Claude's AI raises a Riemann result to 67.2%

Claude did not solve the Riemann hypothesis, one of the most famous open problems in mathematics. But during the attempt, it found a significant improvement: it raised the known minimum proportion of certain zeros of the zeta function found on the line studied by mathematicians from 41.6% to 67.2%.

The result comes from a research version of Claude that has not yet been published. Anthropic says two of its mathematicians reviewed the work, and that external experts, including Brian Conrey and Dan Goldston, also examined the paper.

What 67.2% means

The Riemann hypothesis, formulated in 1859, is related to the distribution of prime numbers. These are numbers that can only be divided by 1 and themselves, such as 2, 3, 5 and 7.

The Riemann zeta function has points called zeros. The hypothesis holds that all relevant zeros lie on the same vertical line in the complex plane. No one has managed to prove that this is true or find a counterexample. Solving it carries a prize of one million dollars.

Claude's work does not prove the complete hypothesis. Instead, it improves a lower bound: it can now be stated that at least 67.2% of those zeros lie on that line, compared with the 41.6% known previously.

That does not mean that 67.2% is a probability, or that the rest of the zeros lie outside the line. It is a proven minimum for one part of the problem.

How Claude reached the result

The model combined results published over decades by several mathematicians, including a 2000 paper by Enrico Bombieri and more recent studies by Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh.

The central idea is to study a quadratic form, a mathematical tool that makes it possible to relate zeros located on the line to those that could lie outside it. Claude found that treating all those parts together, without requiring the form to be diagonal, made it possible to surpass the previous bound.

The full technical explanation is in Anthropic's paper. To check the result, Claude also produced a formalization in Lean, a system that automatically verifies mathematical proofs. Anthropic says that formalization passed its standard validation tool.

An unconventional proof process

The initial assignment was much more ambitious: try to prove the Riemann hypothesis. Jarred Sumner, an Anthropic employee without mathematical training, asked Claude to make a serious attempt and allowed the model to choose the approach.

In a first phase, Claude generated 650 ideas, but none worked. It then coordinated around 60 subagents for a day and a half in Claude Code. Together, they ran 2,400 commands, wrote hundreds of Python scripts and carried out thousands of numerical checks using already known zeros of the zeta function.

When the result emerged, Claude subjected it to further testing: it searched for counterexamples, reviewed 54 arXiv papers to check that the finding had not already been published, requested cross-reviews and proved it again from scratch.

What changes for you

You are not going to get a new app or an immediate answer to an everyday problem. The value lies elsewhere: Claude has shown that a model can combine existing research, detect a connection that had not been used in that way and turn it into a proof that human mathematicians can then review.

The result still needs scrutiny from the mathematical community. Anthropic also does not expect this technique to lead to a proof of the Riemann hypothesis. But it does point to a more concrete use of AI in science: not replacing expert validation, but exploring many paths, finding non-obvious combinations and delivering results that specialists can check.

The important point is not that Claude solved a 167-year-old problem. It did not. The important point is that a failed attempt produced a verifiable improvement in a related problem, and that the next frontier will be determining how many of these findings withstand prolonged, independent mathematical review.