Year: 2026
Type: preprint
Abstract: We study the $β=2$ partition function $\int_{|h| \leq \log ^θ q/2}|L(1/2+ih,χ)|^2dh$ for typical Dirichlet characters modulo a large prime $q$ and $θ\in (-1/3,0]$ motivated by a $q$-analogue of the Saksman--Webb conjectures. When $θ<0$, we use an extension of Harper's randomisation argument with different localisations to recover moment upper bounds consistent with the critical normalisation predicted there. As an application, we prove that for $q(1-o(1))$ Dirichlet characters modulo $q$ with $U(q) \to \infty$, $$\max_{|h| \leq 1/2}|L(1/2+ih,χ)|\ll \frac{e^{U(q)}\log q\log_3 q(\log_4 q)^{1/2}}{(\log_2 q)^{3/4}},$$ establishing an upper bound matching the predictions of the $q$-analogue of the Fyodorov--Hiary--Keating conjectures up to second order. Our proofs adapt Harper's argument in \cite{harper2019partition}.
Source: arXiv (Cornell University)
Author: Atherfold, Christopher
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7202167482