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Landau--Siegel zeros over varying number fields

A mix of human-written and AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Baiying Liu ↗

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

Let KK be a number field of degree nn and absolute discriminant DKD_K. We give a direct arithmetic proof that every real zero β∈(0,1)\beta\in(0,1) of ζK\zeta_K satisfies \[ (1-\beta)\{\log D_K+n\log(2n)+6n+1\}\geq\frac1{1370}. \] The bound is explicit for arbitrary varying fields, including nonnormal fields, and gives (1−β)log⁡DK≫1/log⁡(2n)(1-\beta)\log D_K\gg1/\log(2n) for $K\ne\Q$. The proof combines OpenAI's determinant method, Masser's zero estimate as used by Kerr, and Banks' selected-embedding calculation. The resulting bound is expressed directly in the degree and discriminant of KK, without introducing those of its normal closure. Stronger nonvanishing conclusions follow from recent half-plane results and positivity. The contribution here is an arithmetic proof with explicit field dependence, using neither those half-plane results nor earlier zero-exclusion results for real Dirichlet LL-functions.

Provenance statement

OpenAI Codex (GPT 6.1 Sol Medium-Ultra) and Anthropic Claude (Opus 5.5 Medium) assisted with drafting, reasoning, checking references, and auditing. The Lean formalization is in progress, based on OpenAI’s accompanying Lean sources [OpeLean].

References

  1. W. D. Banks, Landau--Siegel zeros over a base field and effective Brauer--Siegel , arXiv:2610.31402v1 (https://arxiv.org/abs/2610.31402v1), 22 October 2026.arXiv
  2. W. D. Banks, On the OpenAI proof that Landau--Siegel zeros do not exist , arXiv:2610.15483v1 (https://arxiv.org/abs/2610.15483v1), 11 October 2026.arXiv
  3. K. Barreto, An elementary proof of the non-existence of Landau--Siegel zeros , manuscript, 19 October 2026.
  4. R. Brauer, On the zeta-functions of algebraic number fields , Amer. J. Math. 69 (1947), 243--250.
  5. K. Chakhkiev, A conditional improvement of the seven-eighths zero-free threshold , preprint, version 1, 7 October 2026, Zenodo. doi:10.5281/zenodo.23221313 (https://doi.org/10.5281/zenodo.23221313).DOI
  6. J.-M. Couveignes, Enumerating number fields , Ann. of Math. (2) 192 (2020), 487--497, doi:10.4007/annals.2020.192.2.4 (https://doi.org/10.4007/annals.2020.192.2.4).DOI
  7. D. Cureton and J. Thorner, Report: A quasi-Riemann hypothesis for Dedekind zeta functions , hexagon:2610.00200v2 (https://hexagonmath.org/pdf/2610.00200v2), October 2026.hexagon
  8. B. Green, On Landau--Siegel zeros , hexagon:2610.00311v2 (https://hexagonmath.org/pdf/2610.00311v2), October 2026.hexagon
  9. B. Kerr, A simpler proof that there are no Landau--Siegel zeros , arXiv:2610.21099v1 (https://arxiv.org/abs/2610.21099v1), 15 October 2026.arXiv
  10. S. D. Kominers, Small algebraic integers and Frobenius prime sums , hexagon:2610.00451v1 (https://hexagonmath.org/pdf/2610.00451v1), October 2026.hexagon
Show all 27 referencesShow fewer references
  1. B. Liu, Improved zero-free half-planes for the quasi-Riemann hypothesis , arXiv:2610.12234v2 (https://arxiv.org/abs/2610.12234v2), 12 October 2026. The first version, Slightly improved zero-free half-planes for the quasi-Riemann hypothesis , was posted on 8 October 2026: arXiv:2610.12234v1 (https://arxiv.org/abs/2610.12234v1).arXiv
  2. B. Liu, Zero-free half-planes for Hecke LL-functions over number fields and Quasi-Riemann Hypothesis , arXiv:2610.26835v2 (https://arxiv.org/abs/2610.26835v2), 29 October 2026.arXiv
  3. B. Liu, An improved zero-free half-plane for Hecke LL-functions over number fields and Quasi-Riemann Hypothesis , hexagon:2610.00655v3 (https://hexagonmath.org/2610.00655?version=3), 3 November 2026.hexagon
  4. B. Liu, Selected character moments, zero-free half-planes for Hecke LL-functions over number fields, and Quasi-Riemann Hypothesis , hexagon:2611.00075v1, 3 November 2026.hexagon
  5. D. W. Masser, On polynomials and exponential polynomials in several complex variables , Invent. Math. 63 (1981), 81--95, doi:10.1007/BF01389194 (https://doi.org/10.1007/BF01389194).DOI
  6. I. J. Mermet, The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane ℜ(s)>0.8749570698\Re(s)>0.8749570698 , manuscript dated 8 October 2026, with accompanying Lean sources, GitHub repository CrossNox/quasi-riemann-hypothesis-extension (https://github.com/CrossNox/quasi-riemann-hypothesis-extension), accessed 28 October 2026.link
  7. J. S. Milne, Class Field Theory , version 4.03, 6 August 2020, course notes (https://www.jmilne.org/math/CourseNotes/CFT.pdf).link
  8. T. Newton, Prime Errors and the Quasi-Riemann Frontier: A Certified Conditional Boundary and Its Parameter-Optimization Barrier , preprint, 16 October 2026, Zenodo. doi:10.5281/zenodo.23414581 (https://doi.org/10.5281/zenodo.23414581).DOI
  9. OpenAI, The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $\Rea s>7/8$ , preprint, 30 September 2026. Public manuscript (https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf).link
  10. OpenAI, Uniform exclusion of Landau--Siegel zeros , preprint, 1 October 2026. Dated manuscript (https://github.com/openai/math/blob/main/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf).link
  11. OpenAI, The Quasi-Riemann Hypothesis , shorter proof of the 11/1211/12 region, 5 October 2026. Dated manuscript (https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-October-5-2026/paper2.pdf).link
  12. OpenAI, Uniform exclusion of Landau--Siegel zeros , accompanying Lean sources, 2026. OAI.NumberTheory.SiegelZeros.Main (https://github.com/openai/math/blob/main/lean/OAI/NumberTheory/SiegelZeros/Main.lean); formalization scope (https://github.com/openai/math/blob/main/lean/docs/003.md). Accessed 29 October 2026.link
  13. H. Osada, The Galois groups of the polynomials Xn+aXl+bX^n+aX^l+b , J. Number Theory 25 (1987), 230--238.
  14. B. Poonen, Tate's thesis , MIT 18.786 lecture notes, Spring 2015, lecture notes (https://math.mit.edu/\string~poonen/786/notes.pdf).link
  15. C. S. Rajan, Distribution of values of Hecke characters of infinite order , Acta Arith. 85 (1998), 279--291, doi:10.4064/aa-85-3-279-291 (https://doi.org/10.4064/aa-85-3-279-291).DOI
  16. L. Wei, Quasi-Riemann boundary refinement in Lean , GitHub repository, 2026, LW56e2/quasi-riemann-improved (https://github.com/LW56e2/quasi-riemann-improved), accessed 28 October 2026.link
  17. Y. Xu and H. Sang, Amplified moments and a refinement of the quasi-Riemann hypothesis , preprint, revised 15 October 2026, arXiv:2610.15291v2 (https://arxiv.org/abs/2610.15291v2).arXiv

Version history

  1. v1Submitted by Baiying LiuInitial depositCurrentNov 04, 2026