Improving the error term in the mean value of $L(\frac{1}{2},\chi )$ in the hyperelliptic ensemble

Abstract

Andrade and Keating computed the mean value of quadratic Dirichlet $L$--functions at the critical point, in the hyperelliptic ensemble over a fixed finite field ${\mathbb{F}}_q$. Summing $L(1/2,{\chi }_D)$ over monic, square-free polynomials $D$ of degree $2g+1$, the main term is of size $|D|{\log }_q|D|$ (where $|D|=q^{2g+1}$) and Andrade and Keating bound the error term by $|D{|}^{\frac{3}{4}+\frac{{\log }_q(2)}{2}}$. For simplicity, we assume that $q$ is prime with $q\equiv 1(\mathrm{mod}4)$. We prove that there is an extra term of size $|D{|}^{1/3}{\log }_q|D|$ in the asymptotic formula and bound the error term by $|D{|}^{1/4+\epsilon }$.