论文标题
$ k $ - 随机步行中的无晶格点
$k$-free lattice points in random walks
论文作者
论文摘要
令$ \ mathbb {z}^2 $为二维整数晶格。对于整数$ k \ geq 1 $，如果其坐标的最大共同除数为$ k $ free，则非零晶格点为$ k $。我们考虑$ k $ free的比例和双$ k $ - free晶格点，在$ \ mathbb {z}^2 $中的$α$ -random Walker的路径上。使用分析数理论的第二阶段方法和工具，我们证明这两个比例分别为$ 1/ζ（2k）$和$ \ prod_ {p}（1-2p^{ - 2k}）$，其中$ζ$是Riemann Zeta Zeta Zeta功能，而无限制的产品接管了所有普遍量。
Let $\mathbb{Z}^2$ be the two-dimensional integer lattice. For an integer $k\geq 1$, 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 $\mathbb{Z}^2$. Using the second-moment method and tools from analytic number theory, we prove that these two proportions are $1/ζ(2k)$ and $\prod_{p}(1-2p^{-2k})$, respectively, where $ζ$ is the Riemann zeta function and the infinite product takes over all primes.
