On a distribution property of the residual order of a(modp)—II

AbstractLet a be a positive integer which is not a perfect hth power with h⩾2, and Qa(x;4,l) be the set of primes p⩽x such that the residual order of a(modp) in Z/pZ× is congruent to l modulo 4. When l=0,2, it is known that calculations of ♯Qa(x;4,l) are simple, and we can get their natural densities unconditionally. On the contrary, when l=1,3, the distribution properties of Qa(x;4,l) are rather complicated. In this paper, which is a sequel of our previous paper [3], under the assumption of the generalized Riemann Hypothesis, we determine completely the natural densities of ♯Qa(x;4,l) for l=1,3