{"schema":"postcutoff/event@1","as_of":"2026-10-14T07:26:00+02:00","url":"https://postcutoff.com/e/2026-10-11-landau-siegel-zeros-banks-explicit-account/","md":"https://postcutoff.com/e/2026-10-11-landau-siegel-zeros-banks-explicit-account/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-11-landau-siegel-zeros-banks-explicit-account","date":"2026-10-11","date_precision":"day","short_title":"William Banks writes out OpenAI's Landau–Siegel proof with explicit constants","deck":"(1 − β) log q ≥ 0.003 for every real character, and says he verified it in full","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.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.15483: On the OpenAI proof that Landau-Siegel zeros do not exist","url":"https://arxiv.org/abs/2610.15483","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"OpenAI: Uniform exclusion of Landau–Siegel zeros (1 Oct manuscript)","url":"https://github.com/openai/math/blob/main/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"}],"official":2,"filed":"2026-10-14","updated":"2026-10-14","orgs":["OpenAI","Anthropic","University of Missouri"],"title":"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","summary":"arXiv 2610.15483 (11 Oct 2026) by number theorist William D. Banks gives \"a detailed account\" of the argument in OpenAI's 1 Oct manuscript claiming that Landau–Siegel zeros do not exist, as a direct proof with explicit constants: for every primitive real character χ of conductor q ≥ 10¹⁰, a real zero β of L(s, χ) satisfies (1 − β) log q ≥ 3.0 × 10⁻³, and with a computation of Lu, Zaman and Zhao this holds for every q ≥ 3. The argument is credited to OpenAI; Claude produced the first report and expanded draft; Banks says he \"verified the mathematical arguments in their entirety\". It is an unrefereed preprint but the first independent expert write-up of the claim.","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.'"],"key_numbers":[],"tags":["math","number-theory","siegel-zeros","openai","claude","verification","effective-bounds"],"science":{"field":"mathematics","subfield":"analytic number theory","problem":"Landau–Siegel zeros (exceptional real zeros of real Dirichlet L-functions)","result":"Explicit version of OpenAI's claimed exclusion: (1 − β) log q ≥ 3.0 × 10⁻³ for every primitive real character of conductor q ≥ 3 (with Lu–Zaman–Zhao for small q).","open_since":"1935","ai_system":["OpenAI internal model (original proof)","Claude"],"human_role":"Expert reconstruction: the proof is OpenAI's; Claude drafted the report, expansion and constants; Banks checked every line","verification":{"key":"lean","label":"Lean-verified","text":"Expert-checked by the author (unrefereed); OpenAI's original has a Lean formalization without explicit constant"},"status":"pending","shock":"An effective, explicit constant would make Siegel's theorem effective, a goal analytic number theory had pursued for 90 years."},"body_md":"## What happened\n\nOn 11 Oct 2026 William D. Banks posted [arXiv 2610.15483](https://arxiv.org/abs/2610.15483) (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](/e/2026-10-06-quasi-riemann-hypothesis-openai/)) 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.\n\nThe 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.\n\nThe 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\".\n\nNo public reactions from other number theorists were found on 14 Oct (web search; X not searched directly).\n\n## Why it matters\n\nThe 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.","disputed":[{"label":"Verification","text":"Expert-checked by the author (unrefereed); OpenAI's original has a Lean formalization without explicit constant","sources":[]}],"related":[{"id":"2026-10-06-quasi-riemann-hypothesis-openai","url":"https://postcutoff.com/e/2026-10-06-quasi-riemann-hypothesis-openai/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI model claims the quasi-Riemann hypothesis","deck":"No zeta or Dirichlet L-function zeros with Re s > 7/8, and no Landau–Siegel zeros (Lean-formalized)","takeaway":"If correct, this is the largest advance on the zeros of ζ since the 19th century, and it would remove the Siegel-zero obstruction that runs through modern analytic number theory.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":11,"official":9,"filed":"2026-10-07","updated":"2026-10-14","orgs":["OpenAI"]},{"id":"2026-10-06-openai-math-release-722-manuscripts","url":"https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems","deck":"Including quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors","takeaway":"If even a fraction of these results hold up, this is the largest single jump in mathematical knowledge on record, produced by an AI system in about six weeks.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":84,"official":13,"filed":"2026-10-07","updated":"2026-10-14","orgs":["OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-14","type":"filed","text":"Created from the arXiv paper"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-14","run":null,"sources_read":"The arXiv paper","updated":"2026-10-14","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":164,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":103,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":194,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":133,"in_training_data":false}],"short_url":"https://postcutoff.com/s/william-banks-writes-out"}