We investigate the mixed second moment involving the second derivative at the central point of a family of quadratic Dirichlet -functions over , associated to the hyperelliptic curves of genus . We compute the full degree five polynomial in the asymptotic expansion of the mixed second moment when the cardinality of the finite field is fixed and the genus tends to infinity. This is a partial analogue of classical Ingham’s result about the Riemann zeta function.
"The mixed second moment of quadratic Dirichlet -functions over function fields." Rocky Mountain J. Math. 51 (6) 2003 - 2017, December 2021. https://doi.org/10.1216/rmj.2021.51.2003