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

For a fixed rational number g is not an element of {-1, 0, 1} and integers a and d we consider the set N-g (a, d) of primes p for which the order of g(mod p) is congruent to a(mod d). It is shown, assuming the generalized Riemann hypothesis (GRH), that this set has a natural density delta(g)(a, d). Moreover, delta(g) (a, d) is computed in terms of degrees of certain Kummer extensions. Several properties of delta(g) (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).