{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T13:34:49Z","timestamp":1753882489543,"version":"3.41.2"},"reference-count":24,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2006,11,20]],"date-time":"2006-11-20T00:00:00Z","timestamp":1163980800000},"content-version":"vor","delay-in-days":323,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"name":"University of the Thai Chamber of Commerce"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2006,1]]},"abstract":"<jats:p>A generalized Ramanujan sum (GRS) is defined by replacing the usual M\u00f6bius function in the classical Ramanujan sum with the Souriau\u2010Hsu\u2010M\u00f6bius 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\u2032s 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.<\/jats:p>","DOI":"10.1155\/ijmms\/2006\/60528","type":"journal-article","created":{"date-parts":[[2006,12,4]],"date-time":"2006-12-04T07:12:10Z","timestamp":1165216330000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Ramanujan sums via generalized M\u00f6bius functions and applications"],"prefix":"10.1155","volume":"2006","author":[{"given":"Vichian","family":"Laohakosol","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pattira","family":"Ruengsinsub","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nittiya","family":"Pabhapote","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2006,11,20]]},"reference":[{"key":"e_1_2_1_1_2","doi-asserted-by":"publisher","DOI":"10.1215\/S0012\u20107094\u201053\u201002021\u20103"},{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-5579-4"},{"key":"e_1_2_1_3_2","first-page":"72","article-title":"On a certain kind of generalized number-theoretical M\u00f6bius function","volume":"25","author":"Brown T. 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On certain generalized Euler-type totients and M\u00f6bius-type functios Dalian University of Technology China preprint."}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2006\/060528.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/IJMMS\/2006\/60528","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,6]],"date-time":"2024-06-06T21:12:49Z","timestamp":1717708369000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/IJMMS\/2006\/60528"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,1]]},"references-count":24,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2006,1]]}},"alternative-id":["10.1155\/IJMMS\/2006\/60528"],"URL":"https:\/\/doi.org\/10.1155\/ijmms\/2006\/60528","archive":["Portico"],"relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"type":"print","value":"0161-1712"},{"type":"electronic","value":"1687-0425"}],"subject":[],"published":{"date-parts":[[2006,1]]},"assertion":[{"value":"2006-11-20","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"060528"}}