Claude raised a proven bound on the Riemann hypothesis after 650 failed ideas

Claude tried 650 ideas on the Riemann hypothesis, failed with every one of them, and then moved a boundary mathematicians had been stuck on.

What Anthropic published

On August 10, 2026, Anthropic released an account of what happened when it asked an unreleased research version of Claude to attempt the Riemann hypothesis. The model did not solve the problem. What it did was raise the proven lower bound for the fraction of zeta-function zeros lying on the critical line, from 41.6 percent to 67.2 percent.

The scale of the run is part of the story. After the first 650 ideas went nowhere, and after being encouraged to keep going, Claude worked across two sessions in Claude Code, coordinating about 60 subagents that ran 2,400 shell commands and wrote hundreds of Python scripts. The effort consumed 31 million output tokens over roughly a day and a half.

The move that finally worked was conceptual rather than brute force: treating zeros on the critical line and zeros off it as one unified geometric space instead of analyzing them separately, which yields a stronger inequality.

How the result was checked

The verification chain matters as much as the number. Working with Anthropic staff member Eric Easley, Claude produced a Lean formalization of the result that passes the standard validation tool. Anthropic mathematicians Levent Alpoge and Ralph Furman examined the work, and two outside experts in the area, Brian Conrey and Dan Goldston, reviewed it.

What it does and does not mean

The ceiling is worth stating plainly. This is not a proof of the Riemann hypothesis. A proof would require showing that 100 percent of those zeros behave as expected, and 67.2 percent is a long way from there.

Still, the shape of the episode cuts against the usual account of AI and mathematics. The standard failure mode is a model that produces confident nonsense on hard problems. Here the record shows 650 documented dead ends, a genuine conceptual reframing, and a result that survived formal verification and expert review. That sequence, rather than the percentage itself, is what makes the publication worth watching.