Counting r-tuples of positive integers with k-wise relatively prime components

Let r≥k≥2r≥k≥2 be fixed positive integers. Let ϱr,kϱr,k denote the characteristic function of the set of r-tuples of positive integers with k-wise relatively prime components, that is any k   of them are relatively prime. We use the convolution method to establish an asymptotic formula for the sum ∑n1,…,nr≤xϱr,k(n1,…,nr)∑n1,…,nr≤xϱr,k(n1,…,nr) by elementary arguments. Our result improves the error term obtained by J. Hu [5].