International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 60528, 34 pages
doi:10.1155/IJMMS/2006/60528
Ramanujan sums via generalized Möbius functions and applications
1Department of Mathematics, Kasetsart University, Bangkok 10900, Thailand
2Department of Mathematics, University of the Thai Chamber of Commerce, Bangkok 10400, Thailand
Received 22 May 2006; Revised 20 August 2006; Accepted 5 September 2006
Copyright © 2006 Vichian Laohakosol et al. 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.
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.