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On primitive lattice points in planar domains. (English) Zbl 1027.11075

Let \(D\) denote a compact convex subset of \(\mathbb{R}^2\) which contains the origin as an inner point. Suppose that the boundary \(\partial D\) of \(D\) is sufficiently smooth with finite nonzero curvature throughout. Let \(B_D(x)\) denote the number of primitive lattice points \((m,n)\) in \(\sqrt{x}D\), such that \(\gcd(m,n)= 1\). B. Z. Moroz [Monatsh. Math. 99, 37-42 (1985; Zbl 0551.10038)], assuming the Riemann hypothesis to be true, proved that \[ B_D(x)= \frac{6}{\pi^2} \text{ area}(D)x+ O(x^{\vartheta+ \varepsilon}) \] holds with \(\vartheta= 41/91\). M. N. Huxley and W. G. Nowak [Acta Arith. 76, 271-283 (1996; Zbl 0861.11056)] proved that the error term can be sharpened to \(\vartheta= 5/12\) and W. Müller [Proc. Number Theory Conf. 1996, Vienna, W. G. Nowak and J. Schoißengeier (eds.), 189-199 (1996; Zbl 0879.11054)] obtained \(\vartheta= 9/22\). The author now proves \(\vartheta= 33349/84040\) and generalizes the estimation to planar domains \(D\), where the boundary \(\partial D\) is piecewise smooth.
Further E. Krätzel and W. G. Nowak [Acta Arith. 99, 331-341 (2001; Zbl 0984.11047)] considered primitive lattice points in a thin strip along the boundary of the convex planar domain and proved that \[ B_D(x+x^\theta)- B_D(x)= \frac{6}{\pi^2} \text{ area}(D) x^\theta(1+o(1)) \] holds with \(\theta> 11/29\). The author generalizes this result also to planar domains \(D\) with piecewise smooth boundary \(\partial D\).

MSC:

11P21 Lattice points in specified regions
11N37 Asymptotic results on arithmetic functions
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