On the Standard Deviations of Additive Arithmetical Functions

Abstract

(4) f (n1n2) f(n1) + f (n2) whenever (n,, n2 ) =1; (f (1) 0). In fact, the series (1) is convergent for every choice of the double sequence { {al} }, since the series has, for every fixed n, at most a finite number of non-vanishing terms. It is also clear that an additive function f and either of the two sequences of additive functions {ff}, {fk} of n determine each other uniquely. The additive functions tQ be considered will be assumed to be real-valued. For a given y f(n), define y+ f+(n) by placing (5) y+ y or y 1 according, as !y I < 1 or I y ?1.