Claude Pushes the Riemann Zeta Function Forward: An Unreleased Research Model Moves a Key Bound from 41.6% to 67.2%
An unreleased research version of Claude has produced a new, formally verified result on the Riemann hypothesis — the 1859 conjecture that still carries a million-dollar bounty. Claude didn't prove or disprove the hypothesis itself, but it raised the lower bound for the fraction of zeta-function zeros satisfying it from 41.6% to 67.2%, and Anthropic mathematicians, along with outside reviewers, checked the work.
An unreleased research version of Claude has produced a new, formally verified result on the Riemann hypothesis — the 1859 conjecture that still carries a million-dollar bounty. Claude didn't prove or disprove the hypothesis itself, but it raised the lower bound for the fraction of zeta-function zeros satisfying it from 41.6% to 67.2%, and Anthropic mathematicians, along with outside reviewers, checked the work.
Specs
- Improved lower bound: 67.2%
- Previous lower bound: 41.6%
- Output tokens used: 31 million
- Initial ideas attempted: 650
- Subagents coordinated: about 60
- Shell commands run: 2,400
- ArXiv papers checked: 54
What's new
- Claude did not prove or disprove the Riemann hypothesis, but improved a known lower bound on the fraction of zeta zeros that satisfy it.
- The result was reached across two Claude Code sessions totaling 31 million output tokens.
- Claude first tried 650 ideas without success before shifting strategy.
- It then coordinated roughly 60 subagents, which ran 2,400 shell commands, wrote hundreds of Python scripts, and ran thousands of numerical checks against known zeta zeros.
- The subagents also reviewed proofs and searched for counterexamples.
- Claude downloaded 54 arXiv papers to check whether the finding was already known, and then independently reproved it from scratch.
- Claude produced a formally verifiable proof, and worked with Eric Easley to build a Lean formalization that passed the standard validation tool comparator.
- The work drew on prior research by Aryan, Baluyot, Goldston, Suriajaya, Turnage-Butterbaugh, and Bombieri.
- Anthropic mathematicians studied and validated the paper, and mathematicians Brian Conrey, Dan Goldston, Levent Alpöge, and Ralph Furman examined it as well.
Who is it for
- Researchers and analytic number theorists tracking incremental progress on the Riemann hypothesis.
- People evaluating whether large language models can generate genuinely novel mathematical results rather than recombine known ones.
- Engineers and teams curious about multi-agent orchestration patterns — many subagents running parallel numerical checks, code, and proof review under one coordinating session.
- Anyone assessing formal verification pipelines that pair a language model's output with tools like Lean.
How to try it
The model used here is an unreleased research version of Claude, so this specific capability isn't available through any public Claude product yet — there's no subscription tier or API flag that grants it today. What you can do now is study the workflow itself: how Claude split 650 failed attempts into a pivot toward roughly 60 coordinated subagents running shell commands, Python checks, and Lean formalization.
Sources
- Anthropic — Anthropic