On the prime twins problem (Q909707)

The quantitative version of the prime twin conjecture states that \[ | \{p\leq x: p+2=p'\}| \sim Cx/\log^ 2x,\quad\text{ as } x\to \infty. \] Here \(p, p'\) denote prime numbers and \(C=2\prod_{p>2}(1-1/(p-2)^ 2)\). Let \(\pi_{1,2}(x)\) count those primes \(p\leq x\) such that \(p+2\) has at most two prime divisors. In a famous paper published in 1973 \textit{J. Chen} [Sci. Sin. 16, 157--176 (1973; Zbl 0319.10056)] proved that \(\pi_{1,2}(x)>\alpha Cx/\log^ 2x\) for large \(x\), where the constant \(\alpha\) was given the numerical value of 0.335, which he later improved to 0.3772 and also mentioned that complicated calculations might give 0.405. More recently, on the basis of the work by \textit{E. Fouvry} and \textit{H. Iwaniec} [Acta Arith. 42, 197--218 (1983; Zbl 0517.10045)] and \textit{E. Bombieri}, \textit{J. B. Friedlander} and \textit{H. Iwaniec} [Acta Math. 156, 203--251 (1986; Zbl 0588.10042)] \textit{E. Fouvry} and \textit{F. Grupp} [J. Reine Angew. Math. 370, 101--126 (1986; Zbl 0588.10051)] further improved this to 0.71. To be consistent with the prime twin conjecture, one would expect the value of \(\alpha\) to be at least 1. Here the author shows that \(\alpha =1.015\) is admissible. The proof by sieve methods also involves the notion of well-factorable functions introduced by Fouvry and Grupp.