A note on a result of Abercrombie
type: Item
instance of: scholarly article
title: A note on a result of Abercrombie
zbMATH Open document ID: 0895.11008
DOI: 10.1007/BF02882834
published in: Chinese Science Bulletin
publication date: 1997-12-17T00:00:00Z
review text: The author considers the following divisor problem. Given a real number \(\theta\geq 1\), and an integer \(k\geq 2\), put \(D_k(\theta;x)=\sum_{\theta n\leq x}d_k\left (\lfloor n\theta\rfloor\right)\). It is proved in the paper that for almost all \(\theta\geq 1\) we have \[ D_k(\theta;x)=\theta^{-1}D_k(1;x)+ O(x^{(k-1)/k+\varepsilon}), \] where the \(O\)-constant depends on \(\theta\) and \(\varepsilon\). This improves and extends an earlier result by \textit{A. G. Abercrombie} [Acta Arith. 70, 195-207 (1995; Zbl 0817.11013)] who considered only the case \(k=2\) and proved the exponent \(5/7+\varepsilon\) in the exponent of the error term.
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 11J04
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 1091410
zbMATH Keywords: Beatty's sequence
zbMATH Keywords: divisor problem
zbMATH Keywords: rational approximation
zbMATH Keywords: metric diophantine approximation
author: Wenguang Zhai
reviewed by: Štefan Porubský
MaRDI profile type: Publication
cites work: Wythoff pairs as semigroup invariants
cites work: Beatty sequences and multiplicative number theory
cites work: Q5520003
cites work: Some extremal functions in Fourier analysis
cites work: Q4692762
description: scientific article; zbMATH DE number 1091410