On a class of ψ-convolutions characterized by the identical equation
Journal de Théorie des Nombres de Bordeaux, Tome 14 (2002) no. 2, pp. 561-583.

Dans le cadre de la convolution de Dirichlet des fonctions arithmétiques, R. Vaidyanathaswamy a obtenu en 1931 une formule de calcul de f(mn) valable pour toute fonction multiplicative f et tout couple d’entiers positifs m et n. Dans [7], cette formule a été généralisée aux ψ-convolutions appelées convolutions de Lehmer-Narkiewicz, qui, entre autres, conservent la multiplicativité. Dans cet article, nous démontrons la réciproque.

The identical equation for multiplicative functions established by R. Vaidyanathaswamy in the case of Dirichlet convolution in 1931 has been generalized to multiplicativity preserving ψ-convolutions satisfying certain conditions (cf. [7]) which can be called as Lehmer-Narkiewicz convolutions for some reasons. In this paper we prove the converse.

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Nicolas, Jean-Louis; Sitaramaiah, Varanasi. On a class of $\psi $-convolutions characterized by the identical equation. Journal de Théorie des Nombres de Bordeaux, Tome 14 (2002) no. 2, pp. 561-583. http://www.numdam.org/item/JTNB_2002__14_2_561_0/

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