Twins almost prime under a Elliott-Halberstam's conjecture

TL;DR

This paper demonstrates, under the Elliott-Halberstam conjecture, that there are infinitely many twin almost primes, which are primes p where p-2 is either prime or a product of two small primes, with an explicit asymptotic.

Contribution

It improves Bombieri's asymptotic sieve to localize variables and proves the infinitude of twin almost primes under a conjecture.

Findings

Proves infinitely many twin almost primes assuming Elliott-Halberstam conjecture.

Provides explicit asymptotic count of twin almost primes.

Extends sieve methods to localize variables in prime gaps.

Abstract

We improve Bombieri's asymptotic sieve to localise the variables. As a consequence, we prove, under a Elliott-Halberstam conjecture, that there exists an infinity of twins almost prime. Those are prime numbers $p$ such that for all $\varepsilon>0$, $p-2$ is either a prime number or can be written as $p_1 p_2$ where $p_1$ and $p_2$ are prime and $p_1<X^{\varepsilon}$, and we give the explicit asymptotic.