An exhaustion-aware product moment and zero-free half-planes for Hecke L-functions
Abstract
We refine the ordinary product moment in the cubic-theta comparison by allowing a returned prime product to be exhausted. The return stops either when the positive-prime length inequality holds or when the established prime-free ordinary estimate applies. Keeping the extracted central coefficients gives a local normalization inequality on a larger domain. On the critical ordinary-length interval this yields a net first-mean gain of 1/20000. We prove the complementary-class, short-sum and algebraic comparison inequalities directly, and obtain the endpoint B*−1/160000=0.87446205769353…, where B* is Liu's algebraic boundary. The zero-free conclusion is conditional on the cited general-field theta, detector, prior-bound and Mellin input theorems in their stated uniform forms. The proof includes the actual return weights, both approximate-functional-equation arms, the principal-residue normalization and descent to arbitrary number fields.
Related
- Overdetermined equations and support functions on the hyperbolic and de Sitter planes
- Zero-free half-planes for Hecke L-functions over number fields and Quasi-Riemann Hypothesis
- The First Moment of Dirichlet L-functions over Generators of the Character Group
- An asymptotic formula for a cubic moment of GL2 L-functions
- Mixed moment of GL(2) and GL(3) L-functions in the level aspect
BibTeX
@misc{chang2026exhaustionaware,
title = {{An exhaustion-aware product moment and zero-free half-planes for Hecke L-functions}},
author = {Xinrui Chang and HaoMing Wang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00924},
eprint = {2611.00924},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00924},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An exhaustion-aware product moment and zero-free half-planes for Hecke L-functions
AU - Xinrui Chang
AU - HaoMing Wang
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00924
UR - https://arxiv.org/abs/2611.00924
AB - We refine the ordinary product moment in the cubic-theta comparison by allowing a returned prime product to be exhausted. The return stops either when the positive-prime length inequality holds or when the established prime-free ordinary estimate applies. Keeping the extracted central coefficients gives a local normalization inequality on a larger domain. On the critical ordinary-length interval this yields a net first-mean gain of 1/20000. We prove the complementary-class, short-sum and algebraic comparison inequalities directly, and obtain the endpoint B_*−1/160000=0.87446205769353…, where B_* is Liu's algebraic boundary. The zero-free conclusion is conditional on the cited general-field theta, detector, prior-bound and Mellin input theorems in their stated uniform forms. The proof includes the actual return weights, both approximate-functional-equation arms, the principal-residue normalization and descent to arbitrary number fields.
ER -