An explicit version of Bombieri's log-free density estimate and Sárközy's theorem for shifted primes

Jesse Thorner, Asif Zaman

Research output: Contribution to journalArticlepeer-review

Abstract

We make explicit Bombieri's refinement of Gallagher's log-free "large sieve density estimate near σ = 1 {\sigma=1} "for Dirichlet L-functions. We use this estimate and recent work of Green to prove that if N ≥ 2 {N\geq 2} is an integer, A ⊆ { 1, ..., N } {A\subseteq\{1,\ldots,N\}}, and for all primes p no two elements in A differ by p - 1 {p-1}, then | A | ≪ N 1 - 10 - 18 {|A|\ll N^{1-10^{-18}}}. This strengthens a theorem of Sárközy.

Original languageEnglish (US)
JournalForum Mathematicum
DOIs
StateAccepted/In press - 2024

Keywords

  • Zero density

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics

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