Bounded gaps between primes with a given primitive root

Abstract

Fix an integer $g\ne -1$ 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 $m\ge 2$. If $q_1<q_2<q_3<\dots$ is the sequence of primes possessing $g$ as a primitive root, then ${\mathrm{liminf}}_{n\to \infty }(q_{n+(m-1)}-q_n)\le C_m$, 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.

small corrections to the treatment of \sum_1 on pp. 11--12