Abstract
AbstractWe 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
$X^{1/6+\varepsilon}\leq H\leq X^{1-\varepsilon}$
.
Publisher
Cambridge University Press (CUP)
Reference30 articles.
1. [27] Polymath, D. H. J. . Variants of the Selberg sieve, and bounded intervals containing many primes. Res. Math. Sci. 1:Art. 12, 83 (2014).
2. [6] Friedlander, J. and Iwaniec, H. . Opera de Cribro . Amer. Math. Soc. Colloq. Publ. vol. 57. (American Mathematical Society, Providence, RI, 2010).
3. [3] Davenport, H. . Multiplicative Number Theory . Graduate Texts in Math. vol. 74. (Springer-Verlag, New York, third edition, 2000). Revised and with a preface by H. L. Montgomery.
4. An averaged form of Chowla’s conjecture;Matomäki;Algebra Number Theory
5. On twin almost primes;Bombieri;Acta Arith.