On a class of \(\psi\)-convolutions characterized by the identical equation
type: Item
instance of: scholarly article
title: On a class of \(\psi\)-convolutions characterized by the identical equation
zbMATH Open document ID: 1071.11007
author: Varanasi Sitaramaiah
author: Jean-Louis Nicolas
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2005-06-30T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93864
review text: \textit{R.~Vaidyanathaswamy} [Bull. Am. Math. Soc. 36, 762--772 (1930; JFM 56.0873.03)] termed the equation \[ f(mn) = \sum_{\substack{ a| m\\ b| n}} f(m/a)f(n/b)f^{-1}(ab)C(a, b) \] involving the Dirichlet convolution as the identical equation for multiplicative functions. \textit{V.~Sita\-ramaiah} and \textit{M.V.~Subbarao} [Proc. Am. Math. Soc. 124, No. 2, 361--369 (1996; Zbl 0847.11003) and J. Indian Math. Soc., New Ser. 64 , No. 1--4, 131--150 (1997)] proved that if \(\psi\) is a Lehmer-Narkiewicz convolution, then a \(\psi\)-generalization of the identical equation holds for all multiplicative functions. The present authors show the converse: If \(\psi\) satisfies certain conditions and the \(\psi\)-generalization of the identical equation holds for all multiplicative functions, then \(\psi\) is a Lehmer-Narkiewicz convolution.
reviewed by: Pentti Haukkanen
Mathematics Subject Classification ID: 11A25
zbMATH DE Number: 2184600
zbMATH Keywords: identical equation
zbMATH Keywords: multiplicative functions
zbMATH Keywords: Dirichlet convolution
zbMATH Keywords: Lehmer-Narkiewicz \(\psi\)-convolution
zbMATH Keywords: regular convolution
MaRDI profile type: Publication
OpenAlex ID: W2330701459
cites work: Arithmetical functions associated with the unitary divisors of an integer
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cites work: The identical equation in 𝜓-products
cites work: Q5710193
DOI: 10.5802/JTNB.375
description: scientific article; zbMATH DE number 2184600