A Note on the Mean Value of $L$--functions in Function Fields

An asymptotic formula for the sum $\sum L(1,\chi )$ is established for a family of hyperelliptic curves of genus $g$ over a fixed finite field ${\mathbb{F}}_q$ as $g\to \mathrm{\infty }$ making use of the analogue of the approximate functional equation for such $L$–functions. As a corollary, we obtain a formula for the average of the class number of the associated rings ${\mathbb{F}}_q[T,sqrtD]$.