HEXAGONbeta

Joint marked moments and optimized zero-free half-planes for unitary Hecke L-functions

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

Contributed by Baiying Liu ↗

Version 1 / Nov 13, 2026 / CC BY 4.0

Abstract

In this paper, we improve the uniform zero-free bound for the Quasi-Riemann Hypothesis and obtain the half-plane LK (s,ω) ̸= 0 (Re s>B∗ = 0.8733458561067185768...) for every continuous unitary Hecke character ω over every number field K, with the possible norm-character pole permitted. We prove joint moment estimates for inverse and ordinary character sums and apply them to zero detection and the averaged Mellin numerator. The proof uses the earlier number-field theta and moment theorems and the unitary estimates of Hu and Gao. The same half-plane holds for finite-dimensional continuous unitary Weil representations, with pole orders equal to the multiplicities of their norm-character constituents. We determine B∗ asanalgebraicnumberofdegreenineandastheinfimumofthespecifiedpositivecomparison, allowing unequal ordinary moments and variable comparison lengths.

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 [4] and in [1].

References

  1. Axiom Math, Dedekind Zeta Zero-Free Region , Lean formalization, revision 09afbb4ce6570c1615d58509dc197fc9f0956b7d . Pinned source repository (https://github.com/AxiomMath/DedekindZetaZeroFree/tree/09afbb4ce6570c1615d58509dc197fc9f0956b7d).link
  2. Z. Gao, Unitary character moments and a zero-free half-plane for unitary Weil LL-functions , Hexagon 2611.00334v1, 2026. https://hexagonmath.org/2611.00334v1.hexagon
  3. B. Hu, A seven-eighths zero-free half-plane for unitary Weil LL-functions , Hexagon 2610.00216v2, 2026. https://hexagonmath.org/2610.00216v2.hexagon
  4. B. Liu, Lean formalizations , accompanying source-code repository, 2026, Lean 4.34.1. GitHub repository for the Lean formalizations (https://github.com/liubaiying101/Improved-zero-free-half-planes-for-the-quasi-Riemann-hypothesis).link
  5. B. Liu, Selected character moments, zero-free half-planes for Hecke LL-functions over number fields, and Quasi-Riemann Hypothesis , Hexagon 2611.00082v1, 2026. https://hexagonmath.org/2611.00082v1.hexagon
  6. B. Liu, Zero-free half-planes for Hecke LL-functions over number fields and Quasi-Riemann Hypothesis , arXiv:2610.26835v2, 2026. https://arxiv.org/abs/2610.26835v2.arXiv
  7. Z. Qi, Theory of Bessel Functions of High Rank--II: Hankel Transforms and Fundamental Bessel Kernels , arXiv:1411.6710v3, 2017. https://arxiv.org/abs/1411.6710.arXiv
  8. TheSil, Add Dirichlet nonvanishing bound 0.8733468 , QuasiRiemannTracker PR 137 (12 November 2026), revision 9d81a4c0b0e8338ef458f3ba5139657ae5404fde . https://github.com/GTimG/QuasiRiemannTracker/pull/137.link

Version history

  1. v1 · 54 pagesSubmitted by Baiying LiuInitial depositCurrentNov 13, 2026