Learning more about Claude's mathematical capabilities

This post details how an unreleased research version of Claude, through a process involving subagents and extensive computation (31 million output tokens, 2,400 shell commands, hundreds of Python scripts), made a novel contribution to analytic number theory by improving a longstanding lower bound for the fraction of zeros of the Riemann zeta function that satisfy the Riemann hypothesis. The technical contribution lies in Claude's ability to synthesize prior research, form a suitable space of functions with a quadratic form, and write an inequality on the rank of this quadratic form using first- and second-moment information. This involved treating the entire space with positive- and negative-definiteness together, and allowing the quadratic form to be non-diagonal, a step that enabled the breakthrough. The process also involved Claude's self-validation through proof review, counterexample searching, and independent re-proving, culminating in a Lean formalization of the result.