William Banks writes out OpenAI’s Landau–Siegel proof with explicit constants
(1 − β) log q ≥ 0.003 for every real character, and says he verified it in full
Awaiting review
Importance: major (4 of 5)The takeaway
The first detailed human account of OpenAI’s claim that Landau–Siegel zeros do not exist gives an explicit, effective constant. Banks used Claude to draft it but says he worked through every line.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Lean-verified
- Importance
- Major (4 of 5)
- Last verified
- 14 October 2026
Your AI and this story
- GPT-6 Astra164 days after its cutoff
- Claude Opus 5.5103 days after its cutoff
- Gemini 4 Argon194 days after its cutoff
- Grok 4.7133 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 103 days before it.
Key facts
- Paper: ‘On the OpenAI proof that Landau-Siegel zeros do not exist’, William D. Banks, arXiv 2610.15483 [math.NT], 19 pages, submitted 11 Oct 2026 (written at the Max Planck Institute for Mathematics, Bonn)
- Theorem: absolute q₀ = 10¹⁰ and δ₀ = 3.0 × 10⁻³ such that every real zero β ∈ (0,1) of L(s, χ), χ primitive real of conductor q ≥ q₀, satisfies (1 − β) log q ≥ δ₀; with Lu–Zaman–Zhao’s computation the bound holds for all q ≥ 3
- Contrast: OpenAI’s Lean scope note gave no explicit value of the constant; an effective constant matters because Siegel’s 1935 theorem is ineffective (no computable constant), which blocks effective class-number bounds
- Method (abstract): an interpolation-determinant argument in the biquadratic field generated by √2 over the quadratic field of χ; a real zero near 1 forces χ(p) = −1 for most primes p, and the Frobenius congruence at those primes makes a nonzero determinant divisible by more prime powers than its archimedean size allows. The paper does not rely on the 7/8 quasi-Riemann manuscript
- AI use: ‘The argument presented in this paper is due to OpenAI [6]. The large language model Claude (Anthropic) was used throughout the preparation of the paper. It produced a report on the manuscript [6] together with an expanded draft of the proof, which is the basis of the present text. It was also used for … working out the explicit constants … and for proof checking. The author has worked through every line of the paper, has verified the mathematical arguments in their entirety, and takes full responsibility for its content.’
What happened
On 11 Oct 2026 William D. Banks posted arXiv 2610.154831 (listed 13 Oct). He rearranges the argument of OpenAI’s 1 Oct manuscript “Uniform exclusion of Landau–Siegel zeros” (part of the quasi-Riemann family) as a direct proof and makes every constant explicit. The result: for primitive real characters χ of conductor q ≥ 10¹⁰, every real zero β of L(s, χ) satisfies (1 − β) log q ≥ 0.003. A recent computation by Lu, Zaman and Zhao covers smaller conductors, so the bound holds for all q ≥ 3.
The method is algebraic: an interpolation determinant in the biquadratic field generated by √2 over the quadratic field attached to χ. A zero close to 1 forces χ(p) = −1 for most primes, and Frobenius congruences then make a nonzero algebraic integer too divisible for its size. The paper does not use the 7/8 zero-free half-plane.
The acknowledgments are explicit about AI use: the argument “is due to OpenAI”; Claude “produced a report on the manuscript … together with an expanded draft of the proof, which is the basis of the present text”, worked out the explicit constants and helped check proofs; Banks “has worked through every line of the paper, has verified the mathematical arguments in their entirety”.
No public reactions from other number theorists were found on 14 Oct (web search; X not searched directly).
Why it matters
The Siegel-zero claim was the part of OpenAI’s release that specialists called the most consequential (“it settles the question”, Hector Pasten). Banks’s paper is the first written check by an outside number theorist, and it adds what the Lean version lacked: a numerical constant. An explicit lower bound for 1 − β would make classical ineffective results, such as Siegel’s class-number bound, effective. The usual caveat applies: one author’s check, with heavy Claude assistance, in an unrefereed preprint.
What is disputed or not yet verified
| Verification | Expert-checked by the author (unrefereed); OpenAI’s original has a Lean formalization without explicit constant |
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Sources
2 sources from 2 sites. Numbers match the chips in the text.
2 sources: 2 primary
Primary
Changes
- Filed from the arXiv paper