Rough numbers between consecutive primes

TL;DR

This paper proves that almost all gaps between consecutive primes contain a number with a large least prime factor, confirming a prediction of Erdős, and provides asymptotic estimates for the count of exceptional gaps under certain conjectures.

Contribution

It introduces a sieve-theoretic approach to analyze prime gaps, confirming Erdős's prediction and deriving asymptotic formulas under the Hardy-Littlewood conjecture.

Findings

Almost all prime gaps contain a number with a least prime factor at least the gap length.

The number of exceptional gaps up to X is at most O(X / log^2 X).

Under the Hardy-Littlewood conjecture, the count of exceptional gaps is asymptotically c X / log^2 X.

Abstract

Using a sieve-theoretic argument, we show that almost all gaps $(p_n, p_{n+1})$ between consecutive primes $p_n, p_{n+1}$ contain a natural number $m$ whose least prime factor $p(m)$ is at least the length $p_{n+1} - p_n$ of the gap, confirming a prediction of Erd\H{o}s. In fact the number $N(X)$ of exceptional gaps with $p_n \in [X,2X]$ is shown to be at most $O(X/\log^2 X)$. Assuming a form of the Hardy--Littlewood prime tuples conjecture, we establish a more precise asymptotic $N(X) \sim c X / \log^2 X$ for an explicit constant $c>0$, which we believe to be between $2.7$ and $2.8$. To obtain our results in their full strength we rely on the asymptotics for singular series developed by Montgomery and Soundararajan.