Korean J. Math. Vol. 21 No. 4 (2013) pp.365-374
DOI: https://doi.org/10.11568/kjm.2013.21.4.365

Remark on average of class numbers of function fields

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Hwanyup Jung


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 $\mathbb{F}_{q}^*$.
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 $D \in \mathcal{H}_{2g+2}$.

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