Claude Boosts Riemann Hypothesis Zero Bound from 41.6% to 67.2%

Anthropic's unreleased research version of Claude has improved a long-standing lower bound for the proportion of Riemann zeta function zeros that lie on the critical line (where the Riemann hypothesis predicts all zeros are). The bound jumps from 41.6% to 67.2%, a leap that surprised the researchers. While the AI didn't prove the Riemann hypothesis itself, its result is a notable advance in analytic number theory.

The Breakthrough

Claude's key insight was combining results from Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh (works that adapt Montgomery's 1973 techniques without assuming the Riemann hypothesis) with a 2000 paper by Bombieri. In simple terms, Claude constructed a suitable space of functions with a quadratic form induced by Weil, and considered positive- and negative-definite subspaces corresponding to zeros on and off the line. By writing down an inequality on the rank of the quadratic form using first- and second-moment information, it achieved the new bound.

According to the Anthropic note, the approach treats the entire space together, with the quadratic form allowed to be non-diagonal — a step that enables the conclusion from prior work. Two Anthropic mathematicians validated the paper and produced an informal expert note, and external experts Brian Conrey and Dan Goldston also reviewed it on short notice.

How Claude Worked

The result came from two sessions in Claude Code, using a total of 31 million output tokens. Jarred Sumner, an Anthropic staff member (and non-mathematician), prompted Claude to “take a real stab” at the Riemann hypothesis itself, leaving the mathematical choices to the model. Initially, Claude generated 650 ideas that failed, prompting another run. In the second session, Claude coordinated about 60 subagents over a day and a half, running 2,400 shell commands and writing hundreds of Python scripts. The subagents performed thousands of numerical checks against known zeta zeros and peer-reviewed each other's work.

A Formally Verifiable Proof

Notably, Claude also produced a formally verifiable proof of its result, complementing the informal note for experts. This adds to the plausibility of the finding, though the team doesn't expect the techniques to crack the Riemann hypothesis itself.

For developers using AI agents, this is a concrete example of how far mathematical reasoning has come — and how a non-mathematician can steer an agent to make a genuine research contribution.