On the distribution of powerful and r-free lattice points

Abstract

Let $1 < c < 2$. For $m, n \in \mathbb{N}$, a lattice point $(m, n)$ is powerful if and only if $\gcd(m, n)$ is a powerful number, where $\gcd(*, *)$ is the greatest common divisor function. In this paper, we count the number of the ordered pairs $(m, n)$,  $m, n \leq x$ such that the lattice point $(\left\lfloor m^c \right\rfloor, \left\lfloor n^c \right\rfloor)$ is powerful. Moreover, we study $r$-free lattice points analogues of powerful lattice points.