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 $b\ge 2$, $q$ be a prime number and $Q_a(x;q^i,j)$ be the set of primes $p\le x$ such that the residual order of $a\left(\mathrm{m}\mathrm{o}\mathrm{d}p\right)$ in $(\mathbf{Z}/p\mathbf{Z}{)}^{\times }$ is congruent to $j$ modulo $q^i$. 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 $Q_a(x;q^i,j)$ for $i\ge 3$ if $q=2$, $i\ge 1$ 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]).