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

References in corpus (5)

• Bounded gaps between primes with a given primitive root
• Bounded Gaps Between Primes in Chebotarev Sets
• Bounded gaps between primes of the special form
• Consecutive primes in tuples
• Artin prime producing polynomials

Cited by in corpus (17)

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• The "bounded gaps between primes" Polymath project - a retrospective
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• Arithmetic functions at consecutive shifted primes
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• On the Twin Prime Conjecture
• Variants of the Selberg sieve, and bounded intervals containing many primes
• Gaps between primes
• Bounded gaps between prime polynomials with a given primitive root
• The existence of small prime gaps in subsets of the integers
• Topics In Primitive Roots
• Ergodic recurrence and bounded gaps between primes
• Artin Twin Primes
• Small gaps between the Piatetski-Shapiro primes
• Generalized Artin Primitive Root Conjecture
• Addendum to the paper: "Artin Prime Producing Quadratics" [Abh. Math. Sem. Univ. Hamburg 77 (2007), 109--127; MR2379332 (2008m:11194)] by P. Moree
• Consecutive primes and Legendre symbols