Koninklijke Nederlandse Akademie van Wetenschappen. Indagationes Mathematicae. New Series (Indag. Math. (N.S.)) (20230101), 34, no.~1, 168-185. ISSN: 0019-3577 (print).eISSN: 1872-6100.
Subject
11 Number theory -- 11M Zeta and $L$-functions: analytic theory 11M06 $\zeta
60 Probability theory and stochastic processes -- 60G Stochastic processes 60G50 Sums of independent random variables; random walks
A positive integer $n$ is {\it $k$-free} if it is not divisible by $p^k$ for any prime $p$. A non-zero lattice point $(m, n)$ is {\it $k$-free} if $\gcd(m, n)$ is $k$-free. A lattice point $(m, n)$ is {\it visible} from the origin if and only if $\gcd(m, n) = 1$. People have been interested in visible lattice points, of which $k$-free lattice points are a generalization since 1 is the only 1-free integer. J. Cilleruelo~Mateo, J.~L. Fernández~Pérez and P. Fernández~Gallardo [European J. Combin. {\bf 75} (2019), 92--112; MR3862956] were the first to consider visible lattice points from the viewpoint of random walks: For $0 < \alpha < 1$, an $\alpha$-random walk starting at the origin is defined by $$ P_{i+1} = P_i + \cases (1,0) \text{ with probability } \alpha,\\ (0,1) \text{ with probability } 1 - \alpha, \endcases $$ when $i = 0, 1, 2, \ldots$. \par In the paper under review, the authors extend the random walk connection to $k$-free lattice points. Define the random variables $$ X_i = \cases 1, & \text{ if $P_i$ is $k$-free},\\ 0, & \text{ otherwise,} \endcases $$ $$ \overline{S_n} = \overline{S}(n, k, \alpha) \coloneq \frac{X_1 + X_2 + \cdots + X_n}{n}, \text{ proportion of $k$-free lattice points,} $$ and $$ \multline \overline{T_n} = \overline{T}(n, k, \alpha) \coloneq \frac{X_1 X_2 + X_2 X_3 + \cdots + X_n X_{n+1}}{n},\\ \text{ proportion of twin $k$-free lattice points,} \endmultline $$ respectively. The authors derive the following asymptotic proportions: $$ \lim_{n \rightarrow \infty} \overline{S_n} = \frac{1}{\zeta(2k)} \quad \text{and} \quad \lim_{n \rightarrow \infty} \overline{T_n} = \prod_{p} \Bigl(1 - \frac{2}{p^{2k}} \Bigr) $$ almost surely, independent of the probability $\alpha$. Here, $\zeta(s)$ is the Riemann zeta function and the product is over all primes $p$. \par The method of proof uses the following tools from probability theory: the second-moment method, the local central limit theorem, and results of sums of binomial probabilities $$ \sum_{l \equiv r \pmod{d}} \binom{n}{l} \alpha^l (1 - \alpha)^{n - l}. $$ On the number theory side, the authors make use of the Möbius function to detect $k$-free greatest common divisors, estimates for the $l$-fold divisor function $\tau_l(n) = \sum_{d_1 d_2 \cdots d_l = n} 1$, auxiliary arithmetical functions such as $$ g_k(n) \coloneq \sum \Sb r d = n\\d \text{ is $k$-free}\endSb \mu(r), $$ and the Chinese Remainder Theorem.