On the number of prime factors of integers III

In our papers I, II, and III,1) we were concerned with the number of dis tinct prime factors of integers. In this paper, we shall deal with the number of prime factors of integers, multiple factors being counted multiply, and the number of divisors of integers, or more generally, the number of representations of integers as a product of a fixed number of integers. The object of this paper is to prove the following theorems: THEOREM A1. Let k be a positive integer. Let {f1(ƒÌ),•c, fl(ƒÌ)} be a set of polynomials in ƒÌ, and {_??_1,•c,_??_k} a family of sets of rational prime numbers. Suppose that the following conditions are satisfied: (C1) For each i=1,•c,k, the polynomial fi(ƒÌ) is of positive degree with integral coefficients and with positive leading coefficient; (C2) For each pair of integers i, j with 1•...i