Claude proves more than two-thirds of Riemann zeta zeros are simple and on the critical line (up from 41.6%)
Anthropic reported that an unreleased research version of Claude, running in Claude Code with ~60 subagents, raised the unconditional lower bound on the proportion of Riemann zeta zeros that are simple and on the critical line from ~41.6% to 67.2%. The previous 37 years had added only ~0.8 percentage points. Key results were formalised in Lean, reviewed by Brian Conrey and Dan Goldston, and independently re-proved by Youness Lamzouri.
Key facts
- Paper: 'More than two thirds of the zeta zeros are simple and on the critical line' (arXiv 2608.13637)
- Prior record ~41.6% (Levinson–Conrey lineage); the last 37 years had gained ~0.8 points
- Run by Jarred Sumner with mostly encouragement-style prompting; checked by Levent Alpöge and Ralph Furman
- Lean formalisation of key results with Eric Easley; independent new proof by Lamzouri (arXiv 2609.02882)
Science result
- Field
- mathematics / analytic number theory
- Problem
- Proportion of nontrivial zeros of ζ(s) on the critical line (toward the Riemann hypothesis)
- Result
- Unconditional proof that more than 67.2% of zeta zeros are simple and lie on the critical line.
- AI system
- Claude (unreleased research model) in Claude Code
- Human role
- Near-autonomous: non-mathematician prompted; experts verified
- Verification
- Formal proof in Lean (key results); expert review (Conrey, Goldston); independent re-proof
- Status
- confirmed
- Why surprising
- A decades-slow line of research toward the Riemann hypothesis jumped by 25 percentage points in one AI run.
What happened
A large Claude agent swarm refined the mollifier method behind Levinson- and Conrey-style bounds far beyond the prior state of the art. The result was then checked formally and by leading experts.
Why it matters
It does not prove the Riemann hypothesis, but it is a dramatic quantitative advance on the most famous problem in mathematics, and it was independently confirmed.
Changelog
- 2026-09-29: created
- 2026-09-29: added link to Anthropic's formal-math repository (zeta23 Lean project)
Related events
- Claude produces the first complete machine-checked proof of Fermat's Last Theorem in Lean, in 11 days ★★★★★
- Claude-assisted constructions complete Hadamard matrices for every order below 2000, including 668 ★★★
- Claude-assisted construction claims a complex structure on the 6-sphere, answering Hopf's 1947 problem (pending verification) ★★★★★
- Claude-written Lean proof claims the dying percolation conjecture θ(p_c)=0 in every dimension ★★★★★
Sources (4)
- codeanthropics/formal-math: zeta23 Lean formalization
- officialAnthropic: Claude and the zeros of the Riemann zeta function
- paperMore than two thirds of the zeta zeros are simple and on the critical line (arXiv 2608.13637)
- paperLamzouri: independent proof (arXiv 2609.02882)
id: 2026-08-10-claude-riemann-zeta-zeros-two-thirds · updated 2026-09-29 · open in the interactive timeline