The mean value of the class numbers of cubic function fields

Abstract

We compute the mean value of |L(s, chi)|2 evaluated at s = 1 when chi goes through the primitive cubic Dirichlet odd characters of A := Fq[T], where Fq is a finite field with q elements and q equivalent to 1 (mod 3). Furthermore, we find the mean value of the class numbers for the cubic function fields Km = k(3 root m), where k := Fq(T) is the rational function field, m is an element of A is a cube-free polynomial, and deg(m) equivalent to 1 (mod 3). (c) 2022 Elsevier Inc. All rights reserved.