Density of Singular Pairs of Integers

A positive integer n is called cyclic if there is a unique group of order n, which is necessarily cyclic. Using a characterisation of the cyclic integers as those n satisfying gcd(n, φ(n)) = 1, P. Erdős (1947) proved that the number of cyclic integers n ≤ x is asymptotic to z(x) = e−γ x log log log x , as x→∞, where γ is Euler’s constant. An ordered pair of integers (m,n) is called singular if gcd(m,φ(n)) = 1 and gcd(n, φ(m)) = 1, a concept which is relevant to pairwise products of cyclic groups and to embeddings of complete bipartite graphs. In this note we show that the number of singular pairs of integers (m,n), m,n ≤ x, is asymptotic to z(x).