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

Abstract

Let a be a positive integer greater than 1, and Q_a(x;k,j) be the set of primes p less than x such that the residual order of a(mod p) is congruent to j modulo k. In this paper, the natural densities of Q_a(x;4,j) (j=0,1,2,3) are considered. We assume a is square-free and a is congruent to 1 (mod 4). Then, for j=0, 2, we can prove unconditionally that their natural densities are equal to 1/3. On the contrary, for j=1, 3, we assume Generalized Riemann Hypothesis, then we can prove that their densities are equal to 1/6.