Mean values ofL-functions over function fields

Abstract

In modern number theory L-series play a prominent role. They encode many deep properties of number fields and primes and are objects of intense interest. The analogous L-functions over global function fields play an equally prominent role. Here we will prove estimates for mean values of such L-functions, where the averaging is done over quadratic extensions of a fixed global function field. Our estimates cover a much wider range of cases than the similar estimates of Hoffstein and Rosen [1992] and those of Andrade and Keating (for values on the critical line) [2012]. Our methods are akin to those used by Siegel [1944], where he estimates the average number of quadratic forms with given discriminant and signature. For a prime p, let Fp denote the finite field with p elements and let X be transcendental over Fp, so that Fp.X/ is a field of rational functions. Fix algebraic closures Fp of Fp and Fp.X/ Fp of Fp.X/. In what follows, by global function field (or simply function field) we mean a finite algebraic extension K Fp.X/ contained in Fp.X/. For such a field K we have K\FpD FqK for some finite field FqK with qK elements; this field is called the field of constants of K . We write gK for the genus of K and JK for the number of divisor classes of degree 0. We denote the set of places of K by M.K/ and the divisor group (i.e., the free abelian group generated by the places) by Div K . The reader can refer to Chapters I and V of [Stichtenoth 1993] for a thorough background on these notions. We will use capital script German letters to denote divisors A, B, etc., with the sole exception of the zero divisor 0. For any divisor A2 Div K we write AD P ordv.A/ v, where the