On the third moment of $L\big(\frac{1}{2}, \chi_{d}\big)$ I: the rational function field case

TL;DR

This paper establishes the existence of a secondary term in the asymptotic formula for the cubic moment of quadratic Dirichlet L-functions over rational function fields, paralleling known results over the rationals.

Contribution

It proves a secondary term in the asymptotic formula for the cubic moment of quadratic L-functions in the function field setting, extending prior rational number results.

Findings

Secondary term in the asymptotic formula is of order q^{3/4 D}

The secondary term matches the x^{3/4} behavior over the rationals

Provides a bridge between function field and number field L-function moments

Abstract

In this note, we prove the existence of a secondary term in the asymptotic formula of the cubic moment of quadratic Dirichlet L-functions $$\sum_{\substack{d - \mathrm{monic \, \& \, sq. \, free} \mathrm{deg}\, d \, = \, D}} L\big(\tfrac{1}{2}, \chi_{d}\big)^{3}$$ over rational function fields on the order of $q^{\scriptscriptstyle \frac{3}{4} D}.$ This term is in perfect analogy with the $x^{\scriptscriptstyle \frac{3}{4}}$-term indicated in our joint work arXiv:math/0110092v1 for the corresponding asymptotic formula over the rationals.