Learning more about Claude's mathematical capabilities

Why it matters

Anthropic reports that an unreleased research version of Claude, when challenged by a staff member to attempt the Riemann hypothesis, failed to prove that famous conjecture but unexpectedly advanced a related problem: it raised the longstanding lower bound on the fraction of Riemann zeta function zeros that lie on the critical line. The model built on recent work by Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh, which adapted Montgomery's techniques to work without assuming the hypothesis, combined with earlier research by Bombieri. The key step was treating a space of functions with an induced quadratic form holistically, accounting for positive and negative definiteness together. Claude coordinated dozens of subagents, ran extensive numerical checks against known zeros, refereed its own work, and produced both a paper and a formally verifiable Lean proof. Anthropic mathematicians Levent Alpöge and Ralph Furman validated the result, while outside experts Brian Conrey and Dan Goldston reviewed the paper. Anthropic cautions the techniques will not prove the hypothesis itself, but presents the work as evidence of rapid progress in AI mathematical capability.