Remarks on the distribution of the primitive roots of a prime

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 $(\hat{I}¾,\hat{I}{¾}^{\hat{I}}\pm f(\hat{I}¾))$ such that both $\hat{I}¾$ and $\hat{I}{¾}^{\hat{I}}\pm f(\hat{I}¾)$ are primitive roots of $p$ where $\hat{I}\pm$ is a given integer. This answers a question of Han and Zhang.