The number of relatively $r$-prime $k$-tuple integers

TL;DR

This paper investigates the distribution of relatively r-prime k-tuple integers, establishing the precise order of the error term in their count within a k-dimensional cube for certain parameters.

Contribution

It determines the exact order of the error term in counting relatively r-prime k-tuples, refining previous asymptotic estimates.

Findings

Error term order is x^{k-1} for rk ≥ 3 and k ≠ 1.

Confirmed the main term as x^k / ζ(rk).

Provides a precise asymptotic count for relatively r-prime k-tuples.

Abstract

For a fixed integer $r\ge1$, we say $k$-tuple integers $(x_1,\ldots,x_k)$ are relatively $r$-prime if there exists no prime $p$ such that all $k$ integers is multiple of $p^r$. Benkoski proved that the number of relatively $r$-prime $k$-tuple integers in $[1,x]^k$ is $x^k/\zeta(rk)+$(Error term). We showed that the exact order of error term is $x^{k-1}$ for $rk\ge3$ and $k\not=1$.