Hostname: page-component-848d4c4894-ttngx Total loading time: 0 Render date: 2024-06-08T17:33:45.638Z Has data issue: false hasContentIssue false
  • English
  • Français

On a subgroup of the group of multiplicative arithmetic functions

Published online by Cambridge University Press:  09 April 2009

T. B. Carroll  and
A. A. Gioia
T. B. Carroll
Affiliation:
Western Michigan UniversityKalamazoo Michigan 49001, U.S.A.
A. A. Gioia
Affiliation:
Western Michigan UniversityKalamazoo Michigan 49001, U.S.A.
Rights & Permissions [Opens in a new window]

Extract

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.

An arithmetic function f is said to be multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1, where (m, n) denotes as usual the greatest common divisor of m and n. Furthermore an arithmetic function is said to be linear (or completely multiplicative) if f(1) = 1 and f(mn) = f(m)f(n) for all positive integers m and n.The Dirichlet convolution of two arithmetic functions f and g is defined by for all n∈Z+. Recall that the set of all multiplicative functions, denoted by M, with this operation is an abelian group.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 20 , Issue 3 , November 1975 , pp. 348 - 358
Copyright
Copyright © Australian Mathematical Society 1975

References

Apostol, T. M. (1971), ‘Some properties of completely multiplicative arithmetic functions’, Amer. Math. Monthly 78, 266–271.CrossRefGoogle Scholar
Beumer, M. (1962), ‘The arithmetical function rτk(n)’, Amer. Math. Monthly 69, 777–781.CrossRefGoogle Scholar
Carroll, T. B. and Gioia, A. A. (1911), ‘On extended linear functions’, Notices Amer. Math. Soc. 18bf, Abstract #71T-A161,799.Google Scholar
Dickson, L. E. (1919), History of the Theory of Numbers, Volume I (Washington, Carnegie Institution, 1919).Google Scholar
Gioia, A. A. (1962). ‘On an identity for multiplicative functions’, Amer. Math. Monthly 79, 988–991.CrossRefGoogle Scholar
Goldsmith, D. L. (1969), ‘A generalization of some identities of Ramanujan’, Rend. Mat. (Serie vi) 2, 473–479.Google Scholar
McCarthy, P. J. (1960), ‘Busche-Ramanujan identities’, Amer. Math. Monthly 67, 966–970.CrossRefGoogle Scholar
Potter, M. (1968), Extensions of the Sigma and Tau Functions, (Unpublished Specialist's project, Western Michigan University, Kalamazoo, Michigan, 1968).Google Scholar
Ramanathan, K. G. (1943), ‘Multiplicative arithmetic functions’, Indian Math. Soc. 7, 111–116.Google Scholar
Vaidyanathaswamy, R. (1930), ‘The identical equations of the multiplicative function’, Bull. Amer. Math. Soc. 36, 762–772.CrossRefGoogle Scholar
Vaidyanathaswamy, R. (1931), ‘The theory of multiplicative arithmetical functions’, Trans. Amer. Math. Soc. 33, 579–662.CrossRefGoogle Scholar
Wall, C. R. and Hsu, T. S. (1972), ‘A characterization of totients’, Notices Amer. Math. Soc. 19, Abstract # 72T-A175, 567.Google Scholar