On a Distribution Property of the Residual Order of $a \pmod{p}$ with a Quadratic Residue Condition

Abstract

Let $a$ be a positive integer with $a\geq 2$ and $Q_a(k,l)$ be the set of odd prime numbers $p$ such that the residual order of $a$ in $\mathbf{Z}/p\mathbf{Z}^\times$ is congruent to $l \bmod k$. The natural density of the set $Q_a(q,0)$ ($q$ is a prime) is already known. In this paper, we consider the set $S_{a,b}(k,l)$, which consists of the primes $p$ that belong to $Q_a(k,l)$ and satisfy $\big(\frac{b}{p}\big)=1$, where $\big(\frac{b}{p}\big)$ is the Legendre symbol and $b$ is a fixed integer. Heuristically, the natural density of $S_{a,b}(k,l)$ is expected to be half of that of $Q_a(k,l)$, but it is not true for some choices of $a$ and $b$. In this paper, we determine the natural density of $S_{a,b}(k,l)$ for $(k,l)=(2,j), (q,0), (4,l)$, where $j=0,1$, $q$ is an odd prime and $l=0,2$.