Anthropic: Unreleased Claude Improves Zeta Bound to 67.2%

An unreleased research version of Claude, according to Anthropic's own writeup, has pushed a longstanding lower bound on the fraction of Riemann zeta zeros satisfying the Riemann hypothesis from 41.6% to 67.2%. The model did not attack the hypothesis itself. It combined recent results from Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh with the work of Bombieri, so that techniques Montgomery introduced in 1973 (which had assumed the hypothesis was true) could be turned into an unconditional bound.

The interesting part is the shape of the run. Jarred Sumner, described in the post as an Anthropic staff member and non-mathematician, prompted Claude to take a real stab at the problem. Claude generated and tried 650 ideas, none of which worked. Prompted to try again, it spent a day and a half coordinating about 60 subagents inside Claude Code, running 2,400 shell commands, writing hundreds of Python scripts, and burning 31 million output tokens across two sessions. When it hit a candidate result, subagents reviewed the proofs, searched for counterexamples, downloaded 54 papers from the arXiv to check the finding had not already been made, and independently re-proved it from scratch. Anthropic mathematicians Levent Alpöge and Ralph Furman then examined the work, Eric Easley helped produce a Lean formalization that passed the standard validation tool, and external experts Brian Conrey and Dan Goldston looked at the paper on short notice.

Anthropic is careful about what the result does and does not show. "We don't expect that the techniques Claude used will lead to proving the Riemann hypothesis," the post says, positioning the work as evidence of the speed of progress in AI's mathematical capabilities rather than a breakthrough on the underlying conjecture. The writeup does not include direct verdicts on correctness from Conrey or Goldston beyond noting that they generously examined the paper on short notice, so a third-party sign-off is not yet in the public record.

That leaves the result sitting somewhere between a benchmark claim and a bona fide mathematical contribution. If you build agent systems, the operational point may matter more than the percentage: a roughly 60-agent, day-and-a-half orchestration inside Claude Code produced a paper working mathematicians felt was worth reviewing on short notice, at a token bill any serious lab can already afford to run again. It sits at the ambitious end of the 433 agent stories we've logged in the last 90 days, most of which target shipping products or benchmark scores rather than open math.