Visible lattice points in Pólya's walk

2023/07/31 by Meijie Lu, Lu, Meijie, Xianchang Meng +1
Mathematics · #11A05 #11H06 #60F15 #60G50 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2307.16583

openalex publication_date 2023/07/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

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