Unitary character moments and a zero-free half-plane for unitary Weil L-functions

Zhongpai  Gao

Unitary character moments and a zero-free half-plane for unitary Weil L-functions

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theory math.RT — Representation Theory

Contributed by Zhongpai Gao ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

We extend Liu's moment estimate for ordinary ideal-character sums to two independent fixed unitary twists, preserving the full length range of the finite-order theorem. The proof keeps the two character coefficients separate through the Gauss transformations and controls exceptional terms by the poles of their Mellin transforms. The case of two equal twists strengthens the fourth moment used in zero detection. Using the analytic results of Hu and Liu specified in the paper, we obtain the zero-free half-plane

\operatorname{Re}s>(777-\sqrt{1081})/852=0.8733819666\ldots

for every unitary Hecke

L

-function over every number field, including characters of infinite order. This extends the known finite-order bound to the full unitary family. Unitary induction gives the same half-plane for finite-dimensional continuous unitary Weil representations, with the norm-character poles counted exactly.

Provenance statement

Zhongpai Gao directed the research objectives and scope. OpenAI Codex was used to develop the argument, draft and revise the exposition, and perform internal checks. Python and SymPy were used for symbolic calculations and exact numerical certificates. The work builds on the analytic results of Hu and Liu identified in the manuscript. The computations check scalar inequalities; they do not independently certify the analytic proof. The new endpoint has not been fully formalized in Lean.

Tools used

OpenAI: CodexVersion 6.1 Terra

References

B. Hu, A seven-eighths zero-free half-plane for unitary Weil $L$ -functions , 2026. Hexagon 2610.00216v2 (https://hexagonmath.org/2610.00216v2). Mathematical locators follow source revision edbecbec (https://github.com/Rogerhu12/hecke-seven-eighths/blob/edbecbec14e1a2efd50d1f0606b475fa5eab88f8/weil_seven_eighths_revised.tex).hexagon
B. Liu, Selected character moments, zero-free half-planes for Hecke $L$ -functions over number fields, and Quasi-Riemann Hypothesis , 2026. Hexagon 2611.00082v1 (https://hexagonmath.org/2611.00082v1).hexagon
B. Liu, Zero-free half-planes for Hecke $L$ -functions over number fields and Quasi-Riemann Hypothesis , 2026. arXiv:2610.26835v2 (https://arxiv.org/abs/2610.26835v2), Proposition 13.4 and Lemma 13.10.arXiv
OpenAI, The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $\mathrm{Re}(s)>7/8$ , 30 September 2026. OpenAI Math Release (https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/README.md).link

Version history

v1 · 11 pagesSubmitted by Zhongpai GaoInitial depositCurrentNov 11, 2026