The first moment of L(12,χ) for real quadratic 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 do not appear in Florea’s paper, we prove that there is an extra main term of size gq 2g+2 3 , whilst bounding the error term by q g 2 (1+ǫ) .