On the Distribution of Values of Certain Divisor Functions

Let {Ed} be a sequence of nonnegative numbers and f(n) = E Ed , the suns being over divisors d of n. We say that f has the distribution function F if for all c > 0, the number of integers n c x for which f (n) > c is asymptotic to xF(c), and we investigate when F exists and when it is continuous. Let {E d } be a sequence of nonnegative numbers and Is it true that for all c > 0, f (n) _ y Ed. din Y, 1-xF(c) n-x f(n)>c for some function F(c) depending only on the value of c? If so, it is plain that 0 < F(c) < 1 ; moreover, F is nonincreasing. If c. is large enough, say Ed = 1 for all d so that f (n) =-r(n), then F(c) = I identically. Therefore, it is interesting to ask under what circumstances F exists and Lt F(c) = 0. In this case, we say that f has the distribution function F. We prove the following : 52