Almost Primes in Almost All Short Intervals

TL;DR

This paper proves that almost all short intervals of certain lengths contain numbers with exactly two or three prime factors, improving previous bounds and approaching the minimal possible interval length as the number of prime factors increases.

Contribution

The authors establish new bounds for the presence of almost primes in short intervals, improving previous results for $E_2$ numbers and extending to general $E_k$ numbers with minimal interval lengths.

Findings

Almost all intervals of length $ ext{log}^{1+ ext{epsilon}} x$ contain $E_3$ numbers.

Almost all intervals of length $ ext{log}^{3.51} x$ contain $E_2$ numbers.

Results are optimal up to epsilon and improve previous bounds for $E_2$ numbers.

Abstract

Let $E_k$ be the set of positive integers having exactly $k$ prime factors. We show that almost all intervals $[x,x+\log^{1+\varepsilon} x]$ contain $E_3$ numbers, and almost all intervals $[x,x+\log^{3.51} x]$ contain $E_2$ numbers. By this we mean that there are only $o(X)$ integers $1\leq x\leq X$ for which the mentioned intervals do not contain such numbers. The result for $E_3$ numbers is optimal up to the $\varepsilon$ in the exponent. The theorem on $E_2$ numbers improves a result of Harman, which had the exponent $7+\varepsilon$ in place of $3.51$. We will also consider general $E_k$ numbers, and find them on intervals whose lengths approach $\log x$ as $k\to \infty$.