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A Property of Generalized Ramanujan's Sums Concerning Generalized Completely Multiplicative Functions
Publications de l'Institut Mathématique, _N_S_46 (1989) no. 60, p. 43

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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 Ivić and Redmond.
Classification : 11A25
Keywords: generalized completely multiplicative functions, regular convolutions, generalized Ramanujan's sums, Möbius function, Dirichlet series. 
@article{PIM_1989_N_S_46_60_a6,
     author = {Pentti Haukkanen},
     title = {A {Property} of {Generalized} {Ramanujan's} {Sums} {Concerning} {Generalized} {Completely} {Multiplicative} {Functions}},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {43 },
     publisher = {mathdoc},
     volume = {_N_S_46},
     number = {60},
     year = {1989},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PIM_1989_N_S_46_60_a6/}
}
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Pentti Haukkanen. A Property of Generalized Ramanujan's Sums Concerning Generalized Completely Multiplicative Functions. Publications de l'Institut Mathématique, _N_S_46 (1989) no. 60, p. 43 . http://geodesic.mathdoc.fr/item/PIM_1989_N_S_46_60_a6/