Bounded gaps between prime polynomials with a given primitive root

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• 作者：Lee Troupe¹ ; ^{ltroupe@math.uga.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11N36 ; 11T06
• 刊名：Finite Fields and Their Applications
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：37
• 期：Complete
• 页码：295-310
• 全文大小：363 K

文摘

A famous conjecture of Artin states that there are infinitely many prime numbers for which a fixed integer g   is a primitive root, provided g≠−1$g\ne -1$ and g   is not a perfect square. Thanks to work of Hooley, we know that this conjecture is true, conditional on the truth of the Generalized Riemann Hypothesis. Using a combination of Hooley's analysis and the techniques of Maynard–Tao used to prove the existence of bounded gaps between primes, Pollack has shown that (conditional on GRH) there are bounded gaps between primes with a prescribed primitive root. In the present article, we provide an unconditional proof of the analogue of Pollack's work in the function field case; namely, that given a monic polynomial 749ca9ef121c9d4c62b234dc50c2" title="Click to view the MathML source">g(t)$g(t)$ which is not an vth power for any prime v   dividing q−1$q-1$, there are bounded gaps between monic irreducible polynomials P(t)$P(t)$ in 120" title="Click to view the MathML source">F_q[t]${\mathbb{F}}_q[t]$ for which 749ca9ef121c9d4c62b234dc50c2" title="Click to view the MathML source">g(t)$g(t)$ is a primitive root (which is to say that 749ca9ef121c9d4c62b234dc50c2" title="Click to view the MathML source">g(t)$g(t)$ generates the group of units modulo P(t)$P(t)$). In particular, we obtain bounded gaps between primitive polynomials, corresponding to the choice g(t)=t$g(t)=t$.