On a distribution property of the residual order of $a \pmod {p}$, (2)

Abstract

Let $a$ be a positive integer which is not a perfect $b$-th power with $b \geq 2$, and $Q_a(x; k,l)$ be the set of primes $p \leq x$ such that the residual order of $a \pmod{p}$ in $\mathbf{Z}/p\mathbf{Z}^*$ is congruent to $l \pmod{k}$. In this paper, which is a sequel of our previous paper [1], under the assumption of the Generalized Riemann Hypothesis, we prove that for any residue class $l \pmod{k}$ the set $Q_a(x; k,l)$ has the natural density $\Delta_a(k,l)$, and the values of $\Delta_a(k,l)$ are effectively computable. We also consider some number theoretical properties of $\Delta_a(k,l)$ as a number theoretical function of $k$ and $l$.