International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Issue 18, Pages 60528, 34 p.
doi:10.1155/IJMMS/2006/60528
Ramanujan sums via generalized Möbius functions and applications
Vichian Laohakosol1
, Pattira Ruengsinsub1
and Nittiya Pabhapote3
1Department of Mathematics, Kasetsart University, Bangkok 10900, Thailand
3Department of Mathematics, University of the Thai Chamber of Commerce, Bangkok 10400, Thailand
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.