Ramanujan sums via generalized Möbius functions and applications

A generalized Ramanujan sum (GRS) is defined by replacing the usual Möbius function in the classical Ramanujan sum with the Souriau-Hsu-Möbius function. After collecting basic properties of a GRS, mostly containing existing ones, seven aspects of a GRS are studied. The first shows that the unique re...

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Main Authors: Vichian Laohakosol, Pattira Ruengsinsub, Nittiya Pabhapote
Format: Article
Language:English
Published: Wiley 2006-01-01
Series:International Journal of Mathematics and Mathematical Sciences
Online Access:http://dx.doi.org/10.1155/IJMMS/2006/60528
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author Vichian Laohakosol
Pattira Ruengsinsub
Nittiya Pabhapote
author_facet Vichian Laohakosol
Pattira Ruengsinsub
Nittiya Pabhapote
author_sort Vichian Laohakosol
collection DOAJ
description A generalized Ramanujan sum (GRS) is defined by replacing the usual Möbius function in the classical Ramanujan sum with the Souriau-Hsu-Möbius function. After collecting basic properties of a GRS, mostly containing existing ones, seven aspects of a GRS are studied. The first shows that the unique representation of even functions with respect to GRSs is possible. The second is a derivation of the mean value of a GRS. The third establishes analogues of the remarkable Ramanujan's formulae connecting divisor functions with Ramanujan sums. The fourth gives a formula for the inverse of a GRS. The fifth is an analysis showing when a reciprocity law exists. The sixth treats the problem of dependence. Finally, some characterizations of completely multiplicative function using GRSs are obtained and a connection of a GRS with the number of solutions of certain congruences is indicated.
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institution Kabale University
issn 0161-1712
1687-0425
language English
publishDate 2006-01-01
publisher Wiley
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series International Journal of Mathematics and Mathematical Sciences
spelling doaj-art-b15d868e37c8460a83e04044ce6e17bd2025-02-03T01:11:08ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252006-01-01200610.1155/IJMMS/2006/6052860528Ramanujan sums via generalized Möbius functions and applicationsVichian Laohakosol0Pattira Ruengsinsub1Nittiya Pabhapote2Department of Mathematics, Kasetsart University, Bangkok 10900, ThailandDepartment of Mathematics, Kasetsart University, Bangkok 10900, ThailandDepartment of Mathematics, University of the Thai Chamber of Commerce, Bangkok 10400, ThailandA generalized Ramanujan sum (GRS) is defined by replacing the usual Möbius function in the classical Ramanujan sum with the Souriau-Hsu-Möbius function. After collecting basic properties of a GRS, mostly containing existing ones, seven aspects of a GRS are studied. The first shows that the unique representation of even functions with respect to GRSs is possible. The second is a derivation of the mean value of a GRS. The third establishes analogues of the remarkable Ramanujan's formulae connecting divisor functions with Ramanujan sums. The fourth gives a formula for the inverse of a GRS. The fifth is an analysis showing when a reciprocity law exists. The sixth treats the problem of dependence. Finally, some characterizations of completely multiplicative function using GRSs are obtained and a connection of a GRS with the number of solutions of certain congruences is indicated.http://dx.doi.org/10.1155/IJMMS/2006/60528
spellingShingle Vichian Laohakosol
Pattira Ruengsinsub
Nittiya Pabhapote
Ramanujan sums via generalized Möbius functions and applications
International Journal of Mathematics and Mathematical Sciences
title Ramanujan sums via generalized Möbius functions and applications
title_full Ramanujan sums via generalized Möbius functions and applications
title_fullStr Ramanujan sums via generalized Möbius functions and applications
title_full_unstemmed Ramanujan sums via generalized Möbius functions and applications
title_short Ramanujan sums via generalized Möbius functions and applications
title_sort ramanujan sums via generalized mobius functions and applications
url http://dx.doi.org/10.1155/IJMMS/2006/60528
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AT pattiraruengsinsub ramanujansumsviageneralizedmobiusfunctionsandapplications
AT nittiyapabhapote ramanujansumsviageneralizedmobiusfunctionsandapplications