The exact order of the number of lattice points visible from the origin

TL;DR

This paper determines the precise order of the error term in counting lattice points visible from the origin within a hypercube, establishing it as proportional to r^{m-1} for dimensions m ≥ 3.

Contribution

It proves that the error term in the lattice point count has an exact order of r^{m-1} for dimensions m ≥ 3, refining previous asymptotic estimates.

Findings

Error term order is r^{m-1} for m ≥ 3

Number of visible lattice points approximates (2r)^m / ζ(m)

Provides exact asymptotic behavior of lattice point counts

Abstract

We say a lattice point $X=(x_1,\ldots,x_m)$ is visible from the origin, if $\gcd(x_1,...,x_m)=1$. In other word, there are no other lattice point on the line segment from the origin $O$ to $X$. From J.E. Nymann's result, we know that the number of lattice point from the origin in $[-r,r]^m$ is $(2r)^m/\zeta(m)+$(Error term). We showed that the exact order of the error term is $r^{m-1}$ for $m\ge3$.