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 ^19+εX≤ 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 .