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

TL;DR

This paper studies the distribution of linear combinations of logarithms and arguments of $L$-functions over function fields at microscopic shifts, proving a central limit theorem and connecting fluctuations to random matrix theory results.

Contribution

It extends previous Gaussian distribution results to linear combinations of $L$-functions at microscopic shifts and establishes a central limit theorem for zero fluctuations.

Findings

Distribution functions estimated under low lying zeros hypothesis

Proves a central limit theorem for zero fluctuations

Connects zero fluctuation correlations with random matrix theory

Abstract

In the function field setting with a fixed characteristic, it was proven by the second and third authors that the values $\log \big|L\big(\frac12, \chi_D\big)\big|$ 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 $\log\big|L\big(\frac12+it_j, \chi_D\big)\big|$ with real coefficients, and separately, of $\arg L\big(\frac12+it_j, \chi_D\big).$ We provide estimates for their distribution functions under the low lying zeros hypothesis. We similarly study distribution functions of linear combinations of $\log\big|L\big(\frac12+it_j, E\otimes \chi_D\big)\big|$, and separately $\arg L\big(\frac12+it_j, E\otimes\chi_D\big)$, 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.