Asymptotic For Primitive Roots Producing Polynomials

Abstract

Let $x\ge 1$ be a large number, let $f(x)\in \mathbb{Z}[x]$ be a prime polynomial of degree $\text{deg}(f)=m$, and let $u\ne \pm 1,v^2$ be a fixed integer. Assuming the Bateman-Horn conjecture, an asymptotic counting function for the number of primes $p=f(n)\le x$ with a fixed primitve root $u$ is derived in this note.

Keywords: Primitive root; Prime value of polynomial; Primitive root producing polynomial. arXiv admin note: substantial text overlap with arXiv:1504.00843

References in corpus (3)

• Squarefree polynomials and Mobius values in short intervals and arithmetic progressions
• Some experiments on Bateman-Horn
• On Gaps Between Primitive Roots in the Hamming Metric