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

Abstract

Let a be a positive integer which is not a perfect h-th power with h greater than 1, and Q_a(x;4,j) be the set of primes p less than x such that the residual order of a(mod p) is congruent to j modulo 4. When j=0, 2, it is known that calculations of #Q_a(x;4,j) are simple, and we can get their natural densities unconditionally. On the contrary, when j=1, 3, the distribution properties of Q_a(x;4,j) are rather complicated. In this paper, which is a sequel of our previous paper (part I), under the assumption of Generalized Riemann Hypothesis, we determine completely the natural densities of #Q_a(x;4,j) for j=1, 3.