The second and third moment of $L(\frac{1}{2},\chi)$ in the hyperelliptic ensemble

TL;DR

This paper derives asymptotic formulas for the second and third moments of quadratic Dirichlet L-functions at the critical point over function fields, confirming conjectured formulas for these moments.

Contribution

It provides the first rigorous asymptotic formulas for the second and third moments of quadratic Dirichlet L-functions in the function field setting, validating existing conjectures.

Findings

Second moment matches Andrade and Keating's conjecture

Third moment's leading term agrees with the conjecture

Results are for fixed finite field with prime q ≡ 1 mod 4

Abstract

We obtain asymptotic formulas for the second and third moment of quadratic Dirichlet $L$--functions at the critical point, in the function field setting. We fix the ground field $\mathbb{F}_q$, and assume for simplicity that $q$ is a prime with $q \equiv 1 \pmod 4$. We compute the second and third moment of $L(1/2,\chi_D)$ when $D$ is a monic, square-free polynomial of degree $2g+1$, as $g \to \infty$. The answer we get for the second moment agrees with Andrade and Keating's conjectured formula. For the third moment, we check that the leading term agrees with the conjecture.