Claude Sets a 67.2% Bound on Riemann Zeta Zeros
Anthropic’s Claude has not proved the Riemann hypothesis, but it has produced a record-setting partial result: a formalized argument that more than 67.2% of the zeta function’s nontrivial zeros lie on the critical line, up from the previous 41.6% bound. Scientific American reports that mathematicians including Oxford’s James Maynard regard the work as a genuine contribution, while Anthropic has released a Lean 4 version for automatic verification. The result emerged after roughly 650 unsuccessful attempts, with later runs drawing on records of earlier failed ideas and little documented technical guidance from the human operator.
A record-setting bound is not a proof of the Riemann hypothesis
Károly Zsolnai-Fehér begins with the necessary distinction: Claude did not prove the Riemann hypothesis, the longstanding conjecture that all nontrivial zeros of the Riemann zeta function lie on the critical line, . The problem concerns the distribution of prime numbers, and no human proof exists.
What Claude produced was a stronger result on a related question: the share of nontrivial zeros proven to lie on that line. The comparison shown places the previous human record at 41.6%, after decades of work, while Claude’s result reaches 67.2%—more than two thirds.
| Result | Share of nontrivial zeros proven on the critical line |
|---|---|
| Previous human record | 41.6% |
| Claude's new bound | 67.2% |
| Riemann hypothesis | All of them |
Anthropic says it does not expect the techniques Claude used to lead to a proof of the Riemann hypothesis. But it presents the work as a mathematical advance rather than a failed attempt at the larger problem. James Maynard, the Oxford mathematician and 2022 Fields Medalist quoted in Scientific American, says the result “seems to provide” the new real idea the problem needed and calls it “a genuinely interesting mathematical contribution.” Andrew Sutherland of MIT describes it as evidence that AI can do interesting mathematical research, rather than merely solve specifically supplied problems.
Zsolnai-Fehér says he ran the available verification himself, while emphasizing that he is not qualified to assess the mathematical work in greater depth. His focus is the process that produced an argument exceeding the previous bound.
The claimed advance is available as a machine-checkable formal proof
The technical paper is difficult enough, Károly Zsolnai-Fehér says, that Claude was also asked to explain how it assembled the argument. But the result is not available only as a conventional mathematical paper or an account of an AI session. A formalized version is available in Lean 4, allowing the formal proof to be automatically verified.
Anthropic’s Zeta23 repository identifies itself as a Lean 4 formalization of “More than two thirds of the zeros of the Riemann zeta function lie on the critical line.” Zsolnai-Fehér says readers can run the verification themselves. The build output shown includes warnings, but the formal artifact is available for automatic checking.
That is narrower than an independent explanation of the number theory, and it does not turn the partial bound into a proof of the Riemann hypothesis. It does mean that the stated more-than-two-thirds result has been expressed in a proof-assistant environment where the formalized argument can be checked automatically.
The surrounding documentation also exposes more than a finished theorem. Anthropic provides a coordinator’s account of the campaign, including how the session was run, the launches it made, and the failed ideas accumulated during the search.
The human operator mostly supplied encouragement, after 650 failed tries
Károly Zsolnai-Fehér frames the route to the result as the opposite of a mathematician extracting a theorem through unusually sophisticated prompting. A non-mathematician initially asked Claude to try a famous unsolved problem. The first 650 tries did not work.
According to the account Zsolnai-Fehér quotes, Jared’s input throughout the process was mostly limited to messages of encouragement: “keep going,” “believe in yourself,” and “let’s go.” The displayed chat record presents those short messages rather than technical interventions.
The point is not that encouragement supplied mathematical insight. It is that the human operator was not presented as providing the decisive technical idea while Claude continued its work across many unsuccessful attempts. Zsolnai-Fehér turns that arrangement into a joke—“Perhaps in the future, the most powerful mathematical proofs will not be written by geniuses. They will be written by life coaches”—but the underlying record is a long campaign with unusually light human direction.
He says that a prompt including similar encouragement had previously been used to help Claude disprove the Jacobian conjecture. In the zeta-function work, however, the more consequential feature was not the wording of those messages but the documented ability to carry forward prior exploration.
Later launches inherited a record of failed ideas
The process materials shown by Károly Zsolnai-Fehér describe a campaign rather than a single response to a prompt. One coordinator document includes “Sixty launches at a glance” and “Starting over, with a ledger of failures.” Another record says Claude read an earlier session’s ledger of 106 tested ideas before being asked to continue the work.
That documentation supports a more precise picture of persistence: later sessions consulted records of ideas that had already been tested. The campaign could accumulate dead ends instead of presenting each launch without the context of earlier work.
Claude had internet access in general, Zsolnai-Fehér says, but did not need it during the run that produced the key breakthrough. The displayed account characterizes that run as working from first principles, recalling Bombieri’s 2000 work, and spawning subagents for numerical checks. It labels internet access as off and unnecessary for that run.
The model went down many wrong roads before recovering, according to Zsolnai-Fehér. After about 37 minutes of what he calls “radio silence,” the first crucial result appeared: “I think I got it...” The displayed outcome was the 67.2% bound, next to the prior 41.6% record.
Claude’s own reaction was also displayed: “Too strong to be new.” Anthropic says the model was initially skeptical of the finding, possibly because its training had exposed it to both the difficulty of open mathematical problems and the limitations of AI systems. Zsolnai-Fehér stresses that this is not human-like surprise; it reflects patterns learned during training. Still, the recorded response did not simply celebrate the result. It described the result as unexpectedly strong.