$k$-free lattice points in random walks

TL;DR

This paper investigates the distribution of $k$-free lattice points along random walks in the 2D integer lattice, deriving explicit proportions using number theory and probabilistic methods.

Contribution

It provides the first rigorous analysis of the proportions of $k$-free and twin $k$-free points on random walk paths in $Z^2$, connecting number theory with stochastic processes.

Findings

Proportion of $k$-free points is $1/zeta(2k)$.

Proportion of twin $k$-free points is $igl( extstyle ext{product over primes }(1-2p^{-2k})igr)$.

Results link lattice point properties with classical number-theoretic functions.

Abstract

Let $\mathbb{Z}^2$ be the two-dimensional integer lattice. For an integer $k\geq 1$, a non-zero lattice point is $k$-free if the greatest common divisor of its coordinates is a $k$-free number. We consider the proportions of $k$-free and twin $k$-free lattice points on a path of an $\alpha$-random walker in $\mathbb{Z}^2$. Using the second-moment method and tools from analytic number theory, we prove that these two proportions are $1/\zeta(2k)$ and $\prod_{p}(1-2p^{-2k})$, respectively, where $\zeta$ is the Riemann zeta function and the infinite product takes over all primes.