Resolving Grosswald's conjecture on GRH

In this paper we examine Grosswald’s conjecture on $g(p)$, the least primitive root modulo $p$. Assuming the Generalized Riemann Hypothesis (GRH), and building on previous work by Cohen, Oliveira e Silva and Trudgian, we resolve Grosswald’s conjecture by showing that $g(p)<\sqrt{p}-2$ for all $p>409$. Our method also shows that under GRH we have $\widehat{g}(p)<\sqrt{p}-2$ for all $p>2791$, where $\widehat{g}(p)$ is the least prime primitive root modulo $p$.