When the sieve works II

TL;DR

This paper classifies prime sets for which the count of integers with prime factors in the set closely matches its expected value, confirming a recent conjecture and revealing conditions under which the sieve effectively predicts integer distributions.

Contribution

It proves the main conjecture regarding the classification of prime sets with predictable integer counts, extending understanding of sieve methods in number theory.

Findings

Classifies prime sets with predictable integer counts.

Proves the main conjecture by Granville, Koukoulopoulos, and Matomäki.

Identifies conditions for the sieve's effectiveness in predicting integer distributions.

Abstract

For a set of primes $\mathcal{P}$, let $\Psi(x, \mathcal{P})$ be the number of positive integers $n \leq x$ all of whose prime factors lie in $\mathcal{P}$. In this paper we classify the sets of primes $\mathcal{P}$ such that $\Psi(x, \mathcal{P})$ is within a constant factor of its expected value. This task was recently initiated by Granville, Koukoulopoulos and Matom\"aki and their main conjecture is proved in this paper. In particular our main theorem implies that, if not too many large primes are sieved out in the sense that \[ \sum_{\substack{p \in \mathcal{P} \\ x^{1/v} < p \leq x^{1/u}}} \frac{1}{p} \geq \frac{1 + \varepsilon}{u}, \] for some $\varepsilon > 0$ and $v \geq u \geq 1$, then \[ \Psi(x, \mathcal{P}) \gg_{\varepsilon, v} x \prod_{\substack{p \leq x\\ p \notin\mathcal{P}}} \left(1 - \frac{1}{p}\right). \]