PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE (BEOGRAD) (N.S.) Vol. 46(60), pp. 4349 (1989) 

A property of generalized Ramanujan's sums concerning generalized completely multiplicative functionsPentti HaukkanenDepartment of Mathematical Sciences, University of Tampere SF33101 Tampere, FinlandAbstract: Let $A$ be a regular convolution in the sense of Narkiewicz. A necessary and sufficient condition for a multiplicative function to be $A$multiplicative (i.e. such that $f(n)=f(d)f(n/d)$ whenever $d\in A(n)$) is given in terms of generalized Ramanujan's sums. (With the Dirichlet convolution $A$multiplicative functions are completely multiplicative.) In addition, another necessary and sufficient condition for a multiplicative function to be completely multiplicative is given in terms of generalized Ramanujan's sums as well. As an application a representation theorem in terms of Dirichlet series is given. The results of this paper generalize respective results of Ivi\'c and Redmond. Keywords: generalized completely multiplicative functions, regular convolutions, generalized Ramanujan's sums, Möbius function, Dirichlet series. Classification (MSC2000): 11A25 Full text of the article:
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