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arXiv · 1509.02201

Primitive points in rational polygons

Imre Bárány·Greg Martin·Eric Naslund·Sinai Robins
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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 $\phi(n)$.

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Imre Bárány, Greg Martin, Eric Naslund, Sinai Robins. 2015-09-07. Primitive points in rational polygons. https://doi.org/10.4153/s0008439520000090

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