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
- Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math. Z. 74 (1960), 66–80. MR 112861, DOI 10.1007/BF01180473
- A. A. Gioia, On an identity for multiplicative functions, Amer. Math. Monthly 69 (1962), 988–991.
- P. Haukkanen, Classical arithmetical identities involving a generalization of Ramanujan’s sum, Ann. Acad. Sci. Fenn. Ser. A. I Math. Dissertationes 68 (1966), 1–69.
- K. Krishna, A proof of an identity for multiplicative functions, Canad. Math. Bull. 22 (1979), no. 3, 299–304. MR 555157, DOI 10.4153/CMB-1979-036-3
- D. H. Lehmer, Arithmetic of double series, Trans. Amer. Math. Soc. 33 (1931), 945–952.
- Paul J. McCarthy, Introduction to arithmetical functions, Universitext, Springer-Verlag, New York, 1986. MR 815514, DOI 10.1007/978-1-4613-8620-9
- W. Narkiewicz, On a class of arithmetical convolutions, Colloq. Math. 10 (1963), 81–94. MR 159778, DOI 10.4064/cm-10-1-81-94
- V. Sita Ramaiah and M. V. Subbarao, On a class of $\psi$-products preserving multiplicativity, Indian J. Pure Appl. Math. 22 (1991), no. 10, 819–832. MR 1132555
- M. V. Subbarao and A. A. Gioia, Identities for multiplicative functions, Canad. Math. Bull. 10 (1967), 65–73. MR 215803, DOI 10.4153/CMB-1967-007-x
- M. Sugunamma, Contributions to the theory of general arithmetic functions, Ph.D. Thesis, S. V. University, Tirapathi, India, 1965 [4].
- R. Vaidyanathaswamy, The identical equation of the multiplicative functions, Bull. Amer. Math. Soc. 36 (1930), 762–772.
- —, The theory of multiplicative arithmetic functions, Trans. Amer. Math. Soc. 33 (1931), 579–662.
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