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Anthropic's unreleased AI research model, run as a swarm of roughly sixty sub agents inside Claude Code, has proved at least 67.
Anthropic mathematicians Levent Alpöge and Ralph Furman, with help from an unreleased research model run as a swarm of roughly sixty sub-agents inside Claude Code, have proved that at least 67.2% of the non-trivial zeros of the Riemann zeta function lie on the so-called critical line. The previous proven lower bound, 41.6%, had stood since 1989.
The Riemann Hypothesis, formulated by Bernhard Riemann in 1859, says every non-trivial zero of the Riemann zeta function sits on a single vertical line in the complex plane, the "critical line." The zeta function is a sum that, in the right reading, encodes how prime numbers are distributed across the integers. A proof would settle a 167-year-old "crown jewel" of mathematics. The Clay Mathematics Institute puts a $1 million prize on a full proof.
The new result does not collect that prize. It is a narrow but real improvement: a higher proven minimum share of zeros on the critical line, not a statement about the remaining ~32.8%. Anthropic's own research post says the approach probably does not generalize to a full proof of the conjecture.
Inside Claude Code, the model spun up about sixty sub-agents and ran them for roughly a day and a half. The system produced 31 million output tokens, executed 2,400 shell commands, wrote hundreds of Python scripts, and ran thousands of numerical checks against known zeta zeros. A first pass generated 650 candidate approaches; all of them failed. A second pass assigned roles: 2 agents produced the key mathematical ideas, 13 supplied supporting ideas, 30 explored without success, 13 cross-verified results, and 2 wrote up the paper.
The structure of the run, generate, fail, re-run with role specialization, cross-verify, reproduce from scratch, write up, describes a self-directed research loop rather than a calculator. Anthropic staff mathematicians Alpöge and Furman reviewed the argument. External analytic number theorists Brian Conrey and Dan Goldston also reviewed the paper. Eric Easley produced a Lean 4 formalization, checked by the standard Comparator tool, and the team cross-checked the result against 54 prior arXiv papers. The Lean repository is public.
The 41.6% bound that the new result improves was set in 1989, per Anthropic's research post, and had held the record for more than three decades. The 67.2% figure measures a minimum share, not a fraction of the conjecture that is now known to be true; it says nothing about whether the remaining ~32.8% of zeros fail the hypothesis. The leap is meaningful at the level of proof technique, not at the level of "we are now closer to solving RH."
In an informal note for experts, Anthropic calls the result a new kind of research artifact: a Lean-verified, externally reviewed bound improvement produced by a coordinated multi-agent run. It is not a claim that AI has cracked a century-old problem. No independent team has reproduced the result, and open peer review is still pending.
The story points to an operational shape for AI-assisted mathematical research: a self-directed loop that fails fast, specializes, and writes a paper a human can audit. The compute footprint, tens of millions of tokens and roughly a day and a half of coordinated work, is part of the artifact, not a footnote. Whether that loop extends beyond problems structured like the Riemann Hypothesis, and whether the bound itself survives independent re-derivation, are the two open questions worth watching next.