Grosswald’s conjecture on primitive roots

Abstract: 

Very little is known about the distribution of primitive roots of a prime pp. Grosswald conjectured that the least primitive root of a prime p is less than p–√−2p−2 for all p>409p>409. While this is certainly true for all pp 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.