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

Let $r\ge k\ge 2$ be fixed positive integers. Let $\varrho_{r,k}$ 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 $\sum_{n_1,\ldots,n_r\le x} \varrho_{r,k}(n_1,\ldots,n_r)$ by elementary arguments. Our result improves the error term obtained by J. Hu (2013).