Primes with small primitive roots

TL;DR

This paper constructs a large set of primes for which the smallest primitive root is significantly smaller than the prime itself, approaching a quarter power, with the set's density approaching that of all primes.

Contribution

It explicitly defines a set of primes with small primitive roots based on simple functions of their factors, extending understanding of primitive root distribution.

Findings

The set of primes has density approaching 1 as x increases.

For primes in this set, the least primitive root is at most p^{1/4 - δ(p)}.

The set is explicitly constructed using prime factor functions.

Abstract

Let $\delta(p)$ tend to zero arbitrarily slowly as $p\to\infty$. We exhibit an explicit set $\mathcal{S}$ of primes $p$, defined in terms of simple functions of the prime factors of $p-1$, for which the least primitive root of $p$ is $\le p^{1/4-\delta(p)}$ for all $p\in \mathcal{S}$, where $\#\{p\leq x: p\in \mathcal{S}\} \sim \pi(x)$ as $x\to\infty$.