## Average Values of $L$-series for the Prime Hyperelliptic Ensemble

submitted (2014), 58 pages.

**Abstract:**We establish asymptotic formulae for the first and second moments of quadratic Dirichlet $L$-functions,at the centre of the critical strip ($s=1/2$), 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 (first moment) for the class number and for the cardinality of second $K$-group of maximal order of the associated fields for ramified imaginary, real, and inert imaginary quadratic function fields over $\mathbb{F}_{q}$. The strenuous case where the cardinality of the finite field is even also is handled in this article. The limits studied in this paper for the mean value theorems are all with the cardinality of the finite field fixed and when the degree of $P$ is large. These calculations correspond to the difficult problem over number fields in computing moments of $L$-functions when the average is taken over prime numbers.

The methods used to establish the asymptotic formulas in this paper are those based on the classical analytic methods and in the use of the analogue of approximate functional equation for such $L$-functions. The advantage to work over function fields is that the $L$-functions are polynomials and we can use the Riemann Hypothesis for curves to bound non-trivial character sums with square-root cancellation.