AI Model Claude Makes Breakthrough on Riemann Hypothesis, Surpassing Previous Lower Bound

A significant development in the field of mathematics has emerged from an unexpected quarter: AI model Claude. In a recent challenge, Claude was tasked with tackling one of the most famous unsolved problems in mathematics – the Riemann hypothesis. Although it didn’t succeed in resolving the problem itself, its attempt led to an important breakthrough on a related issue.

Researchers at Anthropic have been studying Claude’s capabilities and recently gave it an unreasonable challenge: take a stab at the Riemann hypothesis. This task has been open since 1859, with a million-dollar bounty still unclaimed. The team was aware that Claude might not succeed in resolving the problem itself, but they were eager to see how it would approach this complex issue.

During its attempt, Claude unexpectedly made significant strides on a related problem – improving the lower bound for the fraction of zeros of the Riemann zeta function that satisfy the Riemann hypothesis. This improvement is substantial: from 41.6% to 67.2%. The new result draws heavily on prior research by mathematicians over several decades, combining results from Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh with Bombieri’s work.

The Riemann zeta function describes the distribution of prime numbers – each zero that contributes to this sequence provides finer detail. The Riemann hypothesis states that these zeros exist along a certain vertical line. This conjecture has become one of the most consequential in mathematics, as many results assume it to provide randomness in primes.

No one has yet been able to prove or disprove the Riemann hypothesis, but mathematicians have made progress in related areas. One such direction concerns quantifying a minimum proportion of zeros that are on the line – over time, this known constant proportion has gradually increased to 41.6%. Another area focuses on the distribution of zeros along the line.

Claude’s result shows how AI models can extend and build upon mathematicians’ ideas in new ways. Even though it couldn’t resolve the Riemann hypothesis itself, its breakthrough emerged as an unintended byproduct of that original request. This development highlights the potential for AI to make meaningful contributions to mathematical research – a field often considered one of human ingenuity.

The team at Anthropic is grateful for the input and expertise provided by Brian Conrey and Dan Goldston, two experts in this area who generously examined Claude’s paper on short notice. The full technical explanation of Claude’s finding is available in its paper, while an informal note concisely stating Claude’s proof has been produced by Levent Alpöge and Ralph Furman.

Claude arrived at the new lower bound over two sessions using a total of 31 million output tokens in Claude Code. This achievement demonstrates how AI models can tackle complex mathematical problems with speed and efficiency – an area where human mathematicians often struggle to make significant progress.

The process behind this breakthrough is fascinating, involving multiple subagents working together to run thousands of numerical checks against known zeta zeros. The team at Anthropic used a combination of encouragement and technical guidance to help Claude overcome initial skepticism about its ability to make meaningful contributions.

Claude’s work also highlights the potential for AI models like itself to write formal proofs, recommend human validation, and even produce Lean formalizations – as seen in its collaboration with Eric Easley. This development underscores the importance of exploring how AI can assist mathematicians in their research efforts.