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Proceedings of the American Mathematical Society

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ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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The identical equation in $\psi$-products
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by V. Sitaramaiah and M. V. Subbarao PDF
Proc. Amer. Math. Soc. 124 (1996), 361-369 Request permission

Abstract:

In Bull. Amer. Math. Soc. 36 (1930), 762–772, R. Vaidyanatha- swamy established a remarkable identity valid for any multiplicative arithmetic function and involving Dirichlet convolution. D. H. Lehmer (Trans. Amer. Math. Soc. 33 (1931), 945–952) introduced a very general class of arithmetical convolutions, called $\psi$-products, which include the well-known Dirichlet products, Eckford Cohen’s unitary convolutions, and in fact Narkiewicz’s so-called regular $A$-convolutions. In this paper, we establish an identical equation valid for multiplicative arithmetic functions and Lehmer’s $\psi$-convolutions which yields, as special cases, all known identical equations valid for the Dirichlet and unitary convolutions, besides establishing identical equations for several new convolutions.
References
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Additional Information
  • V. Sitaramaiah
  • Affiliation: Department of Mathematics, Pondicherry Engineering College, Pillaichavady, Pondicherry 605104, India
  • M. V. Subbarao
  • Affiliation: Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
  • Email: m.v.subbarao@ualberta.ca
  • Received by editor(s): April 4, 1994
  • Received by editor(s) in revised form: August 20, 1994
  • Additional Notes: The second author was partly supported by an NSERC grant
  • Communicated by: William W. Adams
  • © Copyright 1996 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 124 (1996), 361-369
  • MSC (1991): Primary 11A25
  • DOI: https://doi.org/10.1090/S0002-9939-96-03104-8
  • MathSciNet review: 1301528