On the distribution of $k$-full lattice points in ${\mathbb{Z}}^2$

Let ${\mathbb{Z}}^2$ be the two-dimensional integer lattice. For an integer $k\ge 2$, we say a non-zero lattice point in ${\mathbb{Z}}^2$ is $k$-full if the greatest common divisor of its coordinates is a $k$-full number. In this paper, we first prove that the density of $k$-full lattice points in ${\mathbb{Z}}^2$ is $c_k=\prod _p(1-p^{-2}+p^{-2k})$, where the product runs over all primes. Then we show that the density of $k$-full lattice points on a path of an $\alpha$-random walk in ${\mathbb{Z}}^2$ is almost surely $c_k$, which is independent on $\alpha$.