HEXAGONbeta

Exceptional zeros of Hecke LL-functions

Primarily human-written textHuman understanding: all partsmath.NT — Number Theory

Contributed by Christopher Frei ↗, Wataru Kai ↗

Version 2 / Oct 22, 2026 / CC BY-SA 4.0

Abstract

We prove that Hecke LL-functions have no exceptional zeros. We do so by generalising a variant due to Kerr of the proof by OpenAI in the case of Dirichlet LL-functions. v2: Added a reference to independent concurrent work by Banks.

Provenance statement

This draft is not intended for publication. We intend to publish its contents as part of another work in progress by the authors. Due to the recent interest in the OpenAI proof, we deemed it valuable to make this part available in advance. This draft and the mathematics in it were created without the use of artificial intelligence tools.

References

  1. Banks, W.D. On the OpenAI proof that Landau-Siegel zeros do not exist. arXiv:2610.15483, 2026arXiv
  2. Banks, W.D. Landau-Siegel zeros over a base field and effective Brauer-Siegel. arXiv:2610.31402arXiv
  3. A quasi-Riemann hypothesis for Dedekind zeta functions. 2026. Contributors: Dean Cureton; Jesse Thorner. Hexagon preprint hexagon:2610.00200v2hexagon
  4. On Landau–Siegel zeros. 2026. Contributors: Ben Green. Hexagon preprint hexagon:2610.00311v2hexagon
  5. Iwaniec, H.; Kowalski, E. Analytic number theory. American Mathematical Society, Providence, RI, vol. 53, pp. xii+615. 2004DOI
  6. Kerr, B. A simpler proof that there are no Landau–Siegel zeros. arXiv:2610.21099, 2026arXiv
  7. Lagarias, J. C.; Montgomery, H. L.; Odlyzko, A. M. A bound for the least prime ideal in the Chebotarev density theorem. Invent. Math., vol. 54, no. 3, pp. 271–296. 1979DOI
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  9. Masser, D. W. On polynomials and exponential polynomials in several complex variables. Invent. Math., vol. 63, no. 1, pp. 81–95. 1981DOI
  10. OpenAI. Uniform exclusion of Landau–Siegel zeros. 2026. OpenAI Math Release preprint OAI:Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026 (https://github.com/openai/math/blob/main/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf)link
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  1. Rosen, M. A generalization of Mertens' theorem. J. Ramanujan Math. Soc., vol. 14, no. 1, pp. 1–19. 1999

Version history

  1. v1Submitted by Wataru KaiInitial depositSupersededOct 22, 2026
  2. v2Submitted by Wataru KaiSubmitted as a new versionCurrentOct 29, 2026