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The identical equation in  -products
Author(s):
V.
Sitaramaiah;
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.
References:
- 1
- E. Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math. Z. 74 (1960), 66--80. MR 22:3707
- 2
- A. A. Gioia, On an identity for multiplicative functions, Amer. Math. Monthly 69 (1962), 988--991.
- 3
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- 4
- K. Krishna, A proof of an identity for multiplicative functions, Canad. Math. Bull 22 (1979), 299--304. MR 81e:10004
- 5
- D. H. Lehmer, Arithmetic of double series, Trans. Amer. Math. Soc. 33 (1931), 945--952.
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- 7
- W. Narkiewicz, On a class of arithmetical convolutions, Colloq. Math. 10 (1963), 81--94. MR 28:2994
- 8
- V. Sitaramaiah and M. V. Subbarao, On a class of
 -products preserving multiplicativity, Indian J. Pure Appl. Math. 22 (1991), 819--832. MR 92h:11006 - 9
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- M. Sugunamma, Contributions to the theory of general arithmetic functions, Ph.D. Thesis, S. V. University, Tirapathi, India, 1965.
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- R. Vaidyanathaswamy, The identical equation of the multiplicative functions, Bull. Amer. Math. Soc. 36 (1930), 762--772.
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- ------, 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:
10.1090/S0002-9939-96-03104-8
PII:
S 0002-9939(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
Copyright of article:
Copyright
1996,
American Mathematical Society
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