Complex Moments and the distribution of Values of $L(1,\chi_D)$ over Function Fields with Applications to Class Numbers

TL;DR

This paper studies the distribution and moments of $L(1, chi_D)$ over function fields, revealing new insights into their extreme values and connections to class numbers, using asymptotic formulas and probabilistic models.

Contribution

It provides the first asymptotic formulas for complex moments of $L(1, chi_D)$ beyond the first moment, and analyzes their distribution and extreme values in the function field setting.

Findings

Distribution of $L(1, chi_D)$ closely matches probabilistic models

Uncovered unique distribution features for large and small $L(1, chi_D)$ values

Derived $ ext{Omega}$-results for extreme values of $L(1, chi_D)$

Abstract

In this paper we investigate the moments and the distribution of $L(1,\chi_D)$, where $\chi_D$ varies over quadratic characters associated to square-free polynomials $D$ of degree $n$ over $\mathbb{F}_q$, as $n\to\infty$. Our first result gives asymptotic formulas for the complex moments of $L(1,\chi_D)$ in a large uniform range. Previously, only the first moment has been computed due to work of Andrade and Jung. Using our asymptotic formulas together with the saddle-point method, we show that the distribution function of $L(1,\chi_D)$ is very close to that of a corresponding probabilistic model. In particular, we uncover an interesting feature in the distribution of large (and small) values of $L(1, \chi_D)$, that is not present in the number field setting. We also obtain $\Omega$-results for the extreme values of $L(1,\chi_D)$, which we conjecture to be best possible. Specializing $n=2g+1$ and making use of one case of Artin's class number formula, we obtain similar results for the class number $h_D$ associated to $\mathbb{F}_q(T)[\sqrt{D}]$. Similarly, specializing to $n=2g+2$ we can appeal to the second case of Artin's class number formula and deduce analogous results for $h_DR_D$ where $R_D$ is the regulator of $\mathbb{F}_q(T)[\sqrt{D}]$.