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

TL;DR

This paper extends Hardy-Ramanujan inequalities to sifted sets, providing bounds on the count of shifted primes with a fixed number of prime factors, uniformly across all positive integers k.

Contribution

It introduces a Hardy-Ramanujan type inequality for sifted sets, specifically bounding shifted primes with a given number of prime factors uniformly for all k.

Findings

Established a bound for shifted primes p+a with k prime factors

Extended Hardy-Ramanujan inequality to sifted sets

Provided uniform bounds for all positive integers k

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.