Average Values of $L$-series for Real Characters in Function Fields

Abstract

We establish asymptotic formulae for the first and second moments of quadratic Dirichlet $L$--functions, at the centre of the critical strip, associated to the real quadratic function field $k(\sqrt{P})$ and inert imaginary quadratic function field $k(\sqrt{\gamma P})$ with $P$ being a monic irreducible polynomial over a fixed finite field $\mathbb{F}_{q}$ of odd cardinality $q$ and $\gamma$ a generator of $\mathbb{F}_{q}^{\times}$. We also study mean values for the class number and for the cardinality of the second $K$-group of maximal order of the associated fields for ramified imaginary, real, and inert imaginary quadratic function fields over $\mathbb{F}_{q}$. One of the main novelties of this paper is that we compute the second moment of quadratic Dirichlet $L$-functions associated to monic irreducible polynomials. It is worth noting that the similar second moment over number fields is unknown. The second innovation of this paper comes from the fact that, if the cardinality of the ground field is even then the task of average $L$-functions in function fields is much harder and, in this paper, we are able to handle this strenuous case and establish several mean values results of $L$-functions over function fields.