On the r-rank Artin Conjecture II

For any finitely generated subgroup \Gamma of Q we compute a formula for the density of the primes for which the reduction modulo p of \Gamma contains a primitive root modulo p. We use this to conjecture a characterization of "optimal" subgroups (i.e. subgroups that have maximal density). We also improve the error term in the asymptotic formula of [9, Theorem 1.1]. 1. INTRODUCTION AND STATEMENT OF THE MAIN RESULTS Let \Gamma be a finitely generated multiplicative subgroup of Q . We denote by supp(\Gamma) the (finite) set of those primes p such that p (a) 6= 0 for some a 2 \Gamma. For all primes p not in supp(\Gamma), we consider the reduction \Gamma p of \Gamma. Precisely, \Gamma p is the subgroup of F p obtained by reducing modulo p all the elements of \Gamma. We let N \Gamma (x) = # \Phi p x; p = 2 supp(\Gamma) j \Gamma p = F p \Psi : (1) The statement that N hai (x) !1 as x !1 (when a is an integer 6= 0, \Sigma1 and not a perfect square), is known as the Artin C..