Description:
It follows from the work of Artin and Hooley that, under assumption of the generalized Riemann hypothesis, the density of the set of primes q for which a given non-zero rational number r is a primitive root modulo q can be written as an infinite product Qp p of local factors p reflectingv the degree of the splitting field of Xp - r at the primes p, multiplied by a somewhat complicated factor that corrects for the 'entanglement'; of these splitting fields. We show how the correction factors arising in Artin's original primitive root problem and several of its generalizations can be interpreted as character sums describing the nature of the entanglement. The resulting description in terms of local contributions is so transparent that it greatly facilitates explicit computations, and naturally leads to non-vanishing criteria for the correction factors.