Gevrey localization for simple and distinct zeros in a prime-modulus Dirichlet family
Abstract
Let q tend to infinity through odd primes, let T=T(q), and put ℓ=log(qT/2π). Suppose that ℓs=o(T) for some fixed s>1 and that λbw:=min{1, liminfsubstackq→∞ q prime (log(q−1))/ℓ}>0. For the unweighted family of the q−2 nonprincipal characters modulo q, we prove unconditional lower bounds, relative to the total zero multiplicity in (T,2T], of 2−1/cλbw* for both simple and distinct critical-line zeros and of 1/2(3−1/cλbw*) for all distinct zeros, where cλ*=√2tan(λ/√2)/(1+(λ/√2)tan(λ/√2)). When log T=o(log q), this gives respectively 0.6725007036… and 0.8362503518…; in particular it covers every fixed T=(log q)A with A>1. As the growing-height companion to the authors' public mesoscopic shrinking-height theorem, this article is organized around the different localization technology required when T grows. An abstract theorem converts complex-strip Gevrey decay, sampling density, and a local multiset count into stretched-exponential localization in trace and nuclear norm. In the Dirichlet specialization the strip growth is X1/4, where X=exp(λℓ), and a buffer D0=(Kℓ)s absorbs the remote-zero contribution uniformly in the character. Quantitative finite-sampling end-effect bounds, a finite explicit-formula matrix, character-averaged first and second traces, hyperbolic functional-equation blocks, and a rank--trace inequality then yield the three zero statistics. The shared finite-inertia mechanism and the Montgomery--Taylor constant are not claimed as new. No form of GRH is assumed.
Related
- Extreme values of Dirichlet L-functions on the critical line with a growing prime modulus
- Poisson-Dirichlet statistics for the log-modulus of the Riemann zeta function on expanding intervals near the critical line
- Results for distinct covering systems with a prime minimum modulus q ≥ 11
- Simple and accurate approximations to Dirichlet L-functions
- A new localization approach for Willmore flows: Asymptotic analysis for the Dirichlet problem
BibTeX
@misc{hua2026gevrey,
title = {{Gevrey localization for simple and distinct zeros in a prime-modulus Dirichlet family}},
author = {Zhixu Hua and Xiufan Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27808},
eprint = {2609.27808},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27808},
}
RIS
TY - UNPB
TI - Gevrey localization for simple and distinct zeros in a prime-modulus Dirichlet family
AU - Zhixu Hua
AU - Xiufan Yang
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27808
UR - https://arxiv.org/abs/2609.27808
AB - Let q tend to infinity through odd primes, let T=T(q), and put ℓ=log(qT/2π). Suppose that ℓ^s=o(T) for some fixed s>1 and that λ_bw:=min{1, liminf_substackq→∞ q prime (log(q−1))/ℓ}>0. For the unweighted family of the q−2 nonprincipal characters modulo q, we prove unconditional lower bounds, relative to the total zero multiplicity in (T,2T], of 2−1/c_λ_bw^* for both simple and distinct critical-line zeros and of 1/2(3−1/c_λ_bw^*) for all distinct zeros, where c_λ^*=√2tan(λ/√2)/(1+(λ/√2)tan(λ/√2)). When log T=o(log q), this gives respectively 0.6725007036… and 0.8362503518…; in particular it covers every fixed T=(log q)^A with A>1. As the growing-height companion to the authors' public mesoscopic shrinking-height theorem, this article is organized around the different localization technology required when T grows. An abstract theorem converts complex-strip Gevrey decay, sampling density, and a local multiset count into stretched-exponential localization in trace and nuclear norm. In the Dirichlet specialization the strip growth is X^1/4, where X=exp(λℓ), and a buffer D_0=(Kℓ)^s absorbs the remote-zero contribution uniformly in the character. Quantitative finite-sampling end-effect bounds, a finite explicit-formula matrix, character-averaged first and second traces, hyperbolic functional-equation blocks, and a rank--trace inequality then yield the three zero statistics. The shared finite-inertia mechanism and the Montgomery--Taylor constant are not claimed as new. No form of GRH is assumed.
ER -