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Totally Multiplicative Functions in Regular Convolution Rings

Published online by Cambridge University Press:  20 November 2018

K. L. Yocom
K. L. Yocom*
Affiliation:
University of Wyoming, Laramie, Wyoming
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McCarthy [4] generalized a necessary and sufficient condition for an arithmetic function to be totally multiplicative to the incidence algebra on a partially ordered set. Several equivalent conditions for an arithmetic function to be totally multiplicative are known [1], [2]. In this paper we generalize several of these (and some apparently new ones) to the regular convolution rings of Narkiewicz [5]. We also investigate the prime factorization of arithmetic functions in a certain subring of some of these regular convolution rings.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 16 , Issue 1 , March 1973 , pp. 119 - 128
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Apostol, T.M., Some properties of completely multiplicative functions, Amer. Math. Monthly 78 (1971), 266–271.Google Scholar
2. Carlitz, L., Problem E2268, Amer. Math. Monthly 77 (1970), p. 1107.Google Scholar
3. Cashwell, E.D., and Everett, C.J., The ring of number theoretic functions, Pacific J. Math. 9 (1956), 975–985.Google Scholar
4. McCarthy, P.J., Arithmetical functions and distributivity, Canad. Math. Bull. 13 (1970), 491–496.Google Scholar
5. Narkiewicz, W., On a class of arithmetical convolutions, Colloq. Math. 10 (1963), 81–94.Google Scholar
6. Scheid, H., Über ordnungstheoretische functionen, J. Reine Angew. Math. 238 (1969), 1–13.Google Scholar
7. Scheid, H., Functionen über lokal endlichen halbordnungen, I, Monatsh. Math. 74 (1970), 336–347.Google Scholar
8. Scheid, H., Einige ringe zahlentheoretisher functionen, J. Reine Angew. Math. 237 (1968), 1–11.Google Scholar
9. Smith, David, Incidence functions as generalized arithmetic functions, I, Duke Math. J. 36 (1967), 617–637.Google Scholar