On the distribution of the order and index of g(modp) over residue classes II

For a fixed rational number g∉{-1,0,1} and integers a and d we consider the set Ng(a,d) of primes p for which the order of g(modp) is congruent to . It is shown, assuming the generalized Riemann hypothesis (GRH), that this set has a natural density δg(a,d). Moreover, δg(a,d) is computed in terms of degrees of certain Kummer extensions. Several properties of δg(a,d) are established in case d is a power of an odd prime. The result for a=0 sheds some new light on the well-researched case where one requires the order to be divisible by d (with d arbitrary).