Anthropic's Claude tried to solve the Riemann hypothesis and found something new instead

Prime numbers do not follow a predictable pattern, but mathematicians can gauge how closely their distribution aligns with an average. The Riemann zeta function, an infinite mathematical expression, contains vital information about this distribution. The Riemann hypothesis posits that all nontrivial zeros of the zeta function lie on a specific line in the complex plane, where the real part is 1/2.

Despite decades of effort, including by Anthropic's Claude, the hypothesis remains unproven. However, in the process of attempting to solve the problem, Claude made a new mathematical discovery. According to Anthropic, Claude determined that at least 67.2% of the zeros lie on the critical line, surpassing the previous lower bound of 41.6%.

This advancement, while not a solution to the Riemann hypothesis itself, is a significant step forward in understanding prime numbers. The research process involved Claude testing 650 ideas that ultimately failed, along with the use of 60 "agentic" instances of Claude to explore various approaches and validate the findings. Human researchers provided encouragement, prompting Claude to "believe in itself" and persist, rather than offering detailed mathematical guidance.

James Maynard, a mathematician at the University of Oxford, believes the result contributes meaningfully to the field. He notes that the AI has introduced a novel idea, suggesting that Claude has made a genuine mathematical contribution. However, the Riemann hypothesis remains unsolved due to the infinite nature of the zeros; discovering that a larger percentage lie on the critical line does not eliminate the possibility of exceptions.

While the work has garnered attention, Andrew Sutherland, a mathematician at MIT, emphasizes that Claude's accomplishment demonstrates AI's potential to engage in meaningful mathematical research, beyond merely solving specific problems.