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

TL;DR

This paper refines the asymptotic formula for the mean value of quadratic Dirichlet L-functions at the critical point in the hyperelliptic ensemble over finite fields, adding a new main term and improving the error bound.

Contribution

It introduces an additional main term of size |D|^{1/3} log_q |D| and establishes a tighter error bound of |D|^{1/4+ε} for the mean value calculation.

Findings

Identified an extra main term of size |D|^{1/3} log_q |D|

Bounded the error term by |D|^{1/4+ε}

Assumed q is prime with q ≡ 1 mod 4

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 $|D|^{\frac 34+ \frac{\log_q(2)}{2}}$. For simplicity, we assume that $q$ is prime with $q \equiv 1 \pmod 4$. 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 $|D|^{1/4+\epsilon}$.