On the distribution of $k$-free numbers on the view point of random walks
| Autor: | Liu, Kui, Lu, Meijie |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 10 pages |
| Databáze: | arXiv |
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