no. 2
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

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.

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.

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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. https://www.numdam.org/item/JTNB_2002__14_2_561_0/

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