Title: Primes with small primitive roots
Abstract: For a prime $p$, the least primitive root $g(p)$ (smallest generator of $\mathbb{Z}_p^*$) has been studied since the earlist works of Vinogradov. The folklore conjecture states that $g(p)=O(p^{\eps})$ for every fixed $\eps>0$. The strongest unconditional result so far is Burgess' estimate $g(p)=O(p^{1/4+\eps})$, established in the 1960s, which has remained unbeaten since.
Our main result is the following. 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 $g(p) \leq p^{1/4-\delta(p)}$ for all $p\in \mathcal{S}$, and $\#\{p\leq x: p\in \mathcal{S}\}=(1+o(1))\pi(x)$ as $x\to\infty$. This is a joint work with Kevin Ford and Andrew Granville.