Sieving by very thin sets of primes, and Pratt trees with missing primes

TL;DR

This paper introduces a new sieve method to analyze prime sets with a recursive property, showing their counting function grows slower than any linear function, leveraging results related to Artin's primitive root conjecture.

Contribution

It develops a novel sieve technique for prime sets defined by recursive prime factor conditions, providing bounds on their counting functions.

Findings

Counting function of P is O(x^{1-c}) for some c>0

Method relies on results connected with Artin's primitive root conjecture

Applicable when P does not contain all primes

Abstract

Suppose P is a set of primes, such that for every p in P, every prime factor of p-1 is also in P. If P does not contain all primes, we apply a new sieve method to show that the counting function of P is O(x^{1-c}) for some c>0, where c depends only on the smallest prime not in P. Our proof makes use of results connected with Artin's primitive root conjecture.