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arXiv · 2307.16583

Visible lattice points in P\'{o}lya's walk

Meijie Lu·Xianchang Meng
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Abstract

In this paper, for any integer $k\geq 2$, we study the distribution of the visible lattice points in certain generalized P\'{o}lya's walk on $\mathbb{Z}^k$: perturbed P\'{o}lya's walk and twisted P\'{o}lya's walk. For the first case, we prove that the density of visible lattice points in a perturbed P\'{o}lya's walk is almost surely $1/\zeta(k)$, where $\zeta(s)$ denotes the Riemann zeta function. A trivial case of our result covers the standard P\'{o}lya's walk. Moreover, we do numerical experiments for the second case, we conjecture that the density is also almost surely $1/\zeta(k)$.

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Meijie Lu, Xianchang Meng. 2023-07-31. Visible lattice points in P\'{o}lya's walk. https://arxiv.org/abs/2307.16583

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