Density Estimates Related to Gauß Periods

Given two integers q and k, for any prime r not dividing q with r ≡ 1 mod k, we denote by indr(q) the index of q mod r. In [2] the question was raised of calculating the density of the primes r for which indr(q) and (r− 1)/k are coprime; this is the condition that the Gauß period in Fq(r−1)/k defined by these data be normal over Fq. We assume the Generalized Riemann Hypothesis and calculate a formula for this density for all q and k. We prove unconditionally that our formula is an upper bound for the density and then express it as an Euler product. Finally we apply the results to characterize the existence of a special type of Gauß periods.