The area within a curve
type: Item
instance of: scholarly article
title: The area within a curve
zbMATH Open document ID: 0656.10046
DOI: 10.1007/BF02837818
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1987-01-01T00:00:00Z
review text: Let C be a simple closed convex and smooth curve with continuous non-zero radius of curvature and area A. For a large positive variable R consider the blow up domain by the factor R such that it has the area \(AR^ 2\). If N denotes the number of lattice points within it, then it is well- known that \[ (1)\quad N=AR^ 2+O(R^{2/3+\epsilon})\quad (\epsilon >0). \] Recently \textit{H. Iwaniec} and \textit{C. J. Mozzochi} [J. Number Theory 29, No.1, 60-93 (1988; Zbl 0644.10031)] obtained the estimation \(R^{7/11+\epsilon}\) for the error term. The author and \textit{N. Watt} [Proc. Lond. Math. Soc., III. Ser. 57, 1-24 (1988; Zbl 0644.10027); Exponential sums and lattice points (to appear)] showed that their method can be generalized to more general domains. In this paper the author gives a strictly elementary proof of (1) using only the continued fraction algorithm and the circle method. This gives an interesting new view on the work of H. Iwaniec and C. J. Mozzochi. Also simple estimates for the mean square error are obtained.
reviewed by: Ekkehard Krätzel
Mathematics Subject Classification ID: 11P21
Mathematics Subject Classification ID: 11P55
Mathematics Subject Classification ID: 11H06
zbMATH DE Number: 4071033
zbMATH Keywords: estimation of area
zbMATH Keywords: blow up domain
zbMATH Keywords: number of lattice points
zbMATH Keywords: elementary proof
zbMATH Keywords: continued fraction algorithm
zbMATH Keywords: circle method
zbMATH Keywords: mean square error
author: M. N. Huxley
MaRDI profile type: Publication
cites work: Q4723831
cites work: Exponential Sums and the Riemann Zeta Function
cites work: Exponential Sums and Lattice Points
cites work: On the divisor and circle problems
cites work: Lectures on a method in the theory of exponential sums
cites work: The number of lattice points on a convex curve
description: scientific article; zbMATH DE number 4071033