Claude did not solve the Riemann Hypothesis, one of the most famous open problems in mathematics. But during the attempt, it found something unexpected: a new lower bound for the proportion of zeros of the Riemann zeta function that lie on the critical line. The result raises the known limit from 41.6% to 67.2%.
The research was conducted with an unpublished version of Claude and reviewed by mathematicians at Anthropic. In addition, the model produced a formalization in Lean that passed the tool’s standard validation. Does this mean an AI solved the million-dollar problem? No. What is interesting is precisely what happened around the failure.
The challenge behind the Riemann Hypothesis
The Riemann zeta function is deeply connected to the distribution of prime numbers. Its zeros point to increasingly precise patterns in the appearance of these numbers, which seem to emerge irregularly even though they follow very deep mathematical structures.
The Riemann Hypothesis, formulated in 1859, states that all the nontrivial zeros of this function lie on the same vertical line in the complex plane, known as the critical line. So far, no one has been able to prove or disprove the conjecture, which comes with a one-million-dollar prize.
Although the complete hypothesis remains unsolved, mathematicians have managed to prove partial results. One of them involves determining what minimum proportion of zeros must lie on the critical line. For years, the known proportion remained at 41.6%.
Claude improves an existing mathematical result
Anthropic’s research combined several lines of work developed by mathematicians over decades. These include results by Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh, which make it possible to use Montgomery’s techniques without first assuming that the Riemann Hypothesis is true.
Claude also drew on work published by Bombieri in 2000. By combining these results, the model identified a way to surpass the previous lower bound and demonstrate that at least 67.2% of the zeros of the zeta function lie on the critical line.
The advance does not show that all zeros lie on the critical line. It shows that a larger minimum proportion of them must be there.
The difference may seem small conceptually, but it is important from a technical perspective. In mathematics, improving a known bound can require new ideas, an unexpected combination of previous results, or a complete reorganization of the tools available.
What exactly did the model find?
In technical terms, Claude constructed a suitable space of functions together with a quadratic form induced by Weil’s criterion. Within this structure, subspaces appear as positive or negative depending on whether the zeros lie on or off the critical line.
The model then established an inequality related to the rank of a quadratic form and to first- and second-moment information. These moments can be calculated using what is known as the dual perspective on prime numbers or by controlling a Hilbert transform.
The decisive element was to consider the complete space, taking the positive and negative parts into account simultaneously, and allowing the quadratic form to be non-diagonal. This choice made it possible to take advantage of previous results in a way that led to the new 67.2% bound.
The full explanation is available in Anthropic’s technical paper, along with an appendix in which Claude describes how it reached its result.
How Claude worked with its subagents
The finding emerged while Claude was trying to tackle the Riemann Hypothesis directly. Jarred Sumner, a member of Anthropic’s team and not a mathematician, asked it to make a genuine attempt, allowing the model to make the mathematical decisions.
In an initial stage, Claude generated and tested 650 ideas without success. It then received a new push and spent approximately a day and a half coordinating around 60 subagents. Together, they ran 2,400 terminal commands and wrote hundreds of Python scripts.
The subagents carried out thousands of numerical checks using known zeros of the zeta function. They also reviewed one another’s work, searched for counterexamples, and downloaded 54 papers from arXiv to check that the result had not already been published.
The process looks less like an instant answer and more like a small automated research team. Claude generated hypotheses, discarded approaches, ran experiments, requested internal reviews, and eventually proposed writing its conclusions as a paper.
Human validation and formal proof
Two mathematicians at Anthropic, Levent Alpöge and Ralph Furman, studied the work to understand the result and verify its relationship to the existing literature. External experts such as Brian Conrey and Dan Goldston also took part, examining the paper at short notice.
At the same time, Claude worked with Eric Easley to produce a formalization in Lean. This type of formalization translates a mathematical proof into a language that an automated verifier can review step by step.
That does not eliminate the need for human review. A formal proof can confirm that a chain of statements follows the system’s rules, but mathematicians still need to assess whether the formulation correctly represents the result, whether the assumptions are properly stated, and whether the connection to previous work is valid.
The result does not solve the Riemann Hypothesis
Anthropic does not expect the techniques used by Claude to lead directly to a proof of the complete hypothesis. The new bound only establishes that a minimum proportion of 67.2% of the zeros lies on the critical line. It is still necessary to prove that the percentage is 100% or to find a zero outside it.
The importance of the case lies elsewhere. Claude did not produce an isolated discovery from scratch. Instead, it extended a chain of previous mathematical ideas and identified a combination that apparently had not been used to improve the known bound.
This detail matters when we try to understand the current role of AI in science. Models can explore vast spaces of possibilities, automate calculations, compare literature, search for errors, and coordinate tasks that previously required substantial human time. But validation, interpretation, and mathematical judgment remain essential.
Claude also showed skepticism about its own ability, possibly because its training data reflects both the historical difficulty of these problems and the known limitations of AI models. Instructions as simple as “keep trying” appear to have changed the direction of the search.
Are we looking at a machine that replaces mathematicians? This case points to a more useful conclusion: we are seeing tools capable of expanding researchers’ ability to explore and of finding connections that may go unnoticed even in problems studied for decades.