#### Vol. 14, No. 1, 2021

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Generalized lattice-point visibility in $\mathbb{N}^k$

### Carolina Benedetti, Santiago Estupiñan and Pamela E. Harris

Vol. 14 (2021), No. 1, 103–118
##### Abstract

A lattice point $\left(r,s\right)\in {ℕ}^{2}$ is said to be visible from the origin if no other integer lattice point lies on the line segment joining the origin and $\left(r,s\right)$. It is a well-known result that the proportion of lattice points visible from the origin is given by $1∕\zeta \left(2\right)$, where $\zeta \left(s\right)={\sum }_{n=1}^{\infty }1∕{n}^{s}$ denotes the Riemann zeta function. Goins, Harris, Kubik and Mbirika generalized the notion of lattice-point visibility by saying that for a fixed $b\in ℕ$ a lattice point $\left(r,s\right)\in {ℕ}^{2}$ is $b$-visible from the origin if no other lattice point lies on the graph of a function $f\left(x\right)=m{x}^{b}$, for some $m\in ℚ$, between the origin and $\left(r,s\right)$. In their analysis they establish that for a fixed $b\in ℕ$ the proportion of $b$-visible lattice points is $1∕\zeta \left(b+1\right)$, which generalizes the result in the classical lattice-point visibility setting. In this paper we give an $n$-dimensional notion of $b$-visibility that recovers the one presented by Goins et. al. in two dimensions, and the classical notion in $n$ dimensions. We prove that for a fixed $b=\left({b}_{1},{b}_{2},\dots ,{b}_{n}\right)\in {ℕ}^{n}$ the proportion of $b$-visible lattice points is given by $1∕\zeta \left({\sum }_{i=1}^{n}{b}_{i}\right)$.

Moreover, we give a new notion of $b$-visibility for vectors

$b=\left({b}_{1}∕{a}_{1},{b}_{2}∕{a}_{2},\dots ,{b}_{n}∕{a}_{n}\right)\in {\left(ℚ\setminus \left\{0\right\}\right)}^{n},$

with nonzero rational entries. In this case, our main result establishes that the proportion of $b$-visible points is $1∕\zeta \left({\sum }_{i\in J}|{b}_{i}|\right)$, where $J$ is the set of the indices $1\le i\le n$ for which ${b}_{i}∕{a}_{i}<0$. This result recovers a main theorem of Harris and Omar for $b\in ℚ\setminus \left\{0\right\}$ in two dimensions, while showing that the proportion of $b$-visible points (in such cases) only depends on the negative entries of $b$.

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##### Keywords
lattice-point visibility, generalized lattice-point visibility, Riemann zeta function
Primary: 11B05
Secondary: 60B10
##### Milestones
Revised: 13 September 2020
Accepted: 28 September 2020
Published: 4 March 2021

Communicated by Stephan Garcia
##### Authors
 Carolina Benedetti Departamento de Matemáticas Universidad de los Andes Bogotá Colombia Santiago Estupiñan Departamento de Matemáticas Universidad de los Andes Bogotá Colombia Pamela E. Harris Department of Mathematics and Statistics Williams College Williamstown, MA United States