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The identical equation in $\psi$-products

Authors: V. Sitaramaiah and M. V. Subbarao
Journal: Proc. Amer. Math. Soc. 124 (1996), 361-369
MSC (1991): Primary 11A25
MathSciNet review: 1301528
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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.

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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

Keywords: Multiplicative arithmetic functions, Lehmer product, identical equation
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
Article copyright: © Copyright 1996 American Mathematical Society