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

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 a(modd). For d=4 and 3 we show that, under the generalized Riemann hypothesis (GRH), these sets have a natural density δg(a,d) and compute it. The results for d=4 generalise earlier work by Chinen and Murata. The case d=3 was apparently not considered before.