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 $q_1<q_2<q_3<\cdots$ is the sequence of primes possessing $g$ as a primitive root, then ${liminf}_{n\to \infty }\left(q_{n+\left(m-1\right)}-q_n\right)\le C_m$, 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.