# On a class of $\psi $-convolutions characterized by the identical equation

Jean-Louis Nicolas; Varanasi Sitaramaiah

Journal de théorie des nombres de Bordeaux (2002)

- Volume: 14, Issue: 2, page 561-583
- ISSN: 1246-7405

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topNicolas, Jean-Louis, and Sitaramaiah, Varanasi. "On a class of $\psi $-convolutions characterized by the identical equation." Journal de théorie des nombres de Bordeaux 14.2 (2002): 561-583. <http://eudml.org/doc/248898>.

@article{Nicolas2002,

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

author = {Nicolas, Jean-Louis, Sitaramaiah, Varanasi},

journal = {Journal de théorie des nombres de Bordeaux},

keywords = {identical equation; multiplicative functions; Dirichlet convolution; Lehmer-Narkiewicz -convolution; regular convolution},

language = {eng},

number = {2},

pages = {561-583},

publisher = {Université Bordeaux I},

title = {On a class of $\psi $-convolutions characterized by the identical equation},

url = {http://eudml.org/doc/248898},

volume = {14},

year = {2002},

}

TY - JOUR

AU - Nicolas, Jean-Louis

AU - Sitaramaiah, Varanasi

TI - On a class of $\psi $-convolutions characterized by the identical equation

JO - Journal de théorie des nombres de Bordeaux

PY - 2002

PB - Université Bordeaux I

VL - 14

IS - 2

SP - 561

EP - 583

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

LA - eng

KW - identical equation; multiplicative functions; Dirichlet convolution; Lehmer-Narkiewicz -convolution; regular convolution

UR - http://eudml.org/doc/248898

ER -

## References

top- [1] E. Cohen, Arithmetical functions associated with the unitary divisors of an integer. Math. Z.74 (1960), 66-80. Zbl0094.02601MR112861
- [2] D.H. Lehmer, Arithmetic of double series. Trans. Amer. Math. Soc.33 (1931), 945-957. Zbl0003.10201MR1501625JFM57.0177.04
- [3] W. Narkiewicz, On a class of arithmetical convolutions. Colloq. Math.10 (1963), 81-94. Zbl0114.26502MR159778
- [4] V. Sitaramaiah, On the ψ-product of D. H. Lehmer. Indian J. Pure and Appl. Math.16 (1985), 994-1008. Zbl0603.10003
- [5] V. Sitaramaiah, On the existence of unity in Lehmer's ψ-product ring. Indian J. Pure and Appl. Math.20 (1989), 1184-1190. Zbl0698.10004
- [6] V. Sitaramaiah, M.V. Subbarao, On a class of ψ-products preserving multiplicativity. Indian J. Pure and Appl. Math.22 (1991), 819-832. Zbl0751.11006
- [7] V. Sitaramaiah, M.V. Subbarao, The identical equation in ψ-products. Proc. Amer. Math. Soc. 124 (1996), 361-369. Zbl0847.11003
- [8] V. Sitaramaiah, M.V. Subbarao, On regular ψ-convolutions. J. Indian Math. Soc.64 (1997), 131-150. Zbl1074.11500
- [9] R. Vaidyanathaswamy, The identical equation of the multiplicative functions. Bull. Amer. Math. Soc.36 (1930), 762-772. Zbl56.0873.03JFM56.0873.03
- [10] R. Vaidyanathaswamy, The theory of multiplicative arithmetic functions. Trans. Amer. Math. Soc.33 (1931), 579-662. (=[11], 326-414.) Zbl0002.12402MR1501607JFM57.0177.03
- [11] R. Vaidyanathaswamy, The collected papers of Prof. R. Vaidyanathaswamy. Madras University, 1957. MR124996

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