The First Moment of $L(\frac{1}{2},χ)$ for Real Quadratic Function Fields (1908.04078v1)

Published 12 Aug 2019 in math.NT

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 $q^{{\frac{g}{2}(1+\epsilon)}$.}