousands of numerical checks against known zeta zeros and reviewed one another's work.theaiinsider+1

Sumner's involvement was largely limited to messages of encouragement — variants of "keep going" and "believe in yourself" — which Anthropic said helped Claude overcome initial skepticism that it could make progress on such a well-studied problem.anthropic

The Mathematical Advance

The result does not prove the Riemann hypothesis, which would require showing that 100% of nontrivial zeros lie on the critical line. Instead, Claude combined recent work by mathematicians Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh — who extended techniques introduced by Hugh Montgomery in 1973 — with a 2000 paper by Enrico Bombieri.theaiinsider+1

The previous 41.6% bound represented decades of incremental progress by human researchers. Claude nearly doubled that floor in a single step. Anthropic's technical description notes that a key insight was treating zeros on and off the critical line within a unified mathematical framework rather than separately.anthropic+1

Verification and Context

Two Anthropic mathematicians, Levent Alpöge and Ralph Furman, examined the work. External number theorists Brian Conrey and Dan Goldston also reviewed the paper on short notice. Claude additionally produced a formal proof in Lean that passes standard verification.neowin+1

Anthropic published Claude's paper, detailed process transcripts, and a GitHub repository containing the Lean formalization. The company cautioned that Claude's techniques are not expected to lead to a proof of the full Riemann hypothesis.anthropic+1

The result adds to an accelerating pattern of AI contributions to open mathematics problems. Earlier in 2026, an Anthropic researcher used a newer Claude model to help disprove the Jacobian conjecture, another longstanding open problem. Anthropic framed the finding less as a solved prize problem and more as evidence that AI systems are beginning to extend existing mathematical research rather than merely reproduce known results.officechai+1