Correlations of almost primes. (9th March 2023) - EVANS, NATALIE
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 $n=p_1p_2 ≤ X$ such that $n+h$ is a product of exactly two primes which holds for almost all $|h|≤ H$ with $\log^{19+ε}X≤ H≤ X^{1-ε}$, 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 $|h| ≤ H$ with $X^{1/6+ε}≤ H≤ X^{1-ε}$ .