Claude Pushes Riemann-Zeta Bound From 41.6% to 67.2% — With 60 Subagents and 31M Tokens

Anthropic disclosed that an unreleased research version of Claude improved the long-standing lower bound on the fraction of Riemann zeta zeros that satisfy the Riemann hypothesis — from 41.6% to 67.2%. Working through Claude Code, the model generated 650 ideas, orchestrated ~60 subagents, and ran thousands of numerical checks before human mathematicians reviewed the argument.

Key Takeaways

Anthropic disclosed that an unreleased research version of Claude improved the long-standing lower bound on the fraction of Riemann zeta zeros that satisfy the Riemann hypothesis — from 41.6% to 67.2%.
Working through Claude Code, the model generated 650 ideas, orchestrated ~60 subagents, and ran thousands of numerical checks before human mathematicians reviewed the argument.

AI synthesis4 sources9

Anthropic disclosed on August 10 that an unreleased research version of Claude has improved a long-standing lower bound in number theory: the fraction of zeros of the Riemann zeta function that satisfy the Riemann hypothesis, previously known to be at least 41.6%, is now claimed to be at least 67.2%. The result does not prove the Riemann hypothesis itself — one of mathematics' most famous open problems — but improving such bounds is how progress on the problem has historically been measured.

The project ran inside Claude Code over two sessions, consuming 31 million output tokens. According to Anthropic, the model generated 650 initial ideas, then orchestrated roughly 60 subagents that executed 2,400 shell commands and thousands of numerical validation checks, synthesizing recent mathematical literature rather than attempting a head-on proof. Two Anthropic mathematicians reviewed the argument internally, with outside experts including leading zeta-function researcher Brian Conrey among the reviewers.

Anthropic is careful about the claim: this is a data point about the agent-orchestration approach to hard mathematics, not a claim that frontier models are ready to do independent mathematical research. The bound improvement builds directly on decades of work by human mathematicians; Claude's contribution was organizing, checking and extending that material at scale — exactly the kind of workload where orchestrated agent systems have been showing unexpected competence.

The result is the latest sign that AI systems are beginning to touch research-grade mathematics. Whether the new bound holds up to full independent verification will be settled in the mathematical literature over the coming months; the number 0.6725, the optimized constant behind the headline figure, is already circulating in the papers describing the work.