Anthropic's Claude Pushes a 160-Year-Old Math Boundary From 41.6% to 67.2%

The Riemann hypothesis is arguably the most famous unsolved problem in mathematics. Posed in 1859, it carries a $1 million Millennium Prize bounty and sits at the heart of how prime numbers are distributed. An Anthropic staff member recently handed it to an unreleased research version of Claude with a simple prompt: take a real stab at it. Claude didn't solve it. What happened instead is worth paying close attention to.

The problem, in plain terms

The Riemann zeta function is a mathematical object that encodes the distribution of prime numbers. It has infinitely many "non-trivial zeros" — points where the function equals zero — and the Riemann hypothesis claims that every one of those zeros lies on a specific vertical line in the complex plane, called the critical line. Nobody has proven or disproven this in over 160 years.

Because a full proof remains out of reach, mathematicians have focused on a more tractable question: what fraction of those zeros can we prove lie on the critical line? Over decades of work, that lower bound had been pushed to 41.6%. Claude moved it to 67.2%.

How Claude actually did it

An unreleased research version of Claude found the new lower bound across two sessions in Claude Code, using 31 million output tokens. The process was far from a single flash of insight.

• Jarred Sumner, an Anthropic staff member and non-mathematician, prompted Claude to "take a real stab" at the hypothesis, leaving the mathematical choices to the model.
• Claude generated and tried 650 ideas. None worked.
• After being prompted to try again, it spent a day and a half coordinating roughly 60 Claude subagents, which collectively ran 2,400 shell commands and wrote hundreds of Python scripts.
• Of those 60 subagents: two developed the key mathematical ideas, 13 contributed supporting ideas to those agents, 30 attempted but failed to generate new ideas, 13 acted as validators checking the correctness of arguments, and the final two helped write the initial paper.

The subagents ran thousands of numerical checks against known zeta zeros and refereed one another's work. Jarred's input throughout was mostly encouragement — variants of "keep going" or "believe in yourself." Those prompts appear to have materially helped Claude push past its own skepticism about whether it could contribute to an open problem.

The math behind the result

Claude's result draws heavily on recent work by mathematicians Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh, who published a series of papers allowing Montgomery's 1973 techniques to work without assuming the Riemann hypothesis is true. That's a crucial distinction: earlier methods for studying the distribution of zeros on the critical line required assuming the very thing you were trying to prove.

The key conceptual move, as described in Anthropic's writeup: Claude forms a suitable space of functions with a quadratic form, and positive- and negative-definite subspaces arising from zeros on and off the critical line. It then writes down an inequality on the rank of a quadratic form in terms of first- and second-moment information. The willingness to treat the entire space together — with positive- and negative-definiteness taken into account simultaneously, and with the quadratic form allowed to be non-diagonal — is what allows Claude to reach its conclusion.

In plainer terms: instead of analyzing zeros on the line and zeros off the line separately, as previous approaches did, Claude treated the full space of zeros as a single geometric object and extracted a stronger inequality from that unified view. The result is a structural insight, not a computational brute-force outcome.

Validation and formal proof

Two mathematicians at Anthropic, Levent Alpoge and Ralph Furman, examined Claude's work to understand the new results and their relationship to prior work. In parallel, Claude worked with another staff member to produce a Lean formalization of the result, which passes the standard validation tool comparator. Lean is a proof assistant language that mechanically checks whether a mathematical argument is logically valid — as close to a machine-verified certificate of correctness as mathematics currently has.

Claude also volunteered to write its findings up as a paper and recommended that a human number theorist validate them. Brian Conrey and Dan Goldston, two experts in the field, examined the paper on short notice. The full paper, an informal note for experts, and the Lean formalization are all publicly available.

Before submitting, Claude had various subagents review the proofs, search for counterexamples, download 54 papers from arXiv to confirm the finding hadn't already been made, and independently re-derive the result from scratch — exactly what a careful human mathematician would do.

What this result is and isn't

Anthropic does not expect that the techniques Claude used will lead to proving the Riemann hypothesis. The jump from "67.2% of zeros are on the line" to "100% of zeros are on the line" requires fundamentally different machinery. But the significance of what happened here doesn't hinge on that gap.

Claude made a genuine, novel contribution to a problem that has occupied professional mathematicians for decades — as a byproduct of being asked to do something it couldn't do. Claude was skeptical of its own finding at first, likely because its training reflects both the difficulty of open problems in mathematics and the known limitations of AI models.

A pattern forming fast

This result doesn't exist in isolation. Since mid-2025, when AI solved five of six problems at the International Mathematical Olympiad for the first time, the pace at which open problems have fallen has accelerated. In May 2026, an OpenAI model disproved the 80-year-old Erdős conjecture in combinatorial geometry. Just weeks before this Riemann result, Levent Alpoge used Claude Fable 5 to generate a counterexample disproving the Jacobian conjecture — a problem open since 1939. The counterexample is almost comically compact: a 216-character polynomial map in three variables.

The emerging pattern is AI as a tireless collaborator that can explore thousands of dead ends, synthesize large bodies of prior literature, and occasionally land on a structural insight that humans hadn't tried. These results show how AI can extend the reach of mathematicians' ideas in directions that weren't previously tractable.

The Lean formalization carries particular weight for the research community. A machine-verifiable proof means the result doesn't depend on anyone trusting Claude's reasoning — it can be checked mechanically, the same way a compiler checks code. As AI systems produce more mathematical results, that kind of formal verification infrastructure becomes the difference between a curiosity and a contribution the field can actually build on.