On a class of ψ-convolutions characterized by the identical equation
Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 2, p. 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.

@article{JTNB_2002__14_2_561_0,
     author = {Nicolas, Jean-Louis and Sitaramaiah, Varanasi},
     title = {On a class of $\psi $-convolutions characterized by the identical equation},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {14},
     number = {2},
     year = {2002},
     pages = {561-583},
     zbl = {1071.11007},
     mrnumber = {2040694},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2002__14_2_561_0}
}
Nicolas, Jean-Louis; Sitaramaiah, Varanasi. On a class of $\psi $-convolutions characterized by the identical equation. Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 2, pp. 561-583. http://www.numdam.org/item/JTNB_2002__14_2_561_0/

[1] E. Cohen, Arithmetical functions associated with the unitary divisors of an integer. Math. Z. 74 (1960), 66-80. | MR 112861 | Zbl 0094.02601

[2] D.H. Lehmer, Arithmetic of double series. Trans. Amer. Math. Soc. 33 (1931), 945-957. | JFM 57.0177.04 | MR 1501625 | Zbl 0003.10201

[3] W. Narkiewicz, On a class of arithmetical convolutions. Colloq. Math. 10 (1963), 81-94. | MR 159778 | Zbl 0114.26502

[4] V. Sitaramaiah, On the ψ-product of D. H. Lehmer. Indian J. Pure and Appl. Math. 16 (1985), 994-1008. | Zbl 0603.10003

[5] V. Sitaramaiah, On the existence of unity in Lehmer's ψ-product ring. Indian J. Pure and Appl. Math. 20 (1989), 1184-1190. | Zbl 0698.10004

[6] V. Sitaramaiah, M.V. Subbarao, On a class of ψ-products preserving multiplicativity. Indian J. Pure and Appl. Math. 22 (1991), 819-832. | Zbl 0751.11006

[7] V. Sitaramaiah, M.V. Subbarao, The identical equation in ψ-products. Proc. Amer. Math. Soc. 124 (1996), 361-369. | Zbl 0847.11003

[8] V. Sitaramaiah, M.V. Subbarao, On regular ψ-convolutions. J. Indian Math. Soc. 64 (1997), 131-150. | Zbl 1074.11500

[9] R. Vaidyanathaswamy, The identical equation of the multiplicative functions. Bull. Amer. Math. Soc. 36 (1930), 762-772. | JFM 56.0873.03

[10] R. Vaidyanathaswamy, The theory of multiplicative arithmetic functions. Trans. Amer. Math. Soc. 33 (1931), 579-662. (=[11], 326-414.) | JFM 57.0177.03 | MR 1501607 | Zbl 0002.12402

[11] R. Vaidyanathaswamy, The collected papers of Prof. R. Vaidyanathaswamy. Madras University, 1957. | MR 124996