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
$n=p_1p_2 \leq X$
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 o… Show more
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