Extreme values of quadratic Hecke L-functions
Abstract
We study large values of quadratic Hecke L-functions in the conductor aspect. Let K be a fixed number field, and assume GRH for its finite-order Hecke L-functions. In a fixed ray class component with conductor norm comparable to X, we prove that maxχL(1/2+A/log2X,χ) ≥exp{(e−A+o(1)) √((log Xlog3X)/log2X)} for every fixed A≥0. Every fixed smaller constant is attained by at least X1−o(1) characters. The same count holds at a suitably slowly moving threshold approaching the displayed constant. The combinatorial input is a sparse squarefree Gál set of cardinality N, retaining the known leading constant $2$ and having square multiplicative energy N2+o(1). The energy bound reflects the low degree of the associated Boolean polynomial. Together with the resonance estimate, it yields the abundance bound. We also prove unconditional analogues for quadratic characters with prime conductor away from one fixed place over any global function field of odd characteristic. Finally, we give bounds in the fixed strip and at s=1, including the dependence on the residue of the Dedekind zeta function and the prescribed local factors.
Related
- Large values of quadratic Hecke L-functions with prime-related moduli of the Gaussian field
- Extreme Values of Quadratic Dirichlet L-functions over Monic Irreducible Polynomials in 𝔽q[t]
- Non-vanishing of central values of angular Hecke L-functions
- Distribution of central values of Hecke L-functions of n-th order characters
- Extreme values of derivatives of Dedekind zeta functions on the $1$-line
BibTeX
@misc{dong2026extreme,
title = {{Extreme values of quadratic Hecke $L$-functions}},
author = {Zikang Dong and Long Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16807},
eprint = {2609.16807},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16807},
}
RIS
TY - UNPB
TI - Extreme values of quadratic Hecke L-functions
AU - Zikang Dong
AU - Long Liu
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16807
UR - https://arxiv.org/abs/2609.16807
AB - We study large values of quadratic Hecke L-functions in the conductor aspect. Let K be a fixed number field, and assume GRH for its finite-order Hecke L-functions. In a fixed ray class component with conductor norm comparable to X, we prove that max_χL(1/2+A/log_2X,χ) ≥exp{(e^−A+o(1)) √((log Xlog_3X)/log_2X)} for every fixed A≥0. Every fixed smaller constant is attained by at least X^1−o(1) characters. The same count holds at a suitably slowly moving threshold approaching the displayed constant. The combinatorial input is a sparse squarefree Gál set of cardinality N, retaining the known leading constant $2$ and having square multiplicative energy N^2+o(1). The energy bound reflects the low degree of the associated Boolean polynomial. Together with the resonance estimate, it yields the abundance bound. We also prove unconditional analogues for quadratic characters with prime conductor away from one fixed place over any global function field of odd characteristic. Finally, we give bounds in the fixed strip and at s=1, including the dependence on the residue of the Dedekind zeta function and the prescribed local factors.
ER -