Zero-free half-planes for Hecke L-functions over number fields and Quasi-Riemann Hypothesis
Abstract
In [Liu26a], varying the length and moment parameters in the cubic-theta argument of [Ope26a] gave the zero-free half-plane Re(s)>Bnew=0.874957019420098946… for finite-order Hecke L-functions over ℚ(√(−3)) and for Dirichlet L-functions, and its optimality in the specified exponent comparison system. In this paper, we extend the argument to finite-order Hecke L-functions over cyclotomic fields containing μ12 and, by class-field descent, to every pair (K,η) whose cyclic class field has metabelian normal closure over ℚ, including every finite-order character over every abelian number field. Then, adapting the averaged numerator estimate developed by Xu and Sang [XS26] for Hecke L-functions over ℚ(√(−3)) and combining it with a limiting contradiction argument, we obtain the boundary B♭=0.874886647039964…, the root near $0.8749$ of 133488B4−376596B3+384534B2−167541B+26231; it is the exact infimum of the refined comparison system, unattained by finite chains starting above it. Since the analytic argument uses only μ12⊂ F, we run it over F=K(ζ12) for an arbitrary number field K and descend: every finite-order Hecke L-function over every number field is zero-free for Re(s)>B♭, giving a uniform Quasi-Riemann Hypothesis over number fields. The analytic argument uses the exceptional-theta constructions and local calculations of the preprint [Ope26b].
Related
- Some applications of the quasi-Riemann hypothesis to class groups and small generators of number fields
- Improved zero-free half-planes for the quasi-Riemann hypothesis
- Ramanujan Conjecture and Riemann Hypothesis for the unitary groups over function fields
- Explicit formula for the discrete Laplace transform of the Möbius function, related special functions, and a criterion for the Riemann hypothesis
- Angular Monotonicity of Completed Dirichlet L-Functions via the Three-Term Zeta Decomposition: A Proof of the Generalized Riemann Hypothesis for Non-Principal Characters
BibTeX
@misc{liu2026zerofree,
title = {{Zero-free half-planes for Hecke $L$-functions over number fields and Quasi-Riemann Hypothesis}},
author = {Baiying Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26835},
eprint = {2610.26835},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26835},
}
RIS
TY - UNPB
TI - Zero-free half-planes for Hecke L-functions over number fields and Quasi-Riemann Hypothesis
AU - Baiying Liu
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.26835
UR - https://arxiv.org/abs/2610.26835
AB - In [Liu26a], varying the length and moment parameters in the cubic-theta argument of [Ope26a] gave the zero-free half-plane Re(s)>B_new=0.874957019420098946… for finite-order Hecke L-functions over ℚ(√(−3)) and for Dirichlet L-functions, and its optimality in the specified exponent comparison system. In this paper, we extend the argument to finite-order Hecke L-functions over cyclotomic fields containing μ_12 and, by class-field descent, to every pair (K,η) whose cyclic class field has metabelian normal closure over ℚ, including every finite-order character over every abelian number field. Then, adapting the averaged numerator estimate developed by Xu and Sang [XS26] for Hecke L-functions over ℚ(√(−3)) and combining it with a limiting contradiction argument, we obtain the boundary B_♭=0.874886647039964…, the root near $0.8749$ of 133488B^4−376596B^3+384534B^2−167541B+26231; it is the exact infimum of the refined comparison system, unattained by finite chains starting above it. Since the analytic argument uses only μ_12⊂ F, we run it over F=K(ζ_12) for an arbitrary number field K and descend: every finite-order Hecke L-function over every number field is zero-free for Re(s)>B_♭, giving a uniform Quasi-Riemann Hypothesis over number fields. The analytic argument uses the exceptional-theta constructions and local calculations of the preprint [Ope26b].
ER -