Lattice point visibility on power functions

Abstract

It is classically known that the proportion of lattice points visible from the origin via functions of the form $f(x)=nx$ with $n\in \mathbb{Q}$ is $\frac{1}{\zeta(2)}$ where $\zeta(s)$ is the classical Reimann zeta function. Goins, Harris, Kubik and Mbirika, generalized this and determined the proportion of lattice points visible from the origin via functions of the form $f(x)=nx^b$ with $n\in \mathbb{Q}$ and $b\in\mathbb{N}$ is $\frac{1}{\zeta(b+1)}$. In this article, we complete the analysis of determining the proportion of lattice points that are visible via power functions with rational exponents, and simultaneously generalize these previous results.