On a distribution property of the residual order of $a \pmod {p}$

Abstract

Let $a$ be a positive integer which is not a perfect $h$-th power with $h \geq 2$, and $Q_a(x; k,l)$ be the set of primes $p \leq x$ such that the residual order of $a$ in $\mathbf{Z} / p\mathbf{Z}^{\times}$ is congruent to $l \bmod{k}$. It seems that no one has ever considered the density of $Q_a(x; k,l)$ for $l \ne 0$ when $k \geq 3$. In this article, the natural densities of $Q_a(x; 4,l)$ ($l = 0, 1, 2, 3$) are considered. When $l = 0, 2$, calculations of $\sharp Q_a(x; 4,l)$ are simple, and we can get these natural densities unconditionally. On the contrary, the distribution properties of $Q_a(x; 4,l)$ for $l = 1, 3$ are rather complicated. Under the assumption of Generalized Riemann Hypothesis, we determine completely the natural densities of $\sharp Q_a(x; 4,l)$ for $l = 1, 3$.