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

2023/09/12 by Liu, Kui, Lu, Meijie
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2309.06210

Abstract

In this paper, we investigate the distribution of k-free numbers in a class of α-random walks on the integer lattice ℤ. In these walks, the walker starts from a non-negative integer r and moves to the right by a units with probability α, or by b units with probability 1-α. For k≥ 3, we obtain the asymptotic proportion of k-free numbers in a path of such α-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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