On the Integers Having Exactly K Prime Factors
Paul L. Erdos · Annals of Mathematics · 1948
(4) 7rk(n) = (1 + o(1)).(n/log n).xI/(k 1)!. I believe that a formula like (4) holds for k < cx, but the proof presents difficulties which I have not yet been able to overcome. In the proof of (4) we will have to use the prime number theorem. It will be relatively easy to prove (3) (Lemma 3), and the prime number theorem will not be required for the proof of (3). Throughout this paper k and 7 will denote integers in I. a k), i = 2 * denotes the integers _ n having exactly k prime factors (multiple factors are counted only once). E (1/ak) will indicate that the summation is extended over t. LEMMA 1.