Anthropic's Claude Pushes a 160-Year-Old Math Boundary From 41.6% to 67.2%
An unreleased Claude research model pushed a 40-year-old mathematical lower bound from 41.6% to 67.2%, using a multi-agent swarm running inside Claude Code
- An unreleased Claude research model increased the proven lower bound for zeros of the Riemann zeta function on the critical line from 41.6% to 67.2%.
- Claude ran as a multi-agent swarm of ~60 subagents inside Claude Code, spending a day and a half, 31M output tokens, and running 2,400 shell commands.
- The key insight: treating zeros on and off the critical line as a unified geometric space rather than analyzing them separately, yielding a stronger inequality.
- The result was formally verified in Lean and reviewed by external number theorists Brian Conrey and Dan Goldston; the Lean proof is public.
- Anthropic does not expect this approach to lead to a full proof of the Riemann hypothesis; the result emerged as an unintended byproduct of asking Claude to attempt the full problem.
- This follows Anthropic's recent streak of AI-assisted math breakthroughs, including this result and Claude Fable 5 disproving the 87-year-old Jacobian conjecture.
The Riemann hypothesis is arguably the most famous unsolved problem in mathematics. Posed in 1859, it carries a $1 million Millennium Prize bounty and sits at the heart of how prime numbers are distributed. An Anthropic staff member recently handed it to an unreleased research version of Claude with a simple prompt: take a real stab at it. Claude didn't solve it. What happened instead is worth paying close attention to.
The problem, in plain terms
The Riemann zeta function is a mathematical object that encodes the distribution of prime numbers. It has infinitely many "non-trivial zeros" — points where the function equals zero — and the Riemann hypothesis claims that every one of those zeros lies on a specific vertical line in the complex plane, called the critical line. Nobody has proven or disproven this in over 160 years.
Because a full proof remains out of reach, mathematicians have focused on a more tractable question: what fraction of those zeros can we prove lie on the critical line? Over decades of work, that lower bound had been pushed to 41.6%. Claude moved it to 67.2%.
How Claude actually did it
An unreleased research version of Claude found the new lower bound across two sessions in Claude Code, using 31 million output tokens. The process was far from a single flash of insight.
- Jarred Sumner, an Anthropic staff member and non-mathematician, prompted Claude to "take a real stab" at the hypothesis, leaving the mathematical choices to the model.
- Claude generated and tried 650 ideas. None worked.
- After being prompted to try again, it spent a day and a half coordinating roughly 60 Claude subagents, which collectively ran 2,400 shell commands and wrote hundreds of Python scripts.