On a distribution property of the residual order of \(a\pmod {pq}\)

Abstract. In the authors' previous papers, for a positive integer \(a\) and a prime number \(p\) such that \((a,p)=1\), the distribution of the residual orders \(D_a(p)\) of \(a\) (mod \(p\)) was considered. Under the generalized Riemann hypothesis (GRH), the authors determined the natural density of the primes \(p\) satisfying \(D_a(p)\equiv l\) (mod \(k\)) when \(k\) is a prime power, and proved the existence of an algorithm for computing the density when \(k\) is a composite integer other than prime powers. In this paper, we consider the distribution of the residual orders \(D_a(pq)\) of \(a\) (mod \(pq\)) where \(p\) and \(q\) are distinct primes. Under GRH and some slight restriction to the base \(a\), we determine the natural density of the prime pairs \((p,q)\) satisfying \(D_a(pq)\equiv l\) (mod \(4\)) for \(l=0,1,2,3\).