Claude Nudges a 60-Year-Old Riemann Bound From 41.6% to 67.2%

An unreleased research build of Anthropic’s Claude, prompted to ‘take a real stab’ at the Riemann hypothesis, failed at the headline problem but produced a genuine side result: it raised the best-known lower bound on the fraction of Riemann zeta zeros lying on the critical line from 41.6% to 67.2%. The proof stitches together recent work by Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh — which frees Montgomery’s 1973 techniques from assuming the hypothesis — with a 2000 paper by Bombieri, treating the full function space with a non-diagonal quadratic form rather than restricting to the usual diagonal case. Two Anthropic mathematicians validated the argument, outside experts Brian Conrey and Dan Goldston reviewed it, and the result was independently checked via a Lean formalization that passes the standard verifier.