Claude found that combining the results from Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh with the work of Bombieri provides a way to surpass the previous state-of-the-art lower-bound proportion of 41.6%, increasing it to 67.2%.
A short technical explanation of Claude’s finding is as follows: Claude forms a suitable space of functions with quadratic form induced by Weil, and positive- (respectively negative-)definite subspaces arising from zeros on (respectively off) the line. Then Claude simply writes down an inequality on the rank of a quadratic form in terms of first- and second-moment information. (The successful computation of the latter in terms of the dual picture over primes, or via control of a Hilbert transform, is no surprise in analytic number theory.) The courage to treat the entire space, with positive- and negative-definiteness taken into account together, and with the quadratic form allowed to be non-diagonal, is in some sense the step that allows Claude to achieve the conclusion based on the important prior work.
The full technical explanation is available in the paper. Claude’s explanation of how it arrived at its result is available in a separate Appendix here.
Claude's methodology
An unreleased research version of Claude found the new lower bound over two sessions in Claude Code, using a total of 31 million output tokens.
Jarred Sumner, an Anthropic staff member (and non-mathematician), prompted Claude to “take a real stab” at the hypothesis itself, leaving the mathematical choices from there up to the model. Initially, Claude generated and tried 650 ideas, none of which worked. Jarred prompted Claude to try again, and it spent a day and a half coordinating about 60 Claude subagents, which this time went much deeper: between them, they ran 2,400 shell commands and wrote hundreds of Python scripts.1 The subagents ran thousands of numerical checks against known zeta zeros and refereed one another’s work. Throughout this process, Jarred's input was mostly limited to sending Claude messages of encouragement (mostly variants of “keep going” or “believe in yourself”).2 This seems to have helped Claude overcome some initial skepticism that it could make meaningful progress.