The Average of the Smallest Prime in a Conjugacy Class

Abstract

Let C be a conjugacy class of S-n and K an S-n-field. Let n(K,C) be the smallest prime, which is ramified or whose Frobenius automorphism Frob(p) does not belong to C. Under some technical conjectures, we show that the average of n(K,C) is a constant. We explicitly compute the constant. For S-3- and S-4-fields, our result is unconditional. Let N-K,N-C be the smallest prime for which Frob(p) belongs to C. We obtain the average of N-K,N-C under some technical conjectures. For n = 3 and C = [(12)], we have the average value of N-K,N-C unconditionally.