On differences of two \(k\)th powers of integers
type: Item
instance of: scholarly article
title: On differences of two \(k\)th powers of integers
zbMATH Open document ID: 0922.11080
DOI: 10.1023/A:1009791425210
published in: The Ramanujan Journal
publication date: 1999-10-19T00:00:00Z
review text: For a fixed positive integer \(k\geq 3\), let \(r^-_k(n)\) denote the number of ways to write a natural number \(n\) as the difference of two \(k\)th powers of (absolute values of) integers. The investigation of the asymptotic behavior of the Dirichlet summatory function of \(r^-_k(n)\) leads in a natural way to a certain error term \(P^-_k(t)\). The author establishes a mean-square upper bound \(\int_ X^ {2X}(P_ k^ -(t))^ 2\,dt\ll X^ 2\) and the lower estimate \(P_ k^ -(t)=\Omega_ +(t^ {1/2}(\log t)^ {1/2k}log\log t)\), where \(X\) and \(t\) are large real variables.
Mathematics Subject Classification ID: 11P21
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11L07
zbMATH DE Number: 1251329
zbMATH Keywords: differences of powers
zbMATH Keywords: lattice points
zbMATH Keywords: exponential sums
zbMATH Keywords: asymptotic behavior of the Dirichlet summatory function
zbMATH Keywords: error term
zbMATH Keywords: mean square bound
author: Werner Georg Nowak
reviewed by: Jiahai Kan
MaRDI profile type: Publication
full work available at URL: a:1009791425210
OpenAlex ID: W93593130
description: scientific article; zbMATH DE number 1251329