On a density property of the residual order of $a \pmod{pq}$

We consider a distribution property of the residual order (the multiplicative order) of the residue class $a \hspace{-.4em} \pmod{pq}$. It is known that the residual order fluctuates irregularly and increases quite rapidly. We are interested in how the residual orders $a \hspace{-.4em} \pmod{pq}$ distribute modulo 4 when we fix $a$ and let $p$ and $q$ vary. In this paper we consider the set $S(x) = \{(p, q); p, q \ \text{are distinct primes,} \ pq \leq x \}$, and calculate the natural density of the set $\{(p, q) \in S(x); \ \text{the residual order of} \ a \hspace{-.4em} \pmod{pq} \equiv l \hspace{-.4em} \pmod{4}\}$. We show that, under a simple assumption on $a$, these densities are $\{5/9,\, 1/18,\, 1/3,\, 1/18 \}$ for $l= \{0, 1, 2, 3 \}$, respectively. For $l = 1, 3$ we need Generalized Riemann Hypothesis.