Primitive points in rational polygons

TL;DR

This paper investigates the distribution of primitive lattice points within dilated rational polygons, establishing asymptotic formulas and bounds for the associated error term, extending known results from specific triangles to a broader class.

Contribution

The paper generalizes error term bounds for primitive lattice points from the isosceles right triangle to all rational star-shaped polygons containing the origin.

Findings

Error term bounds are both $igOmega_ imes(t ext{sqrt(log log t)})$ and $O(t( ext{log} t)^{2/3}( ext{log log} t)^{4/3})$.

Asymptotic count of primitive lattice points matches $rac{6}{ ext{pi}^2}$ times the area for large dilations.

Results extend known bounds for Euler's $ ext{phi}(n)$ to a broader class of polygons.

Abstract

Let $\mathcal A$ be a star-shaped polygon in the plane, with rational vertices, containing the origin. The number of primitive lattice points in the dilate $t\mathcal A$ is asymptotically $\frac6{\pi^2}$ Area$(t\mathcal A)$ as $t\to \infty$. We show that the error term is both $\Omega_\pm\big( t\sqrt{\log\log t} \big)$ and $O(t(\log t)^{2/3}(\log\log t)^{4/3})$. Both bounds extend (to the above class of polygons) known results for the isosceles right triangle, which appear in the literature as bounds for the error term in the summatory function for Euler's $\phi(n)$.