Primitive Roots In Short Intervals

TL;DR

This paper demonstrates the existence of primitive roots within short intervals for large primes, establishing bounds on the least primitive and prime primitive roots, advancing understanding of primitive root distribution.

Contribution

It proves the existence of primitive roots in short intervals of size greater than a logarithmic power, providing explicit bounds on the least primitive and prime primitive roots.

Findings

Primitive roots exist in intervals of size greater than $( ext{log } p)^{1+ ext{epsilon}}$

Least primitive root $g(p)= O(( ext{log } p)^{1+ ext{epsilon}})$

Least prime primitive root $g^*(p)= O(( ext{log } p)^{1+ ext{epsilon}})$

Abstract

Let $p\geq 2$ be a large prime, and let $N\gg ( \log p)^{1+\varepsilon}$. This note proves the existence of primitive roots in the short interval $[M,M+N]$, where $M \geq 2$ is a fixed number, and $ \varepsilon>0$ is a small number. In particular, the least primitive root $g(p)= O\left ((\log p)^{1+\varepsilon} \right)$, and the least prime primitive root $g^*(p)= O\left ((\log p)^{1+\varepsilon} \right)$ unconditionally.