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Selected character moments, zero-free half-planes for Hecke L-functions over number fields, and Quasi-Riemann Hypothesis

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

Contributed by Baiying Liu ↗

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

In this paper, we obtain a zero-free half-plane $\Rea s>B_2$ for every finite-order Hecke LL-function over every number field, where \[ B_2=\frac{777-\sqrt{1081}}{852} =0.873381966613207519929\ldots. \] We adapt selected character-sum estimates and finite moment inductions to ideal-character sums over arbitrary number fields and restore ordinary sixth powers. Two finite moment inductions give an ordinary product estimate with length condition n1+n2+2z≤Mn_1+n_2+2z\le M for sufficiently fine original prime factors. Applying this estimate to zero detection and the averaged numerator in the cubic-theta comparison gives the stated boundary. We determine the exact infimum of the resulting comparison system and verify its scalar inequalities by rational Bernstein bounds. The proof uses the earlier theta, moment and continuation theorems over number fields.

Provenance statement

OpenAI Codex (GPT 6.1 Sol Medium-Ultra) and Anthropic Claude (Opus 5.5 Medium) assisted with drafting, checking references, the search for improved bounds, and auditing. The Lean formalization is in progress, based on the Lean files in [Liu26d] and in [Axi26].

References

  1. ArviFoxy, selected character-sum estimate and induction on the source length, public contribution to QuasiRiemannTracker, pull request 97, October 2026. Source and discussion (https://github.com/GTimG/QuasiRiemannTracker/pull/97).link
  2. Axiom Math, Dedekind Zeta Zero-Free Region , Lean formalization accompanying CT26 , revision 09afbb4ce6570c1615d58509dc197fc9f0956b7d , accessed 15 October 2026. Pinned source repository (https://github.com/AxiomMath/DedekindZetaZeroFree/tree/09afbb4ce6570c1615d58509dc197fc9f0956b7d).link
  3. J. Bebel, All-moduli Dirichlet nonvanishing at 0.87338196661320752 , public contribution to QuasiRiemannTracker, pull request 101, November 2026. Source and discussion (https://github.com/GTimG/QuasiRiemannTracker/pull/101).link
  4. J. Bebel, refinement of the selected estimate with a positive margin, public contribution to QuasiRiemannTracker, pull request 99, October 2026. Source and discussion (https://github.com/GTimG/QuasiRiemannTracker/pull/99).link
  5. J. Bebel, Selected ordinary-source sextic norm , public contribution to QuasiRiemannTracker, pull request 93, October 2026. Source and discussion (https://github.com/GTimG/QuasiRiemannTracker/pull/93).link
  6. J. Bebel, Improve to 0.873381966613207519929026778642 , public contribution to QuasiRiemannTracker, pull request 103, 1 November 2026. Source and discussion (https://github.com/GTimG/QuasiRiemannTracker/pull/103).link
  7. D. Cureton and J. Thorner (investigators), Report: A quasi-Riemann hypothesis for Dedekind zeta functions , Axiom Math, preprint, hexagon:2610.00200v2 (2026). Version 2 (https://hexagonmath.org/2610.00200v2).hexagon
  8. B. Liu, Improved zero-free half-planes for the quasi-Riemann hypothesis , preprint, arXiv:2610.12234v2 (https://arxiv.org/abs/2610.12234). 12 October 2026.arXiv
  9. B. Liu, Zero-free half-planes for Hecke LL-functions over number fields and Quasi-Riemann Hypothesis , preprint, arXiv:2610.26835v2 (https://arxiv.org/abs/2610.26835v2). 29 October 2026.arXiv
  10. B. Liu, An improved zero-free half-plane for Hecke LL-functions over number fields and Quasi-Riemann Hypothesis , preprint, hexagon:2610.00655v3 (https://hexagonmath.org/2610.00655?version=1). 3 November 2026.hexagon
Show all 12 referencesShow fewer references
  1. B. Liu, Lean formalizations , accompanying source-code repository for Parts I--IV of LiuArXiv (Lean 4.34.1), 2026. GitHub repository for the Lean formalizations (https://github.com/liubaiying101/Improved-zero-free-half-planes-for-the-quasi-Riemann-hypothesis).link
  2. J. S. Milne, Class Field Theory , course notes, version 4.03, 6 August 2020. Author's notes (https://www.jmilne.org/math/CourseNotes/CFT.pdf).link

Version history

  1. v1 · 60 pagesSubmitted by Baiying LiuInitial depositCurrentNov 04, 2026