Hostname: page-component-7dd5485656-gs9qr Total loading time: 0 Render date: 2025-10-30T23:49:17.865Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Davison Convolution of Arithmetical Functions

Published online by Cambridge University Press:  20 November 2018

Pentti Haukkanen
Pentti Haukkanen*
Affiliation:
Department of Mathematical Sciences University of Tampere Tampere, Finland
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The Davison convolution of arithmetical functions f and g is defined by where K is a complex-valued function on the set of all ordered pairs (n, d) such that n is a positive integer and d is a positive divisor of n. In this paper we shall consider the arithmetical equations f(r) = g, f(r) = fg, f o g = h in f and the congruence (f o g)(n) = 0 (mod n), where f(r) is the iterate of f with respect to the Davison convolution.

Keywords

10A20

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 32 , Issue 4 , 01 December 1989 , pp. 467 - 473
Copyright
Copyright © Canadian Mathematical Society 1989

References

1. Carlitz, L. and M. V. Subbarao, Transformation of arithmetic functions, Duke Math J. 40 (1973), 949–958.Google Scholar
2. Cohen, E., Arithmetical functions associated with the unitary divisors of an integer, Math. Z. 74 (1960), 66–80.Google Scholar
3. Davison, T. M. K., On arithmetic convolutions, Canad. Math. Bull. 9 (1966), 287–296.Google Scholar
4. Ferrero, M., On generalized convolution rings of arithmetic functions, Tsukuba J. Math. 4 (1980), 161–176.Google Scholar
5. Fotino, LP., Generalized convolution ring of arithmetic functions, Pacific J. Math. 61 (1975), 103– 116.Google Scholar
6. Fredman, M. L., Arithmetical convolution products, Duke Math J. 37 (1970), 231–242.Google Scholar
7. Gessley, M. D., A generalized arithmetic convolution, Amer. Math. Monthly 74 (1967), 1216–1217.Google Scholar
8. Gioia, A. A., The K-product of arithmetic functions, Canad. J. Math. 17 (1965), 970–976.Google Scholar
9. Gioia, A. A. and M. Subbarao, V., Generalized Dirichlet products of arithmetic functions (Abstract), Notices Amer. Math. Soc. 9 (1962), 305.Google Scholar
10. Hanumanthachari, J., On an arithmetic convolution, Canad. Math. Bull. 20 (1977), 301–305.Google Scholar
11. Haukkanen, P., Arithmetical equations involving semi-multiplicative functions and the Dirichlet product (To appear in Rend. Math.).Google Scholar
12. Lahiri, D. B., Hypo-multiplicative number-theoretic functions, Aequationes Math. 9 (1973), 184–192.Google Scholar
13. McCarthy, P. J., Note on some arithmetic sums, Boll. Un. Mat. Ital. 21 (1966), 239–242.Google Scholar
14. McCarthy, P. J., Introduction to arithmetical functions, Springer, 1986.Google Scholar
15. Narkiewicz, W., On a class of arithmetical convolutions, Colloq. Math. 10 (1963), 81–94.Google Scholar
16. Rearick, D., Semi-multiplicative functions, Duke Math. J. 33 (1966), 49–53.Google Scholar
17. Subbarao, M. V., A congruence for a class of arithmetic functions, Canad. Math. Bull. 9 (1966), 571–574.Google Scholar
18. Subbarao, M. V., A class of arithmetical equations, Nieuw Arch. Wisk. 15 (1967), 211–217.Google Scholar
19. Yocom, K. L., Totally multiplicative functions in regular convolution rings, Canad. Math. Bull. 16 (1973), 119–129.Google Scholar