It follows from the work of Artin (1927, 1958) and Hooley (1967) that, under the assumption of the generalized Riemann hypothesis, every non-square rational number r different from -1 is a primitive root modulo infinitely many primes. Moreover, the set of these primes has a natural density that can be written as the product of a `naive density' and a somewhat complicated correction factor reflecting the entanglement of the number fields that underly the density statement. We show how the correction factors arising in Artin's original primitive root problem and some 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.