For a fixed natural number m, an integer t is said to possess kth weak
order (mod m), if there exists a natural number n satisfying t(n + k)
= t(k) (mod m).We write g(k)(m) for the numbers of incongruent elemen
ts possessing kth order (mod m). Here the asymptotic behaviour of G(k)
(x) = SIGMA(m less-than-or-equal-to x) g(k)(m) is studied.