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

TL;DR

This paper derives an improved asymptotic formula for counting r-tuples of positive integers with any k of them being relatively prime, using elementary convolution methods, refining previous error estimates.

Contribution

The paper introduces an elementary convolution approach to obtain a sharper asymptotic count for r-tuples with k-wise coprimality, improving prior error bounds.

Findings

Established an asymptotic formula for the sum over r-tuples with k-wise coprimality.

Improved the error term compared to previous results by J. Hu (2013).

Used elementary arguments to achieve these results.

Abstract

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).