Least Prime Primitive Roots

TL;DR

This paper establishes a sharper upper bound for the least prime primitive root modulo a large prime, improving previous estimates and providing a more precise understanding of primitive roots' distribution.

Contribution

It introduces a new upper bound for the least prime primitive root, refining existing estimates to a smaller, more accurate bound for large primes.

Findings

New upper bound: $g^*(p) \\ll p^{5/\\log \\log p}$ for large primes

Improves upon previous bounds like $g^*(p) \\ll p^c, c>2.8$

Results hold uniformly for all sufficiently large primes

Abstract

This note presents an upper bound for the least prime primitive roots $g^*(p)$ modulo $p$, a large prime. The current literature has several estimates of the least prime primitive root $g^*(p)$ modulo a prime $p\geq 2$ such as $g^*(p)\ll p^c, c>2.8$. The estimate provided within seems to sharpen this estimate to the smaller estimate $g^*(p)\ll p^{5/\log \log p}$ uniformly for all large primes $p\geq 2$.