Reciprocals of Certain Large Additive Functions

1 . Introduction and statement of results Let (3(n) = Y_, In p and B (n) = y_p . j jn ap denote the sum of distinct prime divisors of n and the sum of all prime divisors of n respectively . Both (3(n) and B(n) are additive functions which are in a certain sense large (the average order of B(n) is Tr'n/(6log n), [1]) . For a fixed integer m the number of solutions of B(n) = m, is the number of partitions of m into primes, while the number of solutions of (3 (n) = m, μ 2(n) = 1 is the number of partitions of m into distinct primes . There is a certain analogy between the relation of (3(n) to B(n) and the relation of the well-known additive functions c) (n) = 1pjn1 and fl(n) _ Y,° ,,a . Asymptotic estimates of B(n) were investigated in [1], revealing the connection between B(n) and large prime factors of n. In this paper we turn our attention to sums involving reciprocals of (3(n) and B(n) . We shall prove the following theorems :