Claude Lifts a Decades-Old Riemann Zeta Bound From 41.6% to 67.2%

An unreleased research version of Claude has pushed forward a result that number theorists had inched along for decades, and it did so while failing at a much harder task. Anthropic said the model raised the proven lower bound on the proportion of Riemann zeta zeros lying on the critical line from 41.6 percent to 67.2 percent.

The result came out of an offhand request. Jarred Sumner, an Anthropic staff member who is not a mathematician, asked the model to take a genuine attempt at the Riemann hypothesis, the 1859 conjecture that carries a million-dollar prize and remains unsolved. Claude did not solve it. What it produced instead was a sharper bound on a closely related quantity that specialists have been chipping at since the 1970s.

How the model got there

The first attempt went nowhere. Anthropic said Claude generated and discarded roughly 650 ideas before the researcher prompted it to try again. On the second pass the model spent about a day and a half orchestrating some 60 subagents inside Claude Code, which collectively executed around 2,400 shell commands and wrote hundreds of Python scripts.

The division of labour among those subagents is unusually well documented. Two developed the core mathematical ideas, 13 fed ideas to them, 30 tried and failed to produce anything usable, 13 acted as validators checking arguments for correctness, and two drafted the initial paper. The whole run consumed 31 million output tokens across two sessions.

Technically, the proof leans on recent work that removed a restrictive assumption from techniques Montgomery introduced in 1973, in papers by Aryan and by Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh, and combines it with a 2000 result of Bombieri. Anthropic described the decisive move as a willingness to treat the whole function space at once, allowing the quadratic form to be non-diagonal rather than splitting the problem apart.

Verification did most of the work

Claude did not simply hand over an answer. Its subagents refereed one another, ran thousands of numerical checks against known zeta zeros, hunted for counterexamples, pulled 54 arXiv papers to confirm the result was not already published, and re-derived the finding independently from scratch. The model then volunteered to write the result up and recommended that a human number theorist verify it.

That verification happened. Anthropic mathematicians Levent Alpoge and Ralph Furman studied the paper and produced a short informal note stating the proof concisely. Brian Conrey and Dan Goldston, both specialists in this corner of analytic number theory, examined it on short notice. Separately, staff member Eric Easley worked with Claude to produce a Lean formalization that passes standard validation tooling, meaning the result is machine-checkable rather than merely plausible.

What it does and does not mean

Anthropic was careful to bound the claim. The company does not expect these techniques to lead to a proof of the Riemann hypothesis, and 67.2 percent remains far from the 100 percent the hypothesis asserts. The zeta function governs how prime numbers are distributed, and the hypothesis holds that every zero determining that distribution sits on a single vertical line.

What is notable is the shape of the contribution. The model did not invent a new branch of mathematics. It recognised that three existing lines of research could be combined in a way nobody had assembled, then had the stamina to grind through verification. That is a recognisable form of research labour, and it is one that scales with compute rather than with headcount.

There is a small irony in the writeup. Anthropic said Claude was initially sceptical it could make progress at all, likely because its training taught it both how hard open problems are and how limited AI models tend to be. The prompting that unstuck it was largely encouragement, variants of asking the model to keep going. The company noted a similar approach helped Claude disprove the Jacobian conjecture.

The post was updated on 13 August with a revised version of the paper, rewritten by Claude to present a clearer proof and additional historical context. For researchers, the practical question is no longer whether models can contribute to open mathematics, but how much of the verification burden they can absorb on their own.