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

Abstract

<abstract><p>Let $ \mathbb{Z}^2 $ be the two-dimensional integer lattice. For an integer $ k\geq 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 $.</p></abstract>