On Selberg’s approximation to the twin prime problem
Volume 179 / 2017
Acta Arithmetica 179 (2017), 335-361
MSC: Primary 11N36, 11N37; Secondary 11N99.
DOI: 10.4064/aa8558-3-2017
Published online: 1 August 2017
Abstract
In his classical approximation to the twin prime problem, Selberg proved that for infinitely many $n$, $2^{\varOmega (n)}+2^{\varOmega (n+2)} \leq \lambda $ with $\lambda =14$, where $\varOmega (n)$ is the number of prime factors of $n$ counted with multiplicity. This enabled him to conclude that for infinitely many $n$, $n(n+2)$ has at most five prime factors, with one factor having two and the other having at most three prime factors. The aim of this paper is to revisit Selberg’s approach and improve the value of $\lambda $ by using two-dimensional sieve weights suggested by Selberg. We bring down the value of $\lambda $ to about $12.6$.
