On an error term related to the greatest divisor of \(n\), which is prime to \(k\)
type: Item
instance of: scholarly article
title: On an error term related to the greatest divisor of \(n\), which is prime to \(k\)
zbMATH Open document ID: 0655.10038
author: Sukumar Das Adhikari
author: R. Balasubramanian
author: Ayyadurai Sankaranarayanan
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 1988-01-01T00:00:00Z
review text: For a fixed positive squarefree integer \(k\) let \(\delta_ k(n)\) denote the greatest divisor of \(n\) which is prime to \(k\) and define the error term \(E_ k\) by \[ \sum_{n\leq x}\delta_ k(n)=(k/2 \sigma (k))x^ 2+E_ k(x). \] \textit{V. S. Joshi} and \textit{A. M. Vaidya} [Topics in Classical Number Theory, Colloq. Budapest 1981, Vol. 1, Colloq. Math. Soc. Janos Bolyai 34, 791--806 (1984; Zbl 0546.10038)] obtained \(E_ k(x)=O(x)\) and for a prime \(p\) they showed \(E_ p(x)=\Omega_{\pm}(x)\). At the conference on Number Theory in Oberwolfach the reviewer announced a joint paper with \textit{J. Herzog} [Arch. Math. 50, 145--155 (1988; Zbl 0616.10035)] where among other things the same result is shown in the general case by using the Tauberian theorem of Hardy, Littlewood and Karamata. In the paper under review this result is obtained by the Erdős-Shapiro method of averaging over arithmetic progressions. Reviewer's remark: In the above mentioned paper J. Herzog and the reviewer pointed out that the Tauberian argument may be replaced by elementary estimates.
reviewed by: Thomas Maxsein
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 4068998
zbMATH Keywords: divisors prime to a fixed positive squarefree integer
zbMATH Keywords: greatest divisor
zbMATH Keywords: error term
zbMATH Keywords: Erdős-Shapiro method
zbMATH Keywords: averaging over arithmetic progressions
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4068998