Average values of L-functions in even characteristic

Abstract

Let k = F-q(T) be the rational function field over a finite field F-q, where q is a power of 2. In this paper we solve the problem of averaging the quadratic L-functions L(s, chi(u)) over fundamental discriminants. Any separable quadratic extension K of k is of the form K = k(x(u)), where x(u) is a zero of X-2 + X + u = 0 for some u is an element of k. We characterize the family I (resp. F, F') of rational functions u is an element of k such that any separable quadratic extension K of k in which the infinite prime infinity = (1/T) of k ramifies (resp. splits, is inert) can be written as K = k(x(u)) with a unique u is an element of I (resp. u is an element of F, u is an element of F'). For almost all s is an element of C with Re(s) >= 1/2, we obtain the asymptotic formulas for the summation of L(s,chi(u)) over all k(x(u)) with u is an element of I, all k(x(u)) with u is an element of F or all k(x(u)) with u is an element of F' of given genus. As applications, we obtain the asymptotic mean value formulas of L-functions at s = 1/2 and s = 1 and the asymptotic mean value formulas of the class number h(u) or the class number times regulator h(u)R(u). (C) 2017 Elsevier Inc. All rights reserved.