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 $f(x)=mx^b$, 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 $\bf{b}$-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 ${\bf{b}}=(b_1,b_2,\ldots,b_n)\in\mathbb{N}^n$ the proportion of ${\bf{b}}$-visible lattice points is given by $\frac{1}{\zeta(\sum_{i=1}^nb_i)}$. Moreover, we propose a $\bf{b}$-visibility notion for vectors $\bf{b}\in \mathbb{Q}_{>0}^n$, and we show that by imposing weak conditions on those vectors one obtains that the density of ${\bf{b}}=(\frac{b_1}{a_1},\frac{b_2}{a_2},\ldots,\frac{b_n}{a_n})\in\mathbb{Q}_{>0}^n$-visible points is $\frac{1}{\zeta(\sum_{i=1}^nb_i)}$. Finally, we give a notion of visibility for vectors $\bf{b}\in (\mathbb{Q}^{*})^n$, compatible with the previous notion, that recovers the results of Harris and Omar for $b\in \mathbb{Q}^{*}$ in $2$-dimensions; and show that the proportion of $\bf{b}$-visible points in this case only depends on the negative entries of $\bf{b}$.