TL;DR

A research build of Claude that Anthropic has not released has raised the known lower bound on what fraction of the Riemann zeta function’s zeros satisfy the Riemann hypothesis, taking it from 41.6% to 67.2%. It did not prove the hypothesis itself and Anthropic says the techniques used are unlikely to. Two staff mathematicians validated the result and two external number theorists examined the paper.

What actually happened

A non-mathematician at the company, Jarred Sumner, prompted the model to attempt the hypothesis outright — a problem open since 1859 carrying a million-dollar prize. The first pass produced 650 candidate ideas, every one of them a dead end.

On the retry the model spent about 36 hours directing around 60 subagents, which between them issued some 2,400 shell commands plus hundreds of Python scripts. The subagents checked results numerically against known zeros and reviewed one another’s work. Total cost across two Claude Code sessions: 31 million output tokens.

The result leans on existing work — Bombieri’s 2000 paper, and a series by Goldston, Baluyot, Turnage-Butterbaugh and Suriajaya that removed an assumption from techniques Montgomery introduced in 1973. The improvement combines them rather than inventing new machinery.

The checking is the interesting part

Before writing anything up, the model pulled 54 papers from arXiv to confirm the finding was not already known, searched for counterexamples, and had subagents independently re-derive it. It then recommended a human number theorist verify the work.

Two of Anthropic’s own mathematicians, Ralph Furman and Levent Alpöge, did so. A Lean formalisation was produced separately and passes standard validation. Dan Goldston and Brian Conrey, both specialists in the area, reviewed the paper at short notice.

Looking forward

The honest read is narrower than the headline. This is an incremental improvement to a bound, built on decades of published mathematics, not a breakthrough on the hypothesis.

What should interest UK technical leaders is the shape of the workflow rather than the mathematics. A non-expert set the goal, the system decomposed it across dozens of parallel agents, generated a formal machine-checkable proof, and then explicitly asked for human review before publication. That is a template for verifiable AI work in any domain where being wrong is expensive — and a rare example of a lab publishing the validation chain alongside the claim.