Remark on average of class numbers of function fields
Main Article Content
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 $\mathbb{F}_{q}^*$.
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 $D \in \mathcal{H}_{2g+2}$.
Article Details
References
[1] J.C. Andrade, A note on the mean value of L-functions in function fields. Int. J. Number Theory, 08 (2012), no. 12, 1725–1740. Google Scholar
[2] J.C. Andrade and J.P. Keating, The mean value of L( 1 , χ) in the hyperelliptic 2 ensemble. J. Number Theory 132 (2012), no. 12, 2793–2816. Google Scholar
[3] J. Hoffstein and M. Rosen, Average values of L-series in function fields. J. Reine Angew. Math. 426 (1992), 117–150. Google Scholar
[4] H. Jung, A note on the mean value of L(1, χ) in the hyperelliptic ensemble. To appear in Int. J. Number Theory. Google Scholar
[5] M. Rosen, Number theory in function fields. Graduate Texts in Mathematics 210, Springer-Verlag, New York, 2002. Google Scholar