Cureton and Thorner, with AxiomProver, extend the quasi-Riemann zero-free half-plane to every Dedekind zeta function (Re s > 31/32, Lean-formalized)
Awaiting review
Importance: major (4 of 5)The takeaway
A preprint posted on Hexagon on Oct 12, 2026 by Dean Cureton and Jesse Thorner proves that the Dedekind zeta function of any number field has no zeros with Re s > 31/32, and the same for finite-order Hecke L-functions.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Lean-verified
- Importance
- Major (4 of 5)
- Last verified
- 18 October 2026
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None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 104 days before it.
Key facts
- Paper: ‘A quasi-Riemann hypothesis for Dedekind zeta functions’ (Hexagon 2610.00200, math.NT; v1 Oct 12, v2 Oct 14, 2026)
- Theorem 1.1: for every number field K, ζ_K(s) ≠ 0 for Re(s) > 31/32. Corollary 1.2: the same half-plane is zero-free for finite-order Hecke L-functions over K (wherever holomorphic; s = 1 included for nonprincipal characters)
- AI disclosure (abstract): ‘These results were proved and formalized in Lean using AxiomProver’; Hexagon page: ‘The proofs and Lean formalization were generated by AxiomProver’; ‘Human understanding: some parts’
- Repo README (AxiomMath/DedekindZetaZeroFree, created Oct 12): ‘The investigators, Dean Cureton and Jesse Thorner, directed the research and made the mathematical judgments. AxiomProver generated the proofs and Lean formalization’
- Abstract: ‘the hardest part was the formalization of the work of Kazhdan and Patterson on metaplectic forms in order to accommodate all number fields’
- Lean: formal challenge Solution/DedekindZetaSolution.lean; only the standard axioms propext, Quot.sound, Classical.choice; depends on Mathlib, Tau Ceti and a vendored class-field-theory library, with code adapted from openai/math
- Builds on: ‘recent work on Artin’s primitive root conjecture’ (OpenAI’s Oct 6 release, family 029) that gave a zero-free half-plane for finite-order Hecke L-functions over cyclotomic fields
- Already used: Frei, Sofos and Widmer (arXiv 2610.18491) cite ‘recent breakthroughs of OpenAI and Cureton–Thorner’
What happened
OpenAI’s Oct 6 math release claimed a fixed zero-free half-plane for ζ and Dirichlet L-functions (Re s > 7/8), and separately one for Hecke L-functions over cyclotomic fields in its work on Artin’s primitive root conjecture. On Oct 12 Dean Cureton and Jesse Thorner posted a preprint on Hexagon, the repository for LLM-assisted mathematics, that follows that strategy for every number field. The bound is weaker (31/32), but it covers all Dedekind zeta functions and all finite-order Hecke L-functions. Axiom Math’s AxiomProver wrote the proofs and the Lean formalization, which is public and uses only Lean’s standard axioms. The abstract says the hard step was formalizing Kazhdan and Patterson’s work on metaplectic forms.
The story reached Chinese tech press on Oct 18 (36Kr). That article mixes in other results and loosely calls this a proof of the “Generalized Riemann Hypothesis”. It is not: the result is a zero-free half-plane, and the Riemann hypothesis line Re s = 1/2 remains open. 36Kr also claimed that Bürgisser had proved VP ≠ VNP; we found no source for that and do not repeat it.
Why it matters
It is one of the first results to build directly on OpenAI’s claimed quasi-Riemann breakthrough. It comes with a machine-checked proof and with named mathematicians directing the work, and another paper (Frei–Sofos–Widmer) already uses it. If OpenAI’s base results hold up, the number-field version extends their reach to algebraic number theory. It has not been refereed.
What is disputed or not yet verified
| Verification | Formal proof in Lean (public repo); preprint, not peer-reviewed; no independent expert check found yet |
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Sources
4 sources from 4 sites. Numbers match the chips in the text.
4 sources: 3 primary, 1 press
Primary
- Hexagon: A quasi-Riemann hypothesis for Dedekind zeta functions (Cureton, Thorner)hexagonmath.org, paper
- GitHub: AxiomMath/DedekindZetaZeroFree (Lean 4 formalization)github.com, code
- Frei, Sofos, Widmer (arXiv 2610.18491), citing the resultarxiv.org, paper
Press
Changes
- Filed (the Lean repo had not been cited)