On Selberg's approximation to the twin prime problem

Abstract

In his Classical approximation to the Twin prime problem, Selberg proved that for $x$ sufficiently large, there is an $n \in (x,2x)$ such that $2^{\Omega(n)}+2^{\Omega(n+2)} \leq \lambda$ with $\lambda=14$, where $\Omega(n)$ is the number of prime factors of $n$ counted with multiplicity. This enabled him to show that for infinitely many $n$, $n(n+2)$ has atmost $5$ prime factors, with one having atmost $2$ and the other having atmost $3$ prime factors. By adopting Selberg's approach and using a refinement suggested by Selberg, we improve this value of $\lambda$ to about $\lambda=12.59$.