Remarks on the distribution of the primitive roots of a prime

TL;DR

This paper estimates the number of pairs involving primitive roots in a finite field, where both elements are primitive roots and related through a polynomial, addressing a question posed by Han and Zhang.

Contribution

It provides an estimate for the count of pairs of primitive roots connected via a polynomial, extending understanding of primitive root distributions in finite fields.

Findings

Derived bounds for the number of such pairs

Extended previous results to more general polynomial conditions

Answered a specific open question by Han and Zhang

Abstract

Let $\mathbb{F}_p$ be a finite field of size $p$ where $p$ is an odd prime. Let $f(x)\in\mathbb{F}_p[x]$ be a polynomial of positive degree $k$ that is not a $d$-th power in $\mathbb{F}_p[x]$ for all $d\mid p-1$. Furthermore, we require that $f(x)$ and $x$ are coprime. The main purpose of this paper is to give an estimate of the number of pairs $(\xi,\xi^\alpha f(\xi))$ such that both $\xi$ and $\xi^\alpha f(\xi)$ are primitive roots of $p$ where $\alpha$ is a given integer. This answers a question of Han and Zhang.