On Selberg’s approximation to the twin prime problem

R. Balasubramanian ¹   ; Priyamvad Srivastav ¹

¹ Institute of Mathematical Sciences Taramani, Chennai, India 600113 and Homi Bhabha National Institute Training School Complex Anushakti Nagar, Mumbai, India 400094

Acta Arithmetica, Tome 179 (2017) no. 4, pp. 335-361

Résumé

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 $12.6$.

Détail

DOI : 10.4064/aa8558-3-2017

Keywords: his classical approximation twin prime problem selberg proved infinitely many varomega varomega leq lambda lambda where varomega number prime factors counted multiplicity enabled him conclude infinitely many has five prime factors factor having other having three prime factors paper revisit selberg approach improve value lambda using two dimensional sieve weights suggested selberg bring down value lambda about nbsp