ON A CONJECTURE OF ERDOS ABOUT ADDITIVE FUNCTIONS

For a real-valued additive function f : N -> R and for each n is an element of N we define a distribution function F-n(x) := 1/n #{m <= n : f(m) <= x}. In this paper we prove a conjecture of Erdos, which asserts that in order for the sequence Fn to he (weakly) convergent, it is sufficient that there exist two numbers a < b such that lim(n ->infinity) (F-n(b)-F-n(a)) exists and is positive. The proof is based upon the use of the Stone-Cech compactification beta N of N to mimic the behaviour of an additive function as a sum of independent random variables.