Selberg's Central limit theorem for quadratic Dirichlet L-functions over function fields

TL;DR

This paper proves a version of Selberg's Central Limit Theorem for the logarithm of quadratic Dirichlet L-functions over function fields, showing it follows a Gaussian distribution under certain conditions.

Contribution

It establishes the asymptotic Gaussian distribution of the logarithm of L-values in a symplectic family over function fields, extending Selberg's CLT to this setting.

Findings

Distribution of log L-values is asymptotically Gaussian.

Unconditional bounds are provided for the distribution.

Full Gaussian distribution is obtained under a mild zero distribution assumption.

Abstract

In this article, we study the logarithm of the central value $L\left(\frac{1}{2}, \chi_D\right)$ in the symplectic family of Dirichlet $L$-functions associated with the hyperelliptic curve of genus $\delta$ over a fixed finite field $\mathbb{F}_q$ in the limit as $\delta\to \infty$. Unconditionally, we show that the distribution of $\log \big|L\left(\frac{1}{2}, \chi_D\right)\big|$ is asymptotically bounded above by the Gaussian distribution of mean $\frac{1}{2}\log \deg(D)$ and variance $\log \deg(D)$. Assuming a mild condition on the distribution of the low-lying zeros in this family, we obtain the full Gaussian distribution.