Lattice point visibility on power functions

TL;DR

This paper extends the understanding of lattice point visibility from the origin to power functions with rational exponents, providing a complete analysis and generalization of previous results involving linear and polynomial functions.

Contribution

It completes the analysis of lattice point visibility for power functions with rational exponents and generalizes earlier findings to a broader class of functions.

Findings

Proportion of visible lattice points is 1/ζ(b+1) for functions f(x)=nx^b with rational n.

Generalization to rational exponents in power functions.

Unified framework for visibility across different 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.