Anthropic Claude Raises Riemann Zeta Bound to 67.2%

An unreleased research version of Anthropic's Claude has achieved a significant mathematical breakthrough by raising the proven lower bound for the zeros of the Riemann zeta function on the critical line from 41.6% to 67.2%. The Riemann hypothesis, proposed in 1859, is one of the most famous unsolved problems in mathematics and carries a $1 million Millennium Prize. While the AI did not fully solve the hypothesis, its progress represents a massive leap forward for a boundary that mathematicians have spent decades trying to push.

The breakthrough occurred after Jarred Sumner, an Anthropic staff member and non-mathematician, prompted the model to attempt the hypothesis within Claude Code. Operating as a multi-agent swarm of approximately 60 subagents, the system spent a day and a half executing its task. During this period, the model consumed 31 million output tokens, ran 2,400 shell commands, and generated 650 initial ideas that failed before finding a successful path. The core mathematical insight involved treating the zeros on and off the critical line as a unified geometric space rather than analyzing them as separate entities, which ultimately yielded a stronger inequality.

To ensure the validity of the discovery, the resulting proof was formally verified using the Lean theorem prover, and the proof has been made public. External number theorists Brian Conrey and Dan Goldston also reviewed the findings. This achievement follows other recent mathematical milestones for Anthropic, such as when its Claude Fable 5 model disproved the 87-year-old Jacobian conjecture.

For AI practitioners and mathematicians, this development highlights the viability of test-time compute and multi-agent systems for solving highly complex, abstract problems. Rather than relying on a single prompt and response, the model's ability to self-correct, write hundreds of Python scripts, and coordinate dozens of subagents shows how agentic workflows can tackle open research questions. It suggests that future AI tools will act as collaborative research partners capable of exploring vast mathematical spaces that human researchers have yet to map.