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On Selberg’s approximation to the twin prime problem

Volume 179 / 2017

R. Balasubramanian, Priyamvad Srivastav 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 $12.6$.

Authors

  • R. BalasubramanianInstitute of Mathematical Sciences
    Taramani, Chennai, India 600113
    and
    Homi Bhabha National Institute
    Training School Complex
    Anushakti Nagar, Mumbai, India 400094
    e-mail
  • Priyamvad SrivastavInstitute of Mathematical Sciences
    Taramani, Chennai, India 600113
    and
    Homi Bhabha National Institute
    Training School Complex
    Anushakti Nagar, Mumbai, India 400094
    e-mail

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