A Hardy-Ramanujan type inequality for shifted primes and sifted sets

Abstract

We establish an analog of the Hardy-Ramanujan inequality for counting members of sifted sets with a given number of distinct prime factors. In particular, we establish a bound for the number of shifted primes p+a below x with k distinct prime factors, uniformly for all positive integers k.

arXiv commentv3. Added hypothesis x>2|a| to Cor. 1 and 3; changed hypothesis in Theorem 1 so that s is ANY pos. integer dividing all members of S (this is needed for Cor. 1 and Cor. 3). Added remarks explaining that gcd{p+a:p+a in Q(E)} is not always 1 or 2