Correlations of almost primes

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 = p₁p₂ ≤ X such that n + h is a product of exactly two primes which holds for almost all |h| ≤ H with (log X)^19+ε ≤ 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.

Date of Award	1 Jan 2023
Original language	English
Awarding Institution	King's College London
Supervisor	Stephen Lester (Supervisor), Igor Wigman (Supervisor) & Abhishek Saha (Supervisor)