On the distribution of $k$-free numbers on the view point of random walks

TL;DR

This paper studies the distribution of $k$-free numbers along specific $oldsymbol{ extit{ ext{alpha}}}$-random walks on integers, providing asymptotic results that generalize classical distribution theorems for $k$-free numbers.

Contribution

It introduces a novel analysis of $k$-free number distribution in $oldsymbol{ extit{ ext{alpha}}}$-random walks, extending classical results to a probabilistic path setting.

Findings

Asymptotic proportion of $k$-free numbers is established for $oldsymbol{ extit{ ext{alpha}}}$-random walks.

Results hold almost surely for the paths of the random walks.

Generalizes classical distribution results for $k$-free numbers in arithmetic progressions.

Abstract

In this paper, we investigate the distribution of $k$-free numbers in a class of $\alpha$-random walks on the integer lattice $\mathbb{Z}$. In these walks, the walker starts from a non-negative integer $r$ and moves to the right by $a$ units with probability $\alpha$, or by $b$ units with probability $1-\alpha$. For $k\geq 3$, we obtain the asymptotic proportion of $k$-free numbers in a path of such $\alpha$-random walks in almost surely sense. This provides a generalization of a classical result on the distribution of $k$-free numbers in arithmetic progressions.