On a distribution property of the residual order of a (mod p)

Let a be a positive integer with a≠1 and Qa(x;k,l) be the set of primes p x such that the residual order of a in Z/pZ× is congruent to l mod k. It seems that no one has ever considered the density of Qa(x;k,l) for l≠0 when k 3. In this paper, the natural densities of Qa(x;4,l) (l=0,1,2,3) are considered. We assume a is square free and a≡1 (mod 4). Then, for l=0,2, we can prove unconditionally that their natural densities are equal to 1/3. On the contrary, for l=1,3, we assume the Generalized Riemann Hypothesis, then we can prove their densities are equal to 1/6.