Bounded gaps between primes with a given primitive root, II

Abstract

Let $m$ be a natural number, and let $\mathcal{Q}$ be a set containing at least $\exp (Cm)$ primes. We show that one can find infinitely many strings of $m$ consecutive primes each of which has some $q\in \mathcal{Q}$ as a primitive root, all lying in an interval of length $O_{\mathcal{Q}}(\exp (C^{\prime }m))$. This is a bounded gaps variant of a theorem of Gupta and Ram Murty. We also prove a result on an elliptic analogue of Artin's conjecture. Let $E/\mathbb{Q}$ be an elliptic curve with an irrational $2$-torsion point. Assume GRH. Then for every $m$, there are infinitely many strings of $m$ consecutive primes $p$ for which $E({\mathbb{F}}_p)$ is cyclic, all lying an interval of length $O_E(\exp (C^{\prime \prime }m))$. If $E$ has CM, then the GRH assumption can be removed. Here $C$, $C^{\prime }$, and $C^{\prime \prime }$ are absolute constants.