Linear Combinations of Logarithms of $L$-functions over Function Fields at Microscopic Shifts and Beyond

TL;DR

The paper studies how linear combinations of logarithms and arguments of L-functions in function-field families (hyperelliptic and quadratic twists of elliptic curves) distribute across microscopic, mesoscopic, and macroscopic shifts. It develops a Selberg-type Dirichlet polynomial approximation to control near-critical values and proves central limit theorems and Gaussian-process limits for real and, in part, imaginary parts of these linear combinations, conditioned on low-lying zeros. The results reveal log-correlated Gaussian structures and covariance patterns that align with random matrix theory predictions, and they yield applications to zero fluctuations and nonvanishing proportions, along with insights toward ratio conjectures. The work thus bridges function-field L-functions, probabilistic limit theorems, and RMT-inspired covariance structures in multiple regimes, with explicit quantitative bounds and moment estimates.

Abstract

In the function field setting with a fixed characteristic, it was proven by the second and third authors that the values $\log \big|L\big(\frac12, χ_D\big)\big|$ as $D$ varies over monic and square-free polynomials are asymptotically Gaussian distributed on the assumption of a low lying zeros hypothesis as the degree of $D$ tends to $\infty$. For real distinct shifts $t_j$ all of microscopic size or all of nonmicroscopic size relative to the genus, we consider linear combinations of $\log\big|L\big(\frac12+it_j, χ_D\big)\big|$ with real coefficients, and separately, of $\arg L\big(\frac12+it_j, χ_D\big).$ We provide estimates for their distribution functions under the low lying zeros hypothesis. We similarly study distribution functions of linear combinations of $\log\big|L\big(\frac12+it_j, E\otimes χ_D\big)\big|$, and separately $\arg L\big(\frac12+it_j, E\otimesχ_D\big)$, for quadratic twists of elliptic curves $E$ with root number one as the conductor gets large. As an application of these results, we prove a central limit theorem for the fluctuation of the number of nontrivial zeros of such $L$-functions from its mean, and thus recover previous results by Faifman and Rudnick. Correlations of such fluctuations are in harmony with the results of Bourgade, Coram and Diaconis, and Wieand for zeros of the Riemann zeta function and for eigenangles of unitary random matrices.

Theorems & Definitions (59)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Definition
• Corollary 1.8
• Corollary 1.9