Lattice point visibility on generalized lines of sight

• Published in: arXiv.org 2017-10
• Main Authors: Edray Herber Goins, Harris, Pamela E, Kubik, Bethany, Mbirika, Aba
• Format: Article
• Language: English
• Subjects: Linear functions; Visibility
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1710.04554

For a fixed \(b\in\mathbb{N}=\{1,2,3,\ldots\}\) we say that a point \((r,s)\) in the integer lattice \(\mathbb{Z} \times \mathbb{Z}\) is \(b\)-visible from the origin if it lies on the graph of a power function \(f(x)=ax^b\) with \(a\in\mathbb{Q}\) and no other integer lattice point lies on this curve (i.e., line of sight) between \((0,0)\) and \((r,s)\). We prove that the proportion of \(b\)-visible integer lattice points is given by \(1/\zeta(b+1)\), where \(\zeta(s)\) denotes the Riemann zeta function. We also show that even though the proportion of \(b\)-visible lattice points approaches \(1\) as \(b\) approaches infinity, there exist arbitrarily large rectangular arrays of \(b\)-invisible lattice points for any fixed \(b\). This work specialized to \(b=1\) recovers original results from the classical lattice point visibility setting where the lines of sight are given by linear functions with rational slope through the origin.