Asymptotic For Primitive Roots Producing Polynomials

TL;DR

This paper derives an asymptotic formula for counting primes of the form f(n) with a fixed primitive root u, assuming the Bateman-Horn conjecture, extending understanding of primitive roots in polynomial-generated primes.

Contribution

It provides a new asymptotic estimate for primes produced by polynomials with a fixed primitive root, under the Bateman-Horn conjecture, linking prime values and primitive roots.

Findings

Asymptotic counting function derived for primes of form f(n) with fixed primitive root u

Conditional results based on the Bateman-Horn conjecture

Extends understanding of primitive roots in polynomial-generated primes

Abstract

Let $x \geq 1$ 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 $p=f(n) \leq x$ with a fixed primitve root $u$ is derived in this note.