On the primitive circle problem
type: Item
instance of: scholarly article
title: On the primitive circle problem
zbMATH Open document ID: 0994.11035
author: Jie Wu
published in: Monatshefte für Mathematik
publication date: 2002-06-10T00:00:00Z
review text: Let \(V(x)\) be the number of primitive lattice points \((m,n)\) in the circle given by \(m^2+n^2\leq x\), where \((m,n)\) is said to be primitive if \(\text{h.c.f.}(m,n)=1\). Then it is shown, subject to the Riemann hypothesis, that \[ V(x)= \tfrac {6}{\pi}+ O(x^{221/608+ \varepsilon}) \] for any \(\varepsilon> 0\). Unconditionally the best exponent known is only \(1/2\). Previously the best known conditional exponent was \(11/30+ \varepsilon\), due to \textit{W. Zhai} and \textit{X. Cao} [Acta Arith. 90, 1-16 (1999; Zbl 0932.11066)]. Bounds for multidimensional exponential sums are the main ingredient of the proof.
Mathematics Subject Classification ID: 11P21
Mathematics Subject Classification ID: 11L07
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 1753101
zbMATH Keywords: circle problem
zbMATH Keywords: error term
zbMATH Keywords: bounds for multidimensional exponential sums
zbMATH Keywords: number of primitive lattice points
zbMATH Keywords: Riemann hypothesis
reviewed by: D. R. Heath-Brown
MaRDI profile type: Publication
full work available at URL: s006050200006
OpenAlex ID: W2000829837
DOI: 10.1007/S006050200006
description: scientific article; zbMATH DE number 1753101