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On the distribution of the number of lattice points inside a family of convex ovals. (English) Zbl 0762.11031

Let \(\Omega_ \gamma\) be the domain enclosed by a simple convex smooth curve \(\gamma\) and \(\Omega_ \gamma(R)\) the “blown up” domain by a positive parameter \(R\). Let \(\alpha\in\mathbb{R}^ 2\) be a fixed point in the plane and \(\alpha+\mathbb{Z}^ 2=\{x=\alpha+n,n\in\mathbb{Z}^ 2\}\) a square lattice. Let \(N_ \gamma(R;\alpha)\) be the number of lattice points lying in \(\Omega_ \gamma(R)\) and put \[ F_ \gamma(R;\alpha)=(N_ \gamma(R;\alpha)-\text{Area} \Omega_ \gamma(R))/\sqrt R. \] The author studies the distribution of the error term \(F_ \gamma(R;\alpha)\) generalizing the work of D. R. Heath-Brown [Acta Arith. 60, 389-415 (1992; Zbl 0725.11045)].
Reviewer: E.Krätzel (Jena)

MSC:

11P21 Lattice points in specified regions
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