Average value of the divisor class numbers of real cubic function fields

Abstract

We compute an asymptotic formula for the divisor class numbers of real cubic function fields K = k( m) m 3, where q is a finite field with q elements, q = 1 (mod 3), k. (T) q is the rational function field, and m. [T] q is a cube-free polynomial; in this case, the degree of m is divisible by 3. For computation of our asymptotic formula, we find the average value of |L(s,.)|2 evaluated at s = 1 when. goes through the primitive cubic even Dirichlet characters of [T] q, where L(s,.) is the associated Dirichlet L-function.