Anthropic says research Claude raised a Riemann zeta zero lower bound to 67.2 percent

Anthropic says an unreleased research version of Claude has improved a longstanding lower bound in analytic number theory, raising the known proportion of Riemann zeta function zeros on the critical line from 41.6 percent to 67.2 percent. The model arrived at the result after an Anthropic employee challenged it to make a serious attempt at the Riemann hypothesis. Claude did not solve that famous open problem, and the company is explicit that the techniques it used are not expected to produce such a proof.

The Riemann hypothesis, proposed in 1859, concerns where the nontrivial zeros of the zeta function lie and is closely connected to the distribution of prime numbers. The hypothesis says those zeros all fall on a particular vertical line in the complex plane. Mathematicians have not proved or disproved that statement, but they have established that a certain minimum proportion lies on the line. Before the result described by Anthropic, the published lower bound had reached 41.6 percent.

Claude’s route drew on decades of existing mathematics rather than inventing the subject from nothing. Anthropic says the model combined recent work by Siegfred Alan Baluyot, Daniel Goldston, Ade Irma Suriajaya and Caroline Turnage-Butterbaugh with a 2000 paper by Enrico Bombieri. The company’s technical account describes Claude using a quadratic form induced by André Weil and considering positive- and negative-definite subspaces associated with zeros on and off the critical line. That construction allowed the model to derive a rank inequality from first- and second-moment information.

Verification is the most important part of a claim at this level. Two mathematicians working at Anthropic studied and validated Claude’s paper and prepared a shorter note for specialists. The company says number theorists Brian Conrey and Daniel Goldston also examined the work on short notice. Claude produced a formally verifiable version of the proof in Lean, allowing its logical steps to be checked by a proof assistant rather than relying only on fluent mathematical prose. The paper, expert note, appendix and formalization were made available alongside the announcement.

The computation was substantial. Anthropic reports that the unreleased model worked over two Claude Code sessions and generated 31 million output tokens. Its research workflow decomposed the problem, explored prior work and revised its approach over an extended run. That scale matters because the result should not be interpreted as an ordinary chatbot answering a single prompt. It reflects a costly research process with access to tools, large context and subsequent review by human mathematicians.

The episode also demonstrates why precise framing is essential when AI systems work on open scientific problems. Claude did not prove the Riemann hypothesis, did not show that 67.2 percent is the final answer and did not remove the need for peer scrutiny. It produced a claimed improvement on a related lower bound by synthesizing established techniques in a new way. Anthropic itself says the approach is unlikely to lead to the full conjecture. Publication and examination by the wider mathematics community will provide a stronger test than the company’s internal validation alone.

Even with those limits, the result is evidence that long-running AI research systems can contribute to fields where correctness is unusually demanding. Formal proof tools help because they separate a checkable logical object from a persuasive explanation, though they do not automatically settle whether definitions, assumptions and the claimed mathematical significance have been framed correctly. The next questions are whether independent specialists confirm the argument, whether the method generalizes to other problems and how efficiently similar work can be reproduced. For AI laboratories, the case suggests that scientific progress may depend as much on tool-supported verification and expert review as on increasing a model’s ability to generate candidate ideas.