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Published online by Cambridge University Press: 24 October 2008
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.