Anthropic’s Claude failed to solve the Riemann hypothesis, one of mathematics’ most important unsolved problems, but its attempt inspired mathematician Youness Lamzouri to make progress on a related question, tells Live Science.
The Riemann hypothesis, proposed by Bernhard Riemann in 1859, concerns the distribution of prime numbers and the zeros of the Riemann zeta function. It predicts that all nontrivial zeros lie on a particular vertical line known as the critical line. Proving this remains one of the Clay Mathematics Institute’s Millennium Prize Problems.
Anthropic tasked an unreleased version of Claude with tackling the hypothesis. After exploring more than 600 approaches without finding a proof, researchers shifted to a narrower problem: determining the minimum proportion of zeros known to lie on the critical line. Since 1989, mathematicians had established a lower bound of 40%. Claude produced an argument raising that figure to 67.25%.
The result does not bring mathematicians closer to proving the Riemann hypothesis itself because even a tiny number of exceptions would invalidate it. However, mathematician and Fields Medal winner James Maynard found the new ideas within Claude’s work significant.
Lamzouri initially found Claude’s proof overly complicated. Rather than reconstructing it directly, he examined the mathematical papers Claude had relied upon. That investigation revealed a simpler route. By introducing an additional constraint into the equations, Lamzouri reproduced the 67.25% result with a more intuitive proof.
His approach also produced new information. Lamzouri showed that the proportion of zeros that are neither simple nor located on the critical line is at most 11%.
The episode illustrates a potentially valuable role for AI in mathematics. An AI system may generate ideas without producing an understandable final solution, while human mathematicians can reinterpret those ideas, simplify the reasoning, and use them to uncover new mathematical insights.
