Lattice points in bodies of revolution in \({\mathbb R}^3\): An \(\Omega \_\)-estimate for the error term
type: Item
instance of: scholarly article
title: Lattice points in bodies of revolution in \({\mathbb R}^3\): An \(\Omega \_\)-estimate for the error term
zbMATH Open document ID: 0958.11066
author: Manfred Kühleitner
published in: Archiv der Mathematik
publication date: 2000-12-14T00:00:00Z
review text: Let \(A_{\mathcal B}(t)\) denote the number of triples in \(\mathbb Z^3\) in the ``blown up'' region \(\sqrt{t}\mathcal B\), where \(\mathcal B\) is a compact, convex body of revolution in \(\mathbb R^3\) which contains the origin as an inner point and satisfies some additional (reasonable) conditions. The author proves that, as \(t\to+\infty\), \[ A_{\mathcal B}(t) = \text{ vol} ({\mathcal B}) t^{3/2} + \Omega_-\left(t^{1/2}(\log t)^{1/3}(\log\log t)^{\log 2\over 3} \text{ e}^{-C\sqrt{\log\log\log t}}\right) \] for some absolute, positive constant \(C\). This is achieved by combining the approaches of \textit{W. G. Nowak} [Arch. Math. 47, 232-237 (1986; Zbl 0594.10042)] and \textit{J. L. Hafner} [Invent. Math. 63, 181-186 (1981; Zbl 0458.10031)]. A crucial point in the proof is to choose the axis of revolution to be one of the coordinate axes. The above result is slightly sharper than the general result of W. G. Nowak (op. cit.), who had \(\Omega_-\left(t^{1/2}(\log t)^{1/3}\right)\).
reviewed by: Aleksandar Ivić
Mathematics Subject Classification ID: 11P21
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11L07
Mathematics Subject Classification ID: 11L40
zbMATH DE Number: 1449205
zbMATH Keywords: lattice points
zbMATH Keywords: body of revolution
zbMATH Keywords: unit sphere
zbMATH Keywords: asymptotic expansion
MaRDI profile type: Publication
full work available at URL: s000130050436
OpenAlex ID: W2073912088
DOI: 10.1007/S000130050436
description: scientific article; zbMATH DE number 1449205