# The First Moment of $L(\frac{1}{2},\chi)$ for Real Quadratic Function Fields

## TL;DR

This paper refines the asymptotic understanding of the first moment of quadratic Dirichlet L-functions over function fields, identifying new main terms and bounding the error with advanced techniques.

## Contribution

It extends Florea's methods to include additional main terms for the first moment of quadratic L-functions over function fields, addressing technical challenges.

## Key findings

- Identifies new main terms of size $(2g+2)q^{(2g+2)/3}$, $q^{g/6+[rac{g}{2}]}$, and $q^{g/6+[rac{g-1}{2}]}$.
- Bounds the error term by $q^{g/2(1+\epsilon)}$.
- Improves asymptotic formulas for the first moment of quadratic L-functions in function fields.

## Abstract

In this paper we use techniques first introduced by Florea to improve the asymptotic formula for the first moment of the quadratic Dirichlet L-functions over the rational function field, running over all monic, square-free polynomials of even degree at the central point. With some extra technical difficulties that doesn't appear in Florea's paper, we prove that there are extra main terms of size $(2g+2)q^{\frac{2g+2}{3}}, q^{\frac{g}{6}+\left[\frac{g}{2}\right]}$ and $q^{\frac{g}{6}+\left[\frac{g-1}{2}\right]}$, whilst bounding the error term by $q^{\frac{g}{2}(1+\epsilon)}$.