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Relations between arithmetical functions

Published online by Cambridge University Press:  24 October 2008

C. J. A. Evelyn
C. J. A. Evelyn
Affiliation:
33 Grosvenor Square, London, W. 1
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Extract

In a recent note(1) I proved that if μ(n) denotes the usual Möbius function, N denotes a fixed positive integer and if

then

where T runs through all natural numbers ≤ x which are not divisible by an Nth power. In the present paper I shall establish some further relations of this character and, in particular, I shall prove that if

where

then

Thus, in some respects, L(x) appears more regular than M(x), the sum over L(x/T) being multiplicative, whereas M(x1/N) is not.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 63 , Issue 4 , October 1967 , pp. 1027 - 1029
Copyright
Copyright © Cambridge Philosophical Society 1967

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References

REFERENCE

(1)Evelyn, C. J. A.A relationship for the Möbius function. Quart. J. Math. Oxford, Ser. 17 (1960), 281.CrossRefGoogle Scholar