$ cat wiki/papers/2026/anthropic-2026-08-10-zeta-critical-line.md
More than two thirds of the zeros of the Riemann zeta function lie on the critical line
TL;DR
An unreleased research version of Claude raised the proven lower bound on the fraction of nontrivial zeros of the Riemann zeta function lying on the critical line from 41.6% to 67.2% — reported as the largest single improvement to this bound in the problem's history, formally verified in Lean, and reviewed by two external number theorists. It is not a proof of the Riemann hypothesis, which requires 100% (source).
Authors & Org
Reported author line: Claude; Anthropic, San Francisco, 2026. The authorship as reported is itself notable and is recorded as reported rather than endorsed — no source read discusses how it was decided, and no human co-authors are named (source).
External review by number theorists Brian Conrey and Dan Goldston (source).
Method
The mathematical move reported: treating zeros on and off the critical line as a unified geometric space, rather than analysing the two cases separately, which yields a stronger inequality (source).
The run that produced it, as reported:
| Quantity | Value |
|---|---|
| Harness | Claude Code |
| Wall time | ~a day and a half, across two sessions |
| Output tokens | 31M |
| Initial ideas generated | 650 |
| Subagents orchestrated | ~60 |
| Shell commands run | 2,400 |
| The shape matters as much as the total: **650 ideas generated, then ~60 subagents | |
| orchestrated to pursue them**, with thousands of numerical validation checks along | |
| the way. This is search plus filtering at a scale a person cannot run by hand, not | |
| a single model emitting a proof — and it is the same agentic | |
| pattern this wiki tracks for coding work, pointed at mathematics | |
| (source). |
Results
| Bound on zeros satisfying the hypothesis | Value |
|---|---|
| Previous best, from decades of human work | 41.6% |
| This result | 67.2% |
| Required for the Riemann hypothesis | 100% |
| The 41.6% figure is described as representing decades of incremental progress | |
| by human researchers, which is what makes the size of the step the story | |
| (source). |
Significance
The verification is the significance, not the bound. This wiki has recorded three prior AI-mathematics claims — An OpenAI model has disproved a central conjecture in discrete geometry, OpenAI Parameter Golf — What It Taught Us and the ten results announced with Astra — and AI for Mathematics exists because the recurring problem is the gap between a model producing something and anyone being able to check it. Here the check is public and machine-verifiable: a Lean proof in an open repository, plus two named human reviewers. A reader does not have to take Anthropic's word for the mathematics.
The second point is what Anthropic says against its own result. The bound emerged as an unintended byproduct of asking Claude to attempt the full hypothesis, and Anthropic does not expect this approach to lead to a full proof (source). That is a lab publishing the ceiling of its own result alongside the headline, and it is the opposite of how the Astra mathematics claims were framed.
Open Questions
- Which model? "An unreleased research version of Claude" is all that is stated. No relationship to Claude Opus 5 or Claude Fable 5 is given, and no capability figures.
- What did the human reviewers do? Conrey and Goldston are named as reviewers; nothing read describes what they changed, or whether the human editing step that both OpenAI mathematics results depended on was needed here.
- Is the Lean proof complete, or does it depend on unverified lemmas? The repository is public and was not readable this run.
- How much of the 31M tokens was wasted? 650 ideas producing one result is a hit rate, and no source read gives the shape of the other 649.
- No arXiv ID surfaced. Whether the paper is going to peer review is unstated.
Cite
Claude. More than two thirds of the zeros of the Riemann zeta function lie on the critical line. Anthropic, San Francisco, 2026. https://www.anthropic.com/research/riemann-zeta
Lean formalisation: https://github.com/anthropics/zeta-23-lean