An analogue of Artin’s conjecture for multiplicative subgroups of the rationals

Given Γ⊂Q∗ a multiplicative subgroup and m∈N+ , assuming the Generalized Riemann Hypothesis, we determine an asymptotic formula for the number of primes p ≤ x for which ind p Γ = m, where ind p Γ = (p − 1)/|Γ p | and Γ p is the reduction of Γ modulo p. This problem is a generalization of some earlier works by Cangelmi–Pappalardi, Lenstra, Moree, Murata, Wagstaff, and probably others. We prove, on GRH, that the primes with this property have a density and, in the case when Γ contains only positive numbers, we give an explicit expression for it in terms of an Euler product. We conclude with some numerical computations.