Visible lattice points in Pólya's walk

Abstract

In this paper, for any integer $k\ge 2$, we study the distribution of the visible lattice points in certain generalized Pólya's walk on ${\mathbb{Z}}^k$: perturbed Pólya's walk and twisted Pólya's walk. For the first case, we prove that the density of visible lattice points in a perturbed Pólya's walk is almost surely $1/\zeta (k)$, where $\zeta (s)$ denotes the Riemann zeta function. A trivial case of our result covers the standard Pólya's walk. Moreover, we do numerical experiments for the second case, we conjecture that the density is also almost surely $1/\zeta (k)$.

23 pages