## The first moment of $L\bigl(\frac{1}{2},\chi\bigr)$ for real quadratic function fields

### Volume 198 / 2021

Acta Arithmetica 198 (2021), 1-35
MSC: Primary 11M38; Secondary 11M06, 11G20.
DOI: 10.4064/aa190924-3-8
Published online: 15 December 2020

#### Abstract

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 do not appear in Florea’s paper, we prove that there is an extra main term of size $gq^{\frac {2g+2}{3}}$, while the error term is bounded by $q^{\frac {g}{2}(1+\epsilon )}$.