The Moments and statistical Distribution of Class number of Primes over Function Fields

TL;DR

This paper studies the statistical distribution and moments of class numbers and associated L-values in quadratic function fields over finite fields, adapting number field techniques and introducing a random Euler product model.

Contribution

It computes integral and complex moments of class numbers and L-values in quadratic function fields, extending previous number field results to the function field setting.

Findings

Calculated moments of class numbers for quadratic function fields.

Analyzed the distribution of L(1,χ_P) using a random Euler product.

Extended number field results to the function field context.

Abstract

We investigate the moment and the distribution of $L(1,\x_P),$ where $\x_P$ varies over quadratic characters associated to irreducible polynomials $P$ of degree $2g+1$ over $\mathbb{F}_q[T]$ as $g\to\infty$. In the first part of the paper we compute the integral moments of the class number $h_{P}$ associated to quadratic function fields with prime discriminants $P$ and this is done by adapting to the function field setting some of the previous results carried out by Nagoshi in the number field setting. In the second part of the paper we compute the complex moments of of $L(1,\x_P)$ in large uniform range and investigate the statistical distribution of the class numbers by introducing a certain random Euler product. The second part of the paper is based on recent results carried out by Lumley when dealing with square-free polynomials.