Almost primes in almost all short intervals II

TL;DR

This paper proves that almost all short intervals of the form (x, x+ (log x)^{2.1}] contain numbers that are products of exactly two primes, improving previous bounds through a new type II estimate and Heath-Brown's mean value theorem.

Contribution

The paper introduces a new type II estimate and applies Heath-Brown's mean value theorem to improve bounds on almost primes in short intervals.

Findings

Almost all intervals (x, x+ (log x)^{2.1}] contain semiprimes.

Improved the previous bound of 3.51 to 2.1.

Developed a new type II estimate for this problem.

Abstract

We show that, for almost all $x$, the interval $(x, x+(\log x)^{2.1}]$ contains products of exactly two primes. This improves on a work of the second author that had $3.51$ in place of $2.1$. To obtain this improvement, we prove a new type II estimate. One of the new innovations is to use Heath-Brown's mean value theorem for sparse Dirichlet polynomials.