Correlations of Almost Primes

TL;DR

This paper proves average-case analogues of the Hardy-Littlewood twin prime conjecture for almost primes, establishing asymptotic formulas for certain prime factorization patterns within specified ranges.

Contribution

It introduces new asymptotic formulas for correlations involving almost primes and primes, extending prime conjecture analogues to broader contexts.

Findings

Asymptotic formula for integers with two prime factors within range

Correlation results for primes and almost primes for large shifts

Results hold for almost all shifts within specified bounds

Abstract

We prove that analogues of the Hardy-Littlewood generalised twin prime conjecture for almost primes hold on average. Our main theorem establishes an asymptotic formula for the number of integers $n=p_1p_2 \leq X$ such that $n+h$ is a product of exactly two primes which holds for almost all $|h|\leq H$ with $\log^{19+\varepsilon}X\leq H\leq X^{1-\varepsilon}$, under a restriction on the size of one of the prime factors of $n$ and $n+h$. Additionally, we consider correlations $n,n+h$ where $n$ is a prime and $n+h$ has exactly two prime factors, establishing an asymptotic formula which holds for almost all $|h| \leq H$ with $X^{1/6+\varepsilon}\leq H\leq X^{1-\varepsilon}$.