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On the structure of certain basic sequences associated with an arithmetic function

Published online by Cambridge University Press:  20 January 2009

Donald L. Goldsmith
Donald L. Goldsmith
Affiliation:
Western Michigan University, Kalamazoo, Michigan 49001
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We have previously studied in some detail the multiplicative properties of a given arithmetic function f with respect to a fixed basic sequence (see, for example, (1), (2)). We investigate here the structure of M(f), the collection of all basic sequences such that f is multiplicative with respect to , and in particular we focus our attention on the maximal members of M(f). Our principal result will be a proof that each maximal member of M(f) contains the same set of type II primitive pairs. Moreover, we will give a simple criterion for determining, in terms of the behaviour of f, whether or not a particular primitive pair (p, p) is in any (and therefore every) maximal member of M(f).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 17 , Issue 4 , December 1971 , pp. 305 - 310
Copyright
Copyright © Edinburgh Mathematical Society 1971

References

REFERENCES

(1) Goldsmith, D. L., On the multiplicative properties of arithmetic functions, Pacific J. Math. 27 (1968), 283–304.CrossRefGoogle Scholar
(2) Goldsmith, D. L., A note on sequences of almost-multiplicative arithmetic functions, Rendiconti di Matematica 3 (6) (1970), 167–170.Google Scholar