Bounded gaps between primes with a given primitive root

Pollack, Paul

doi:10.2140/ant.2014.8.1769

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Bounded gaps between primes with a given primitive root

Paul Pollack

Vol. 8 (2014), No. 7, 1769–1786

DOI: 10.2140/ant.2014.8.1769

Abstract

Fix an integer $g \neq - 1$ that is not a perfect square. In 1927, Artin conjectured that there are infinitely many primes for which $g$ is a primitive root. Forty years later, Hooley showed that Artin’s conjecture follows from the generalized Riemann hypothesis (GRH). We inject Hooley’s analysis into the Maynard–Tao work on bounded gaps between primes. This leads to the following GRH-conditional result: Fix an integer $m \geq 2$ . If $q_{1} < q_{2} < q_{3} < \dots$ is the sequence of primes possessing $g$ as a primitive root, then ${liminf}_{n \to \infty} (q_{n + (m - 1)} - q_{n}) \leq C_{m}$ , where $C_{m}$ is a finite constant that depends on $m$ but not on $g$ . We also show that the primes $q_{n}, q_{n + 1}, \dots, q_{n + m - 1}$ in this result may be taken to be consecutive.

Keywords

primitive root, Artin's conjecture, bounded gaps, Maynard–Tao theorem

Mathematical Subject Classification 2010

Primary: 11A07

Secondary: 11N05

Milestones

Received: 27 April 2014

Revised: 21 June 2014

Accepted: 19 July 2014

Published: 21 October 2014

Authors

Paul Pollack
Department of Mathematics University of Georgia Boyd Graduate Studies Building Athens, GA 30602 United States

Vol. 8, No. 7, 2014