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Ramanujan sums via generalized Möbius functions and applications

Author

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  • Vichian Laohakosol
  • Pattira Ruengsinsub
  • Nittiya Pabhapote

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.

Suggested Citation

  • Vichian Laohakosol & Pattira Ruengsinsub & Nittiya Pabhapote, 2006. "Ramanujan sums via generalized Möbius functions and applications," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2006, pages 1-34, November.
    • Handle: RePEc:hin:jijmms:060528
    DOI: 10.1155/IJMMS/2006/60528
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