k-free lattice points in random walks

2022/02/05 by Kui Liu, Liu, Kui, Shunqi Ma +1
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2202.02449

openalex publication_date 2022/02/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let ℤ² be the two-dimensional integer lattice. For an integer k≥ 1, a non-zero lattice point is k-free if the greatest common divisor of its coordinates is a k-free number. We consider the proportions of k-free and twin k-free lattice points on a path of an α-random walker in ℤ². Using the second-moment method and tools from analytic number theory, we prove that these two proportions are 1/ζ(2k) and ∏_p(1-2p^-2k), respectively, where ζ is the Riemann zeta function and the infinite product takes over all primes.

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