Grosswald’s conjecture on primitive roots

Very little is known about the distribution of primitive roots of a prime $p$. Grosswald conjectured that the least primitive root of a prime p is less than $\sqrt{p} - 2$ for all $p> 409$. While this is certainly true for all $p$ sufficiently large, Grosswald’s conjecture in still open. I shall outline some recent work which resolves the conjecture completely under the Generalised Riemann Hypothesis and which almost resolves the conjecture unconditionally.