Primitive lattice points in starlike planar sets (Q1367610)

Let \(D\subseteq\mathbb{R}^2\), and suppose that \(D\) is starlike, that is \(\lambda u\in D\) as soon as \(0<\lambda<1\) and \(u\in D\). Assuming that \[ \text{card}\{u\mid u\in \mathbb{Z}^2, {u\over \sqrt x}\in D\}= \sum^n_{i=0} c_ix^{ \alpha_i} +O(x^\alpha) \] with \(\alpha_0=1> \alpha_1> \cdots>\alpha_n> \alpha\), \(\alpha <1/2\), the author deduces from the Riemann hypothesis that \[ \text{card}\bigl\{ u\mid u\in \mathbb{Z}^2\cap \sqrt xD,\text{ gcd }(u_1,u_2) =1 \bigr\} =\sum_{\alpha_i>\theta}{c_i\over\zeta(2\alpha_i)}x^{\alpha_i }+O (x^{\theta + \varepsilon}),\varepsilon>0, \text{ with }\theta=\tfrac{4-\alpha}{11-8\alpha}. \] This theorem generalises and strengthens an earlier result of the reviewer [\textit{B. Moroz}, Monatsh. Math. 99, 37-42 (1985; Zbl 0551.10038)]. The author discusses several applications of his theorem.