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

Abstract

Let a be a positive integer which is not a perfect b-th power with b≥2, q be a prime number and Q_a(x;qⁱ,j) be the set of primes p≤x such that the residual order of a (mod p) in (Z/pZ)^× is congruent to j modulo qⁱ. In this paper, which is a sequel of our previous papers [1] and [6], under the assumption of the Generalized Riemann Hypothesis, we determine the natural densities of Q_a(x;qⁱ,j) for i≥3 if q=2, i≥1 if q is an odd prime, and for an arbitrary nonzero integer j (the main results of this paper are announced without proof in [3], [7] and [2]).