Improving the Error Term in the Mean Value of in the Hyperelliptic Ensemble
Alexandra Florea · International Mathematics Research Notices · 2016
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 Fq. Summing L(1/2,χD) over monic, square-free polynomials D of degree 2g+1, the main term is of size |D|logq|D| (where |D|=q2g+1) and Andrade and Keating bound the error term by |D|34+logq(2)2. For simplicity, we assume that q is prime with q≡1 (mod4). We prove that there is an extra term of size |D|1/3logq|D| in the asymptotic formula and bound the error term by |D|1/4+ϵ.