Claude Raises Lower Bound of Riemann Hypothesis: From 41.6% to 67.2% of Zeros on Critical Line

Anthropic reported on August 10 that its research Claude model, tasked with seriously attacking the Riemann Hypothesis, managed to raise the proven lower bound for the proportion of zeros of the zeta function lying on the critical line from 41.6% to 67.2%, a massive jump compared to previous incremental improvements over decades. The model did not invent new mathematics; it noticed that existing human results, including recent works by Baluyot, Goldston, Suryajaya, and Turneage-Butterbaugh adapting Montgomery's 1973 technique without assuming the hypothesis, plus a 2000 Bombieri paper, could be combined in a way no one had tried. The task was given by Jared Sumner, an Anthropic employee who is not a mathematician, who mostly sent encouraging prompts like 'keep going' and 'believe in yourself' to overcome the model's initial skepticism. In the second attempt, Claude coordinated about 60 subagents over 1.5 days, executing 2400 shell commands, writing hundreds of Python scripts, and running thousands of numerical checks, using two Claude Code sessions and 31 million output tokens. The model then organized its own verification: subagents reviewed each other's proofs, searched for counterexamples, re-proved the result from scratch, and downloaded 54 arXiv papers to ensure novelty. Human review involved two in-house Anthropic mathematicians, Levent Alpoge and Ralph Furman, and external experts Brian Conrey, whose 1989 record stood for over 20 years, and Dan Goldston, co-author of the relied-upon works, plus Eric Izley formalized the proof in Lean, passing machine verification. Anthropic notes that these techniques are unlikely to lead to a full proof of the Riemann Hypothesis, the paper hasn't undergone independent journal review, and the heavy math is human-derived, but the case shows AI's potential to find overlooked connections in existing literature.