Exploring Claude's Mathematical Capabilities Further

An unreleased research version of Anthropic's Claude did not prove the Riemann Hypothesis itself, but achieved a new result in a related problem. It raised the known lower bound for the proportion of zeros of the Riemann zeta function that satisfy the Riemann Hypothesis from 41.6% to 67.2%.

In its attempt to tackle the Riemann Hypothesis, Claude combined recent research by Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh with Bombieri's 2000 study. This allowed it to apply Montgomery's zero distribution analysis techniques without assuming the Riemann Hypothesis.

The core approach involved constructing a function space using a quadratic form derived by Weil, and distinguishing between positive definite and negative definite subspaces for zeros on the critical line and zeros off the critical line, respectively. Claude established inequalities between the coefficients of the quadratic form and first- and second-moment information to calculate the lower bound for the proportion of zeros.

Two mathematicians at Anthropic reviewed Claude's paper and verified the results, and experts in the field Brian Conrey and Dan Goldston also examined the paper. Claude also wrote a formally verifiable Lean proof of the result.

While this technique is not expected to lead to a proof of the Riemann Hypothesis, it demonstrates the mathematical reasoning capabilities and rapid progress of AI models by combining decades of accumulated mathematical research to improve upon existing results.