Resolving Grosswald's conjecture on GRH

Abstract

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 $g(p) 409$. Our method also shows that under GRH we have $\hat{g}(p) 2791$, where $\hat{g}(p)$ is the least prime primitive root modulo $p$.