Geodesic
Simultaneous distribution of primitive lattice points in convex planar domain
Čebyševskij sbornik, Tome 16 (2015) no. 1, pp. 163-175.

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Let $\Omega$ denote a compact convex subset of $\mathbf{R}^2$ which contains the origin as an inner point. Suppose that $\Omega$ is bounded by the curve $\partial \Omega,$ parametrized by $x=r_{\Omega}(\theta)\cos \theta,$ $y =r _{\Omega}(\theta)\sin \theta,$ where $r_{\Omega}$ is continuous and piecewise $C^3$ on $[0,\pi/4]$. For each real $R\ge 1$ we consider the domain $\Omega_R=\{(Rx,Ry) \vert (x,y) \in \Omega\}$ and we consider $\mathcal F (\Omega,R)=\{A\in \Omega_R\cap \mathbf{Z}^2 \vert A=(x,y), \text{НОД}(x,y)=1 \}$ — integer lattice points from $Q_R,$ which are visible from the origin. In this paper we study the simultaneous distribution for the lengths of the segments connecting the origin and a primitive lattice points from $\mathcal F (\Omega,R)$. Actually, we give an asymptotic formula $$\frac{\#\Phi(R)}{\#\mathcal F (\Omega,R)} =2\int_0^{\beta}\!\!\!\int_{0}^{\alpha} [\alpha'+\beta'\ge 1]d\alpha' d\beta'+O\big(R^{-\frac{1}{3}}\log^{\frac{2}{3}} R\big),$$ where $[A]=1,$ if $A$ is true, $[A]=0,$ if $A$ is false and for $\alpha,\beta\in [0,1]$ the value $\#\Phi(R)$ is equal to the number of fundamental parallelograms of the lattice $\mathbf{Z}^2$ for which the lengths $d_1,d_2$ of the segments do not exceed $\alpha \cdot R\cdot r_{\Omega}(\theta_1)$, $\beta \cdot R\cdot r_{\Omega}(\theta_2)$. Bibliography: 4 titles.
Keywords: primitive lattice points, simultaneous distribution. 
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     author = {O. A. Gorkusha},
     title = {Simultaneous distribution of primitive lattice points in convex planar domain},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {163--175},
     publisher = {mathdoc},
     volume = {16},
     number = {1},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2015_16_1_a7/}
}
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O. A. Gorkusha. Simultaneous distribution of primitive lattice points in convex planar domain. Čebyševskij sbornik, Tome 16 (2015) no. 1, pp. 163-175. http://geodesic.mathdoc.fr/item/CHEB_2015_16_1_a7/

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