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 $\phi(n)$.

中文翻译：

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有理多边形中的基本点

令$\mathcal A$ 为平面中的星形多边形，具有有理顶点，包含原点。膨胀$t\mathcal A$ 中的原始格点数渐近$\frac6{\pi^2}$ Area$(t\mathcal A)$ as $t\to \infty$。我们表明误差项既是 $\Omega_\pm\big( t\sqrt{\log\log t} \big)$ 和 $O(t(\log t)^{2/3}(\log\记录 t)^{4/3})$。两个边界都扩展了（到上述多边形类）等腰直角三角形的已知结果，这些结果出现在文献中作为欧拉 $\phi(n)$ 求和函数中误差项的边界。