Generalized Lattice Point Visibility

Abstract

It is a well-known result that the proportion of lattice points visible from the origin is given by $\frac{1}{\zeta (2)}$, where $\zeta (s)={\sum }_{n=1}^{\infty }\frac{1}{n^s}$ denotes the Riemann zeta function. Goins, Harris, Kubik and Mbirika, generalized the notion of lattice point visibility by saying that for a fixed $b\in \mathbb{N}$, a lattice point $(r,s)\in {\mathbb{N}}^2$ is $b$-visible from the origin if no other lattice point lies on the graph of a function $f(x)=m{x}^b$, for some $m\in \mathbb{Q}$, between the origin and $(r,s)$. In their analysis they establish that for a fixed $b\in \mathbb{N}$, the proportion of $b$-visible lattice points is $\frac{1}{\zeta (b+1)}$, which generalizes the result in the classical lattice point visibility setting. In this short note we give an $n$-dimensional notion of $\mathbf{b}$-visibility that recovers the one presented by Goins et. al. in $2$-dimensions, and the classical notion in $n$-dimensions. We prove that for a fixed $\mathbf{b}=(b_1,b_2,\dots ,b_n)\in {\mathbb{N}}^n$ the proportion of $\mathbf{b}$-visible lattice points is given by $\frac{1}{\zeta ({\sum }_{i=1}^n{b}_i)}$. Moreover, we propose a $\mathbf{b}$-visibility notion for vectors $\mathbf{b}\in {\mathbb{Q}}_{>0}^{\mathbf{n}}$, and we show that by imposing weak conditions on those vectors one obtains that the density of $\mathbf{b}=(\frac{b_1}{a_1},\frac{b_2}{a_2},\dots ,\frac{b_n}{a_n})\in {\mathbb{Q}}_{>0}^n$-visible points is $\frac{1}{\zeta ({\sum }_{i=1}^n{b}_i)}$. Finally, we give a notion of visibility for vectors $\mathbf{b}\in ({\mathbb{Q}}^{\ast }{)}^{\mathbf{n}}$, compatible with the previous notion, that recovers the results of Harris and Omar for $b\in {\mathbb{Q}}^{\ast }$ in $2$-dimensions; and show that the proportion of $\mathbf{b}$-visible points in this case only depends on the negative entries of $\mathbf{b}$.