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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| Format: | Article |
| Language: | English |
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Wiley
2006-01-01
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| 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. |
| format | Article |
| id | doaj-art-b15d868e37c8460a83e04044ce6e17bd |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2006-01-01 |
| publisher | Wiley |
| record_format | Article |
| 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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