---
id: "2026-08-10-claude-riemann-zeta-zeros-two-thirds"
url: "https://postcutoff.com/e/2026-08-10-claude-riemann-zeta-zeros-two-thirds/"
as_of: "2026-11-13T13:49:00+01:00"
date: "2026-08-10"
date_precision: day
category: science
importance: 5
confidence: high
status: [Result confirmed]
verification: Lean-verified
sources: 10
editor: human editor
human_review: null
version: "2026-10-23"
---

As of: 2026-11-13 13:49 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-08-10-claude-riemann-zeta-zeros-two-thirds/

# 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)
- Further follow-ups: Biao Wang extended the results to short intervals using Lamzouri's method, with ChatGPT-6 Astra implementing the proof ideas (arXiv 2609.07918, 7 Sep); Kristian Muri Knausgård raised the distinct-zeros proportion to more than 83.69%, calling the paper 'primarily an experiment in AI-assisted mathematical research' done with OpenAI Codex and Anthropic Claude, with a Lean formalisation (arXiv 2609.33043, 27 Sep)
- 6 Oct 2026: Knausgård again (arXiv 2610.08965): more than 83.9% of zeros distinct (liminf ≥ 1645064/1960733) and more than 67.35% simple and on the critical line (≥ 0.67353; 'earlier work proves 0.67250'), improving Claude's two-thirds bound. 'The arguments, the computations, the Lean formalisation, the figures and much of the text were developed with substantial assistance from AI systems (OpenAI Codex and Anthropic Claude) under the author's direction.' Proved in Lean 4 with 'three axioms beyond those of Mathlib, one per inequality. Each records that a search program returned true; Lean's kernel does not check the run itself'
- 17 Oct 2026: Juan Arias de Reyna (Seville) posted 'Human Exposition of Claude's 2/3 Theorem on the zeros of zeta' (arXiv 2610.24612), a self-contained re-proof of N0(T) ≥ 0.6725 N(T) for a future book: 'This proof is one of the first important results obtained by a machine: Claude the AI by Anthropic.' He calls the original arXiv paper 'like the notes of a poor student. Filled with bad notations, wrong assertions and unneeded statements', but the argument 'actually very simple and much easier to follow than the previous ones'; 0.6725 matches Montgomery–Taylor's constant 'and not by coincidence'
- arXiv comment on Alpöge & Furman, arXiv 2608.13637: 'Proof discovered autonomously by Claude (Anthropic); verified and communicated by the listed authors. See §1 for provenance.'
- Expert survey on Terence Tao's blog (22 Oct 2026, guest post by Caroline Turnage-Butterbaugh, a co-author of the BGST-B pair-correlation papers Claude built on): 'On August 10, 2026, Anthropic announced that an unreleased research model of Claude had connected the BGST-B approach above with a 2000 paper by Bombieri to unconditionally prove' the 67.25% bound, 'soon after verified by Alpöge and Furman'; she calls Lamzouri's re-proof 'conceptually simpler' and stresses that even 100% would not prove RH. Tao notes the post 'was initially written in a different file format and converted using AI'

## 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.

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 102 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 41 days after its cutoff
- Gemini 4 Hydrogen (training cutoff March 2026): 132 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 71 days after its cutoff

## Sources

1. [Farzad Aryan: On the Imaginary Part of ζ′/ζ, with 'An Expository Note on Claude's Argument' (arXiv 2610.12853, 9 Oct 2026)](https://arxiv.org/abs/2610.12853) (arxiv.org, paper)
2. [Arias de Reyna: Human Exposition of Claude's 2/3 Theorem on the zeros of zeta (arXiv 2610.24612, 17 Oct 2026)](https://arxiv.org/abs/2610.24612) (arxiv.org, paper)
3. [anthropics/formal-math: zeta23 Lean formalization](https://github.com/anthropics/formal-math) (github.com, code)
4. [Anthropic: Claude and the zeros of the Riemann zeta function](https://www.anthropic.com/research/riemann-zeta) (anthropic.com, official)
5. [More than two thirds of the zeta zeros are simple and on the critical line (arXiv 2608.13637)](https://arxiv.org/abs/2608.13637) (arxiv.org, paper)
6. [Lamzouri: independent proof (arXiv 2609.02882)](https://arxiv.org/abs/2609.02882) (arxiv.org, paper)
7. [arXiv 2609.07918: Simple critical zeros and distinct zeros of the Riemann zeta-function in short intervals (Wang)](https://arxiv.org/abs/2609.07918) (arxiv.org, paper)
8. [arXiv 2609.33043: More than 83.69% of the zeros of the Riemann zeta function are distinct (Knausgård)](https://arxiv.org/abs/2609.33043) (arxiv.org, paper)
9. [arXiv 2610.08965: More than 83.9% of the zeros are distinct and more than 67.35% are simple and on the critical line (Knausgård)](https://arxiv.org/abs/2610.08965) (arxiv.org, paper)
10. [Tao's blog: Simple and critical zeros of the Riemann zeta-function (guest post by Turnage-Butterbaugh, 22 Oct)](https://terrytao.wordpress.com/2026/10/22/simple-and-critical-zeros-of-the-riemann-zeta-function/) (terrytao.wordpress.com, discussion)

## Changes

- 2026-10-23: Added Turnage-Butterbaugh's survey guest post on Tao's blog (22 Oct)
- 2026-10-20: Added Arias de Reyna's independent expository re-proof (arXiv 2610.24612)
- 2026-10-19: Added the AI-provenance statement verbatim from the arXiv record
- 2026-10-19: Added video(s) fireship-summer-math-fell-to-the-machines
- 2026-10-13: Related: added [October 2026: the arXiv stream of conjectures settled with disclosed AI help continues](https://postcutoff.com/e/2026-10-13-ai-assisted-results-october-2026/) (October catalogue, split out of the summer catalogue)
- 2026-10-13: Added Knausgård's second paper (arXiv 2610.08965): 83.9% distinct, 67.35% simple and on the line, Lean 4 with three search axioms
- 2026-10-12: Added Farzad Aryan's paper (arXiv 2610.12853), an expository account of Claude's argument by the author of the unconditional pair-correlation framework it built on; it says Claude "replaced the termwise positivity argument by a global linear-algebraic analysis"
- 2026-10-07: Science block: short label for the open-problems table
- 2026-09-30: Added AI-assisted follow-ups (Wang 2609.07918; Knausgård 2609.33043, 83.69% distinct zeros)
- 2026-09-29: Added link to Anthropic's formal-math repository (zeta23 Lean project)
- 2026-09-29 (filed): Created

## Related

- 2026-10-13: [October 2026: the arXiv stream of conjectures settled with disclosed AI help continues](https://postcutoff.com/e/2026-10-13-ai-assisted-results-october-2026/index.md)
- 2026-09-30: [Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help](https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/index.md)
- 2026-09-04: [Claude produces the first complete machine-checked proof of Fermat's Last Theorem in Lean, in 11 days](https://postcutoff.com/e/2026-09-04-claude-formalizes-fermats-last-theorem/index.md)
- 2026-08-12: [Claude-assisted constructions complete Hadamard matrices for every order below 2000](https://postcutoff.com/e/2026-08-12-claude-hadamard-matrices-below-2000/index.md)
- People: [Terence Tao](https://postcutoff.com/person/terence-tao/), [Levent Alpöge](https://postcutoff.com/person/levent-alpoge/)
- Archived: 2 videos, listed in https://postcutoff.com/e/2026-08-10-claude-riemann-zeta-zeros-two-thirds/index.json