Large values of derivatives of quadratic Dirichlet L-functions
Abstract
e study large values of fixed derivatives of quadratic Dirichlet L-functions on the real segment 1/2≤σ≤1. Assuming GRH, we prove lower bounds in terms of GCD sums and finite Euler products. At the central point, every fixed derivative order is compared with the same square-free GCD sum. In the mesoscopic band σ=1/2+A/log2X, a renormalized resonator estimates contributions from both halves of the differentiated approximate functional equation. The resulting lower bound has a relative error. On a smaller boundary scale, a common finite product appears in both contributions to this lower bound. This allows us to locate the crossover between their derivative weights. The coarser exponential bounds have constant $1$ at the center and e−A in the mesoscopic band. For fixed 1/2<σ<1, the lower bound includes the full typical logarithmic weight of the resonating Euler product. Burgess truncation improves the exponential constant when a shorter cutoff allows a longer resonator. Near the $1$-line, our lower bounds involve Dickman constants on the 1/log2X scale. We also prove a uniform transition estimate outside a shrinking neighborhood of the central point. All results hold in the positive even subfamily of fundamental discriminants.
Related
- Large values of cubic and quartic Dirichlet L-functions of prime moduli
- Extreme Values of Quadratic Dirichlet L-functions over Monic Irreducible Polynomials in 𝔽q[t]
- Large values of quadratic Hecke L-functions with prime-related moduli of the Gaussian field
- Extreme values of Dirichlet L-functions on the critical line with a growing prime modulus
- Linear independence of values of Dirichlet L functions
BibTeX
@misc{dong2026large,
title = {{Large values of derivatives of quadratic Dirichlet $L$-functions}},
author = {Zikang Dong and Ruihua Wang and Weijia Wang and Hao Zhang and Shengbo Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24958},
eprint = {2610.24958},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24958},
}
RIS
TY - UNPB
TI - Large values of derivatives of quadratic Dirichlet L-functions
AU - Zikang Dong
AU - Ruihua Wang
AU - Weijia Wang
AU - Hao Zhang
AU - Shengbo Zhao
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.24958
UR - https://arxiv.org/abs/2610.24958
AB - e study large values of fixed derivatives of quadratic Dirichlet L-functions on the real segment 1/2≤σ≤1. Assuming GRH, we prove lower bounds in terms of GCD sums and finite Euler products. At the central point, every fixed derivative order is compared with the same square-free GCD sum. In the mesoscopic band σ=1/2+A/log_2X, a renormalized resonator estimates contributions from both halves of the differentiated approximate functional equation. The resulting lower bound has a relative error. On a smaller boundary scale, a common finite product appears in both contributions to this lower bound. This allows us to locate the crossover between their derivative weights. The coarser exponential bounds have constant $1$ at the center and e^−A in the mesoscopic band. For fixed 1/2<σ<1, the lower bound includes the full typical logarithmic weight of the resonating Euler product. Burgess truncation improves the exponential constant when a shorter cutoff allows a longer resonator. Near the $1$-line, our lower bounds involve Dickman constants on the 1/log_2X scale. We also prove a uniform transition estimate outside a shrinking neighborhood of the central point. All results hold in the positive even subfamily of fundamental discriminants.
ER -