Least primitive root and simultaneous power-non residues

TL;DR

This paper establishes an improved upper bound for the least primitive root modulo a prime for most primes with specific factorization properties, and characterizes the exceptional set of primes.

Contribution

It provides a new bound on the least primitive root for a large class of primes and describes the exceptional set explicitly.

Findings

Bound g(p) mp;lt; p^{1/(4sqrt{e})+\u03b5} for most primes p

Most primes with p-1 lacking small prime factors have primitive roots below this bound

Explicit description of the set of primes not satisfying the bound

Abstract

Let $p$ be a prime and let $g(p)$ be the least primitive root modulo $p$. We prove that for any $\epsilon>0$ and $p$ large enough the bound \begin{align} g(p)\ll p^{\frac{1}{4\sqrt{e}}+\epsilon} \nonumber \end{align} holds for most prime $p$ such that $p-1$ does not have small prime factors, but $2$. We also give an explicit description of the exceptional set.