View source for Uniform exclusion of Landau–Siegel zeros

'''Uniform exclusion of Landau–Siegel zeros''' concerns real zeros of Dirichlet ''L''-functions of real characters close to ''s'' = 1; excluding them uniformly in the conductor is the Landau–Siegel zero problem. A manuscript dated 1 October 2026 in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]), part of the group of manuscripts described on [[The quasi-Riemann hypothesis]], claims an absolute constant ''c'' &gt; 0 such that every real zero β of every primitive nonprincipal real Dirichlet ''L''-function of conductor ''q'' ≥ 3 satisfies (1 − β) log ''q'' ≥ ''c''. The collection includes a Lean formalization of this statement. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

== Background ==
* As recounted in the introduction of the manuscript:
** The classical zero-free region of Dirichlet ''L''-functions excludes zeros near Re ''s'' = 1 except a possible simple real zero attached to a real character. Siegel's theorem of 1935 gives ''L''(1, χ) ≫<sub>ε</sub> ''q''<sup>−ε</sup> with an ineffective constant, which leaves open a sequence of real zeros with (1 − β) log ''q'' tending to zero.
** Page's theorem allows, for each ''Q'', at most one primitive real character of conductor at most ''Q'' with a real zero in a window of width proportional to 1/log ''Q''. The Deuring–Heilbronn phenomenon, developed quantitatively by Linnik and Heath-Brown, forces other zeros away from 1, with a recent explicit form by Benli, Goel, Twiss and Zaman, and Friedlander and Iwaniec discuss the logarithmic zero-gap conjecture and its relation to lower bounds for ''L''(1, χ).

== The original result ==
* '''Claim.''' Theorem 1 of ''Uniform exclusion of Landau–Siegel zeros'' states that there is an absolute constant ''c'' &gt; 0 such that every real zero β in (0, 1) of every primitive nonprincipal real Dirichlet ''L''-function of conductor ''q'' ≥ 3 satisfies (1 − β) log ''q'' ≥ ''c''. The manuscript gives no explicit value of ''c''. It notes that such zeros obstruct uniform estimates for primes in arithmetic progressions and affect quadratic class numbers through the values ''L''(1, χ).
* ''Method, as described:'' a real zero with (1 − β) log ''q'' small forces a shortage of primes with χ(''p'') = 1. The algebraic part works in the biquadratic field obtained by adjoining √2 to the quadratic field of the character, forms an interpolation determinant from conjugates of integer combinations of a basis, and extracts powers of the primes with χ(''p'') = −1 from its rows by taking ''p''-th powers modulo ''p''; the contradiction comes from comparing this divisibility, to which the primes up to the degree scale ''U'' contribute log ''U'', with a size bound of order (3/4) log ''U''.
* '''Sources.''' The manuscript [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026 Uniform exclusion of Landau–Siegel zeros] at commit <code>1581d4c</code>.
* '''Formalization, as described by the repository.''' The [https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/003.md scope note], shared with [[The quasi-Riemann hypothesis]], states that the formalization gives one constant ''c'' &gt; 0 such that every primitive nonprincipal real Dirichlet character of conductor ''q'' ≥ 3 and every real zero 0 &lt; β &lt; 1 of its ''L''-function satisfy 1 − β ≥ ''c''/log ''q'', with both character parities included. It states that no explicit value of ''c'' is given, and that the result excludes real zeros in 1 − ''c''/log ''q'' &lt; β &lt; 1 but does not rule out real zeros elsewhere in (0, 1). The Comparator statement is <code>SiegelZeros.lean</code>.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 13–20 October 2026)
|-
| 2026-10-08 || [https://github.com/mathvet/mathvet MathVet] (GitHub; review of openai/math, tables at repository commit <code>c76f1e4</code>, 14 October 2026) || Review (fidelity of the formal statements) || Rates the family's Lean statements "full" and records a Comparator pass for <code>SiegelZeros</code>, the uniform real-zero gap; its review note concerns only the Hecke half-plane statement of the family. || MathVet states that it does not claim any theorem false and reports what was machine-checked as stated.
|-
| 2026-10-10 || [https://github.com/carlostoledo1891/cert-machine/blob/d93dd5b48e798372da73a42ac2275a3eeacd9a5e/corpus/openai-math/kernel-runs/38019899278/SiegelZeros.json cert-machine], openai/math lane K, run 38019899278 || Verification (two kernels) || Records that Lean's default kernel and the nanoda kernel accept the <code>SiegelZeros</code> solution (1,029 s). The project reports two-kernel certification of 398 of 416 challenges of the collection. || Repository.
|-
| 2026-10-10 to 2026-10-14 || [https://github.com/leanprover/con-leche/blob/90577d9e49ee3aed45da3d421bdf5dc780192304/DESIGN.md con-leche development journal] (task #348) || Verification (second kernel) || In its run over all Comparator challenges of the collection at commit <code>fd4aeeb</code>, the journal's table of exceptions (time-outs, memory, errors) lists no entry for <code>SiegelZeros</code>, so it was accepted by con-leche and by Lean's official kernel. || Development journal.
|-
| 2026-10-10 || [https://github.com/AolinHAN/landau-siegel-lean/tree/919e60f4630399ba27e1f3c3d4a39d754842cf9c AolinHAN/landau-siegel-lean] (GitHub, commit <code>919e60f</code>) || Verification (independent Lean formalization) || A separate Lean formalization of the nine-page manuscript, stated with Mathlib's Dirichlet ''L''-functions and no added analytic or arithmetic hypotheses: there is ''c'' &gt; 0 such that every real zero β of ''L''(''s'', χ) for a primitive real nonprincipal character of conductor ''q'' ≥ 3 satisfies (1 − β) log ''q'' ≥ ''c''; a coverage table maps the manuscript's equations to Lean declarations. || Repository; the README reports that the whole library compiled on 10 October 2026 and that all 406 public theorems passed an axiom audit with only the standard axioms.
|-
| 2026-10-11 || W. D. Banks, ''On the OpenAI proof that Landau–Siegel zeros do not exist'' ([https://arxiv.org/abs/2610.15483 arXiv:2610.15483], v1) || Exposition (with explicit constants) || A detailed account of the argument of the Landau–Siegel manuscript, arranged as a direct proof with explicit constants: for every primitive real Dirichlet character χ of conductor ''q'' ≥ 10<sup>10</sup>, every real zero β ∈ (0, 1) of ''L''(''s'', χ) satisfies (1 − β) log ''q'' ≥ 3.0 × 10<sup>−3</sup>; combined with a computation of Lu, Zaman and Zhao, the same bound holds for every ''q'' ≥ 3. The abstract states that the underlying ideas are those of the OpenAI manuscript and that the explicit constants and the presentation are new. || arXiv v1.
|-
| 2026-10-15 (submitted) || B. Kerr, ''A simpler proof that there are no Landau–Siegel zeros'' ([https://arxiv.org/abs/2610.21099 arXiv:2610.21099]) || Simplification || States that it "simplifies the recent OpenAI proof that there are no Landau–Siegel zeros": an absolute ''c'' &gt; 0 with (1 − β) log |''q''| ≥ ''c'' for every real zero β of ''L''(''s'', χ), χ = (''q''/·) primitive and real. It keeps OpenAI's field '''Q'''(√''q'', √2) and evaluation points, uses the pigeonhole principle to find a polynomial of bounded degree vanishing at them in place of OpenAI's interpolation and determinant analysis, and derives the contradiction from Masser's 1981 estimate for polynomials vanishing on a finitely generated subgroup of '''C'''<sup>''n''</sup>. || arXiv preprint (v1). The acknowledgement states that ChatGPT assisted with the mathematical development and preparation of the paper, and thanks W. D. Banks for a draft of his exposition of the OpenAI proof.
|-
| 2026-10-15 (submitted) || I. Kimura, ''Imaginary quadratic fields of class number at most 2000'' ([https://arxiv.org/abs/2610.21511 arXiv:2610.21511]) || Use (explicit constants) || Makes explicit "a recent argument, released by OpenAI and not refereed, that excludes Landau–Siegel zeros": every real zero β of ''L''(''s'', χ) for a primitive real character χ of conductor ''q'' ≥ 3 satisfies (1 − β) log ''q'' ≥ 5.8·10<sup>−4</sup>. With it, determines all imaginary quadratic fields of class number at most 2000 unconditionally: 16,453,493 fields, the largest |''d''<sub>''K''</sub>| being 1,062,350,083; the abstract notes that such a list was previously known only for class number at most 100. The comments link code, a Lean formalization and the lists. || arXiv preprint. The abstract describes the OpenAI argument it uses as "released by OpenAI and not refereed". For class number at most 1000, the paper's count (4,115,897 fields) and largest |''d''<sub>''K''</sub>| (227,932,027, with class number 996) are the figures that zeta-lab's hunt <code>qrh_class_number</code> reported on 8 October under the 7/8 theorem (see [[The quasi-Riemann hypothesis]]); the paper itself relies on the Landau–Siegel manuscript and Lean, not on the 7/8 theorem.
|-
| 2026-10-17 || B. Green, [https://hexagonmath.org/2610.00311v1 ''On Landau–Siegel zeros''], Hexagon 2610.00311v1 || Variant of the proof || A variant of the argument of the Landau–Siegel manuscript. It works entirely in the quadratic field '''Q'''(√''d'') instead of passing to the biquadratic field '''Q'''(√''d'', √2), and replaces the manuscript's linear-algebraic interpolation by an explicit Gram determinant built from derivatives of the monomials ''z''<sup>''r''</sup>''z''′<sup>''s''</sup> at the points (''x'' + ''y''√''d'', ''x'' − ''y''√''d''); it compares the archimedean size of this determinant with its ''p''-adic valuations at the primes ''p'' with χ(''p'') = −1. Its main theorem bounds the sum of log ''p''/''p'' over the primes ''p'' ≤ ''X'' = ''q''<sup>''C''</sup> with χ(''p'') = −1 by (3/4) log ''X'' + ''O''<sub>''C''</sub>(1), which the note states implies that no Landau–Siegel zero exists. It cites the expositions of Banks and Kerr listed above. || Hexagon version 1, labelled primarily human-written text. Its provenance statement says that the mathematical arguments are all due to ChatGPT, prompted in various ways by the author, and that the text is entirely human-written; the introduction states that no human originality is claimed. A version 2 was posted on 18 October 2026.
|-
| 2026-10-20 || R. van Dijk, ''A Frobenius determinant proof of the non-existence of Landau–Siegel zeros'' ([https://hexagonmath.org/2610.00409v1 hexagon:2610.00409v1]) || Variant of the proof || States that it gives a variant of the OpenAI proof, following the simplifications of Kerr, Green and K. Barreto. A Frobenius congruence proves both divisibility and nonvanishing of the determinant; at a large inert prime supplied by the hypothetical zero, nonvanishing reduces to a polynomial root count, which replaces resultants and Wronskians. || Hexagon version 1, CC BY 4.0. The record states that the argument was discovered in discussions with GPT-6 Astra and Claude Opus 5.5 and that the new arguments should be attributed to these models; it marks human understanding as none of the parts.
|}

== Notes ==
* The earlier results are due to the authors named in the Background section, as recounted by the manuscript.
* Several follow-ups state substantial AI assistance in producing their proofs and texts, as recorded in the table.
* Until 20 October 2026 these follow-ups were listed on [[The quasi-Riemann hypothesis]], which covers the zero-free half-planes of the same group of manuscripts.
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]