Did Claude prove the Riemann hypothesis?

No. Claude improved a lower bound on how many nontrivial zeros of the Riemann zeta function sit on the critical line, moving it from 41.6% to 67.2%. A full proof of the Riemann hypothesis needs 100%. Anthropic states plainly that it does not expect the techniques Claude used here to lead to proving the hypothesis itself.

Can I read and check Claude's proof myself?

Yes. Anthropic published Claude's full paper as a PDF, an informal note explaining the result, the working transcripts, and the Lean formalization in the public repo github.com/anthropics/zeta-23-lean under Apache-2.0. The Lean version passes comparator, the standard validation tool, so the proof can be machine-checked rather than taken on trust.

Which Claude model did this, and can I use it?

Anthropic credits an unreleased research version of Claude, so the exact model is not available in the API or in Claude apps. What is public is the method and the output: the work ran inside Claude Code across two sessions, and every artifact Claude produced has been released for anyone to inspect.

How much compute did the Riemann zeta result take?

Claude used 31 million output tokens in total across two Claude Code sessions. Roughly 60 Claude subagents coordinated the search, running 2,400 shell commands and writing hundreds of Python scripts, including numerical checks against known zeta zeros before the final argument was written up.

Who verified the result besides Anthropic?

Two mathematicians at Anthropic, Levent Alpöge and Ralph Furman, examined the work. Anthropic also credits Brian Conrey and Dan Goldston, two experts in this area, who reviewed the paper on short notice. The Lean formalization gives an independent machine check on top of that human review.

Anthropic · 2026-08-10 · seismic

Claude raises a Riemann zeta bound to 67.2% — Anthropic ships the Lean proof

An unreleased research version of Claude raised the proven lower bound for Riemann zeta zeros on the critical line from 41.6% to 67.2%. Anthropic published the paper plus a Lean proof that passes the comparator tool.

Anthropic research post on Claude's mathematical capabilities and the Riemann zeta function

A research version of Claude moved a classical number-theory bound and left behind a machine-checkable Lean proof.

Key specs

New lower bound	67.2%
Output tokens used	31 million

Quick facts

Maker	Anthropic
Model	Unreleased research version of Claude
Result	Lower bound raised from 41.6% to 67.2%
Setup	Two Claude Code sessions, ~60 subagents
Compute	31M output tokens, 2,400 shell commands
Verification	Lean proof passing the comparator tool
Code	github.com/anthropics/zeta-23-lean, Apache-2.0

Benchmarks

Proven lower bound on Riemann zeta zeros on the critical line

Claude (Anthropic, 2026)		67.2%
Previous published bound		41.6%

source ↗

What is it?

Anthropic asked an unreleased research version of Claude to attack the Riemann hypothesis. Claude did not solve it, but it raised the proven lower bound for the fraction of nontrivial zeros of the Riemann zeta function lying on the critical line from 41.6% to 67.2%. Anthropic published the paper, an informal note, the transcripts, and the Lean formalization.

How does it work?

The search ran inside Claude Code over two sessions and 31 million output tokens. About 60 subagents coordinated the work, issuing 2,400 shell commands and writing hundreds of Python scripts, including thousands of numerical checks against known zeta zeros. Claude then wrote the argument up as a formally verifiable Lean proof, which passes comparator, the standard validation tool.

Why does it matter?

Bounds like this normally move by small amounts over decades of human work, so a jump from 41.6% to 67.2% is a concrete data point on what frontier models can now do in research mathematics. The Lean artifact matters as much as the number: it lets any mathematician machine-check the claim instead of arguing about whether a model hallucinated a step.

Who is it for?

mathematicians and AI researchers

Frequently asked questions

Did Claude prove the Riemann hypothesis?: No. Claude improved a lower bound on how many nontrivial zeros of the Riemann zeta function sit on the critical line, moving it from 41.6% to 67.2%. A full proof of the Riemann hypothesis needs 100%. Anthropic states plainly that it does not expect the techniques Claude used here to lead to proving the hypothesis itself.
Can I read and check Claude's proof myself?: Yes. Anthropic published Claude's full paper as a PDF, an informal note explaining the result, the working transcripts, and the Lean formalization in the public repo github.com/anthropics/zeta-23-lean under Apache-2.0. The Lean version passes comparator, the standard validation tool, so the proof can be machine-checked rather than taken on trust.
Which Claude model did this, and can I use it?: Anthropic credits an unreleased research version of Claude, so the exact model is not available in the API or in Claude apps. What is public is the method and the output: the work ran inside Claude Code across two sessions, and every artifact Claude produced has been released for anyone to inspect.
How much compute did the Riemann zeta result take?: Claude used 31 million output tokens in total across two Claude Code sessions. Roughly 60 Claude subagents coordinated the search, running 2,400 shell commands and writing hundreds of Python scripts, including numerical checks against known zeta zeros before the final argument was written up.
Who verified the result besides Anthropic?: Two mathematicians at Anthropic, Levent Alpöge and Ralph Furman, examined the work. Anthropic also credits Brian Conrey and Dan Goldston, two experts in this area, who reviewed the paper on short notice. The Lean formalization gives an independent machine check on top of that human review.

Try it

git clone https://github.com/anthropics/zeta-23-lean