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_{1}p_{2} ≤
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 |
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Original language | English |
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Awarding Institution | |
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Supervisor | Stephen Lester (Supervisor), Igor Wigman (Supervisor) & Abhishek Saha (Supervisor) |
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Correlations of almost primes
Evans, N. (Author). 1 Jan 2023
Student thesis: Doctoral Thesis › Doctor of Philosophy