Lattice points in domains \(| x|\) \(p+| y|\) p\(\leq R\) p
type: Item
instance of: scholarly article
title: Lattice points in domains \(| x|\) \(p+| y|\) p\(\leq R\) p
zbMATH Open document ID: 0627.10032
DOI: 10.1007/BF01194154
author: Wolfgang Müller
author: Werner Georg Nowak
published in: Archiv der Mathematik
publication date: 1988-01-01T00:00:00Z
review text: For arbitrary real \(p\geq 2\), let \(A_ p(R)\) denote the number of lattice points in the planar domain \(D_ p(R):\) \(| x| ^ p+| y| ^ p\leq R^ p\). As a generalization of the classical circle problem \((p=2)\), one can ask for the infimum \(\theta\) (p) of all exponents \(\lambda\) for which the asymptotic \(A_ p(R)=a_ pR^ 2+O(R^{\lambda})\) is true \((a_ p\) the area of \(D_ p(1))\). It was proven that \((*)\quad \theta (p)=1-1/p\) for \(p>3\) by \textit{J. G. van der Corput} [Thesis Groningen 1919], for \(p=3\) by \textit{E. Krätzel} [Math. Ann. 179, 90-96 (1969; Zbl 0165.363)] and for \(p>41/14\) by the first author [Sitzungsber., Abt. II, Österr. Akad. Wiss., Math.-Naturwiss. Kl. 191, 125-132 (1982; Zbl 0486.10038)]. In the present paper, the recently developd ``discrete Hardy-Littlewood method'' in the version of \textit{M. Huxley} and \textit{N. Watt} [Exponential sums and the Riemann zeta-function (submitted to Proc. Lond. Math. Soc.)] is applied in order to verify (*) for any \(p>38/13=2,92307..\). In fact, the asymptotic formula \[ A_ p(R)=a_ pR^ 2+F_ p(R)R^{1- 1/p}+O(R^{25/38+\epsilon}) \] (with an explicitly given almost periodic function \(F_ p(R))\) is established for this range of p.
Mathematics Subject Classification ID: 11P21
Mathematics Subject Classification ID: 11L03
zbMATH DE Number: 4019168
zbMATH Keywords: number of lattice points
zbMATH Keywords: planar domain
zbMATH Keywords: discrete Hardy-Littlewood method
zbMATH Keywords: exponential sums
zbMATH Keywords: asymptotic formula
MaRDI profile type: Publication
cites work: New omega theorems for two classical lattice point problems
cites work: Q4192121
cites work: Exponential Sums and the Riemann Zeta Function
cites work: On the divisor and circle problems
cites work: Remarks on a lattice point problem
cites work: Q3946250
cites work: Q5803461
full work available at URL: https://doi.org/10.1007/bf01194154
OpenAlex ID: W2066559838
description: scientific article; zbMATH DE number 4019168