Primes with small primitive roots

Abstract

Let $\hat{I}´(p)$ 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 $\le p^{1/4-\hat{I}´(p)}$ for all $p\in \mathcal{S}$, where $\#\{p\le x:p\in \mathcal{S}\}\sim \ddot{I}€(x)$ as $x\to \infty$.