Anthropic details an unreleased Claude model's attempt to solve the Riemann hypothesis; it didn't solve it but "unexpectedly" made strides on a related problem (Anthropic)

Anthropic revealed an unreleased version of the Claude model tackling a challenging mathematical problem. The task was to tackle the Riemann hypothesis, a centuries-old unsolved issue in mathematics first proposed in 1859. Claude made an attempt at the problem, but as expected given the complexity, it did not succeed in proving the hypothesis.

However, during its efforts, Claude made significant progress on a related problem. The research version of Claude has improved on a long-standing lower bound for the fraction of zeros of the Riemann zeta function that satisfy the Riemann hypothesis. This improvement increased the bound from 41.6% to 67.2%. The findings were validated by two mathematicians at Anthropic, Brian Conrey and Dan Goldston, who examined the paper on short notice.

Claude also provided a formally verifiable proof of its result. While Anthropic does not anticipate Claude's techniques leading to a proof of the Riemann hypothesis, the work demonstrates the rapid advancements in AI models' mathematical capabilities. The article discusses Claude's approach to the problem and its unexpected success.

The Riemann zeta function plays a crucial role in describing the distribution of prime numbers, with the Riemann hypothesis asserting that the zeros determining the primes lie along a specific vertical line. Proving or disproving the hypothesis has been a significant challenge, but progress has been made in related areas studying the Riemann zeta function and its zeros.

Claude's result builds upon this research, combining various techniques and findings to achieve the enhanced lower bound. The process involved extensive numerical checks, collaboration between subagents, and cross-referencing with existing literature to ensure the validity of the findings. Claude's attempt not only showcases the power of AI in advancing mathematical knowledge but also highlights the potential for future breakthroughs as AI models continue to evolve.