On a class of ψ -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

Abstract

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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.

How to cite

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Nicolas, 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

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  1. [1] E. Cohen, Arithmetical functions associated with the unitary divisors of an integer. Math. Z.74 (1960), 66-80. Zbl0094.02601MR112861
  2. [2] D.H. Lehmer, Arithmetic of double series. Trans. Amer. Math. Soc.33 (1931), 945-957. Zbl0003.10201MR1501625JFM57.0177.04
  3. [3] W. Narkiewicz, On a class of arithmetical convolutions. Colloq. Math.10 (1963), 81-94. Zbl0114.26502MR159778
  4. [4] V. Sitaramaiah, On the ψ-product of D. H. Lehmer. Indian J. Pure and Appl. Math.16 (1985), 994-1008. Zbl0603.10003
  5. [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. [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. [7] V. Sitaramaiah, M.V. Subbarao, The identical equation in ψ-products. Proc. Amer. Math. Soc. 124 (1996), 361-369. Zbl0847.11003
  8. [8] V. Sitaramaiah, M.V. Subbarao, On regular ψ-convolutions. J. Indian Math. Soc.64 (1997), 131-150. Zbl1074.11500
  9. [9] R. Vaidyanathaswamy, The identical equation of the multiplicative functions. Bull. Amer. Math. Soc.36 (1930), 762-772. Zbl56.0873.03JFM56.0873.03
  10. [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. [11] R. Vaidyanathaswamy, The collected papers of Prof. R. Vaidyanathaswamy. Madras University, 1957. MR124996

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