arXiv subjects

Meijie Lu

Publications and source records attributed to Meijie Lu.

3 recordsLinked to original sources

arXiv2023

On the distribution of $k$-free numbers on the view point of random walks

In this paper, we investigate the distribution of $k$-free numbers in a class of $α$-random walks on the integer lattice $\mathbb{Z}$. In these walks, the walker starts from a non-negative integer $r$ and moves to the right by $a$ units with probability $α$, or by $b$ units with probability $1-α$. For $k\geq 3$, we obtain the asymptotic proportion of $k$-free numbers in a path of such $α$-random walks in almost surely sense. This provides a generalization of a classical result on the distribution of $k$-free numbers in arithmetic progressions.

math.NT↗

arXiv2023

Visible lattice points in Pólya's walk

In this paper, for any integer $k\geq 2$, we study the distribution of the visible lattice points in certain generalized Pólya's walk on $\mathbb{Z}^k$: 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 $1/ζ(k)$.

math.NT↗

arXiv2022

Visible lattice points in higher dimensional random walks and biases among them

For any integers $k\geq 2$, $q\geq 1$ and any finite set $\mathcal{A}=\{{\boldsymbolα}_1,\cdots,{\boldsymbolα}_q\}$, where ${ \boldsymbolα_t}=(α_{t,1},\cdots,α_{t,k})~(1\leq t\leq q)$ with $0<α_{t,1},\cdots,α_{t,k}<1$ and $α_{t,1}+\cdots+α_{t,k}=1$, this paper concerns the visibility of lattice points in the type-$\mathcal{A}$ random walk on the lattice $\mathbb{Z}^k$. We show that the proportion of visible lattice points on a random path of the walk is almost surely $1/ζ(k)$, where $ζ(s)$ is the Riemann zeta-function, and we also consider consecutive visibility of lattice points in the type-$\mathcal{A}$ random walk and give the proportion of the corresponding visible steps. Moreover, we find a new phenomenon that visible steps in both of the above cases are not evenly distributed. Our proof relies on tools from probability theory and analytic number theory.

math.NT↗