Let f be a complex valued
additive number theoretic function (i.e., f(mn) = f(m) + f(n) if m and n
are relatively prime). This paper shows that ∑D2(pα)p−α= O(D2(n)) or
∑|f(pα)|p−α= O(D(n)) (where the summations are over those pα≦ n, pα being a
prime raised to a power) is sufficient to guarantee that the following analogue of
Kolmogorov’s inequality holds:
where, if pα∥m denotes the fact that pα divides m but pα+1 does not (i.e., pα exactly
divides m), then