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

Kui Liu; Meijie Lu

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arXiv · 2309.06210

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

Kui Liu·Meijie Lu

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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.

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Kui Liu, Meijie Lu. 2023-09-12. On the distribution of $k$-free numbers on the view point of random walks. https://arxiv.org/abs/2309.06210

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