Ramanujan sums via generalized Möbius functions and applications

Laohakosol, Vichian; Ruengsinsub, Pattira; Pabhapote, Nittiya

doi:https://doi.org/10.1155/IJMMS/2006/60528

International Journal of Mathematics and Mathematical Sciences

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Volume 2006 | Article ID 060528 | https://doi.org/10.1155/IJMMS/2006/60528

Ramanujan sums via generalized Möbius functions and applications

Vichian Laohakosol,¹Pattira Ruengsinsub,¹and Nittiya Pabhapote²

Received22 May 2006

Revised20 Aug 2006

Accepted05 Sept 2006

Published20 Nov 2006

Abstract

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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Copyright © 2006 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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