The distribution of the generalized greatest common divisor and visibility of lattice points

Abstract

For a fixed $b\in \mathbb{N}=\{1,2,3,\dots\}$, Goins et al. \cite{Harris} defined the concept of $b$-visibility for a lattice point $(r,s)$ in $L=\mathbb{N}\times \mathbb{N}$ which states that $(r,s)$ is $b$-visible from the origin if it lies on the graph of $f(x)=ax^b$, for some positive $a\in \mathbb{Q}$, and no other lattice point in $L$ lies on this graph between $(0,0)$ and $(r,s)$. Furthermore, to study the density of $b$-visible points in $L$ Goins et al. defined a generalization of greatest common divisor, denoted by $\gcd_b$, and proved that the proportion of $b$-visible lattice points in $L$ is given by $1/\zeta(b+1)$, where $\zeta(s)$ is the Riemann zeta function. In this paper we study the mean values of arithmetic functions $\Lambda:L\to \mathbb{ C}$ defined using $\gcd_b$ and recover the main result of \cite{Harris} as a consequence of the more general results of this paper. We also investigate a generalization of a result in \cite{Harris} that asserts that there are arbitrarily large rectangular arrangements of $b$-visible points in the lattice $L$ for a fixed $b$, more specifically, we give necessary and sufficient conditions for an arbitrary rectangular arrangement containing $b$-visible and $b$-invisible points to be realizable in the lattice $L$. Our result is inspired by the work of Herzog and Stewart \cite{Herzog} who proved this in the case $b=1$.