The identical equation in -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. Vaidyanathaswamy 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 -products, which include the well-known Dirichlet products, Eckford Cohen's unitary convolutions, and in fact Narkiewicz's so-called regular -convolutions. In this paper, we establish an identical equation valid for multiplicative arithmetic functions and Lehmer's -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.

**1**Eckford Cohen,*Arithmetical functions associated with the unitary divisors of an integer*, Math. Z.**74**(1960), 66–80. MR**0112861****2**A. A. Gioia,*On an identity for multiplicative functions*, Amer. Math. Monthly**69**(1962), 988--991.**3**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.**4**K. Krishna,*A proof of an identity for multiplicative functions*, Canad. Math. Bull.**22**(1979), no. 3, 299–304. MR**555157**, 10.4153/CMB-1979-036-3**5**D. H. Lehmer,*Arithmetic of double series*, Trans. Amer. Math. Soc.**33**(1931), 945--952.**6**Paul J. McCarthy,*Introduction to arithmetical functions*, Universitext, Springer-Verlag, New York, 1986. MR**815514****7**W. Narkiewicz,*On a class of arithmetical convolutions*, Colloq. Math.**10**(1963), 81–94. MR**0159778****8**V. Sita Ramaiah and M. V. Subbarao,*On a class of 𝜓-products preserving multiplicativity*, Indian J. Pure Appl. Math.**22**(1991), no. 10, 819–832. MR**1132555****9**M. V. Subbarao and A. A. Gioia,*Identities for multiplicative functions*, Canad. Math. Bull.**10**(1967), 65–73. MR**0215803****10**M. Sugunamma,*Contributions to the theory of general arithmetic functions*, Ph.D. Thesis, S. V. University, Tirapathi, India, 1965.**11**R. Vaidyanathaswamy,*The identical equation of the multiplicative functions*, Bull. Amer. Math. Soc.**36**(1930), 762--772.**12**------,*The theory of multiplicative arithmetic functions*, Trans. Amer. Math. Soc.**33**(1931), 579--662.

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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

DOI:
https://doi.org/10.1090/S0002-9939-96-03104-8

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