A Machine’s Record Proof About Zeta Zeros Now Has a Version People Can Follow

Some people spend whole careers on a single fraction. For 80 years, mathematicians have tried to show that more of the Riemann zeta function’s zeros sit where the Riemann hypothesis says they must, raising that share a sliver at a time. Nobody has proved the hypothesis, and nobody in this story claims to have.

This summer the share jumped. An unreleased research version of Anthropic’s Claude produced a proof that more than 67.25 percent of the nontrivial zeros are both simple and on the critical line, up from a record of about 41.7 percent, or about 40.8 percent if you also insist the zeros be simple, meaning not repeated. Two mathematicians at the company checked it and posted it to arXiv. Three weeks later Youness Lamzouri, a mathematician in Nancy, France, proved the same bounds by a shorter route of his own. Both papers are preprints, and neither has been peer reviewed.

For the people who have spent decades on this

The history is a ladder of hard-won rungs. Atle Selberg showed in 1942 that a positive proportion of the zeros lie on the critical line. Norman Levinson got past one third in 1974, Brian Conrey more than two fifths in 1989, and in 2020 Pratt, Robles, Zaharescu and Zeindler nudged it to more than 5/12. Against that, the result Anthropic announced on August 10, 2026, more than two thirds simple and on the line and at least 83.62 percent distinct, moves a headline bound by 25 percentage points.

The news is honest and still good. About a third of the zeros remain unaccounted for and the hypothesis has not moved, but anyone pushing past 67.25 percent next has a higher floor, and two proofs of it.

For the students who come next

Lamzouri, of Universite de Lorraine, CNRS and the Institut Elie Cartan de Lorraine, posted a new proof of the same bounds on September 2, 2026. He described the original argument as technically intricate, with its main mechanism not immediately transparent, a polite way of saying the idea was hard to see. His version replaces a finite dimensional matrix framework with one inequality in a Hilbert space, letting Montgomery’s pair correlation theorem be applied directly. He also proved two estimates the original paper neither states nor proves, including that at least 88.76 percent of zeros are simple or on the critical line, or both, and presented it at the Max Planck Institute for Mathematics in Bonn on September 3.

He is candid about what he did not do: the two proofs are variants of the same second moment approach and arrive at the same constants. His is a clearer route to the same result, not a stronger bound. That matters most to whoever learns this next. A proof you can follow is one you can teach, argue with and build on, and there is real pleasure in seeing why something is true.

For everyone who will check what machines produce

How the proof turned up comes only from Anthropic, which has an obvious interest in the story. By its account, staff member Jarred Sumner asked the model to take a real stab at the Riemann hypothesis. The work ran across two sessions in Claude Code, used about 31 million output tokens, and began with roughly 650 ideas that went nowhere. A later push of about a day and a half coordinated around 60 subagents, which hunted for counterexamples, proved the result again from scratch, ran numerical checks against known zeros and read 54 arXiv papers to confirm the bound was new.

Levent Alpoge and Ralph Furman, the two Anthropic mathematicians, posted the paper on August 13 under their names, noting that the proof was discovered autonomously by Claude and verified and communicated by them. According to Anthropic, outside experts Brian Conrey and Dan Goldston examined it on short notice.

A bound came out of a model, and the field did the right thing with it: read it, doubted it, and wrote a version a person can follow. Somewhere there is a student who will one day try to push past two thirds, and that student will most likely start from Lamzouri’s pages rather than the machine’s.