Artin's primitive root conjecture and a problem of Rohrlich

Abstract

Let K be a number field, Gamma a finitely generated subgroup of K , for instance the unit group of K, and kappa > 0. For an ideal a of K let ind(Gamma)(a) denote the multiplicative index of the reduction of Gamma in (O-K / a) ( whenever it makes sense). For a prime ideal p of K and a positive integer gamma let I-gamma(kappa) (p) be the average of ind(< a1,..., a gamma >) (p)(kappa) over all tupels (a(1),..., a(gamma)) is an element of (O-K / p) (gamma) Motivated by a problem of Rohrlich we prove, partly conditionally on fairly standard hypotheses, lower bounds for Sigma(N) (a <= x) ind(Gamma) (a)(kappa) and asymptotic formulae for Sigma(N) (p <= x) I-gamma(kappa)(p).