Remark on average of class numbers of function fields

Abstract

Let $k={\mathbb{F}}_q(T)$ be a rational function field over the finite field ${\mathbb{F}}_q$, where $q$ is a power of an odd prime number, and $\mathbb{A}={\mathbb{F}}_q[T]$.
Let $\gamma$ be a generator of ${\mathbb{F}}_q^{\ast }$.
Let ${\mathcal{H}}_n$ be the subset of $\mathbb{A}$ consisting of monic square-free polynomials of degree $n$.
In this paper we obtain an asymptotic formula for the mean value of $L(1,{\chi }_{\gamma D})$ and
calculate the average value of the ideal class number $h_{\gamma D}$ when the average is taken over $D\in {\mathcal{H}}_{2g+2}$.