Asymptotic results for a class of arithmetical functions
type: Item
instance of: scholarly article
title: Asymptotic results for a class of arithmetical functions
zbMATH Open document ID: 0551.10035
DOI: 10.1007/BF01300735
author: J. Chidambaraswamy
author: R. Sita Rama Chandra Rao
published in: Monatshefte für Mathematik
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178210
review text: In this paper, we obtain asymptotic formulae for the summatory functions of a class of arithmetical functions. These extend, and in certain cases refine, earlier results due to \textit{S. S. Pillai} [J. Annamalai Univ. 2, 243--248 (1933; Zbl 0008.19603)] and \textit{H. G. Kopetzky} [Monatsh. Math. 84, 213--217 (1977; Zbl 0551.10034)]. Let \(f_1, f_2,\cdots,f_r\) be arbitrary nonconstant polynomials with integer coefficients and let \(g(n)\) be an arithmetical function. We investigate the function \[ \Psi^g_{k,r}(n)=\Psi^g_{k,r}(f_1,f_2,\cdots,f_r;n)= \sum_{1\leq s_i\leq n^k, 1\leq i\leq r} g((f_1(s_1),\cdots,f_r(s_r);n^k)_k). \] Here \((a_1,\cdots,a_r;n^k)_k\) is the greatest common divisor of \(a_1,\cdots,a_r\) and \(n^k\) which is a \(k\)th power. The proofs use, among other things, a theorem of E. Wirsing.
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 3877290
zbMATH Keywords: largest higher power common divisor
zbMATH Keywords: asymptotic formulae
zbMATH Keywords: summatory functions
zbMATH Keywords: arithmetical functions
reviewed by: R. Sita Rama Chandra Rao
MaRDI profile type: Publication
cites work: On the Sum ∑k=1xd(f(k))
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description: scientific article; zbMATH DE number 3877290