More than two thirds of the zeros of the Riemann zeta function lie on the critical line

In a recent demonstration, a Claude AI model made progress on a related problem to the Riemann hypothesis. Anthropic's staff member provided Claude with an unreasonable challenge, which the AI model accepted. Though it did not succeed in solving the Riemann hypothesis, Claude made strides in understanding the zeros of the Riemann zeta function.

The Riemann zeta function is used to describe the distribution of prime numbers, and the Riemann hypothesis posits that its zeros all lie on a specific vertical line. Up until this point, mathematicians had only been able to establish a lower bound of 41.6% for the fraction of zeros that lie on this line. Claude's research increased this bound to 67.2%.

Two Anthropic mathematicians validated Claude's findings and created an informal note for experts. Claude also produced a formally verifiable proof of the result. Experts Brian Conrey and Dan Goldston reviewed the paper. Claude approached the problem by combining techniques from prior research with Bombieri's 2000 work. The key step was treating the entire space of functions with positive and negative-definite subspaces together, allowing for a non-diagonal quadratic form.

Claude tested its work through extensive numerical checks, peer reviews, and independent re-proofs. The result highlights the rapid progress in AI models' mathematical capabilities, even if it could not resolve the Riemann hypothesis itself.