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

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 $m\ge3$.