Large gaps between consecutive prime numbers

Abstract

Let $G(X)$ denote the size of the largest gap between consecutive primes below $X$. Answering a question of Erdős, we show that \[ G(X) \geq f(X) \frac{\log X \log \log X \log \log \log \log X}{(\log \log \log X)^2},\] where $f(X)$ is a function tending to infinity with $X$. Our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions. As such, we rely on recent work on the existence and distribution of long arithmetic progressions consisting entirely of primes.

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      }
  • [West] E. Westzynthius, "Über die Verteilung der Zahlen, die zu den $n$ ersten Primzahlen teilerfremd sind," Commentationes Physico–Mathematicae, Societas Scientarium Fennica, Helsingfors, vol. 5, pp. 1-37, 1931.
    @article{West,
      author = {Westzynthius, E.},
      title = {Über die Verteilung der Zahlen, die zu den $n$ ersten Primzahlen teilerfremd sind},
      journal = {Commentationes Physico--Mathematicae, Societas Scientarium Fennica, Helsingfors},
      volume = {5},
      year = {1931},
      pages = {1--37},
      zblnumber = {0003.24601},
      }

Authors

Kevin Ford

University of Illinois at Urbana-Champaign Urbana, IL

Ben Green

Mathematical Institute Oxford England

Sergei Konyagin

Steklov Mathematical Institute Moscow, Russia

Terence Tao

University of California Los Angeles CA