Resolving Grosswald's conjecture on GRH

TL;DR

This paper proves Grosswald's conjecture on the least primitive root modulo p under GRH, establishing bounds for all sufficiently large primes and advancing understanding of primitive roots in number theory.

Contribution

It confirms Grosswald's conjecture under GRH and extends bounds to prime primitive roots, improving previous results in analytic number theory.

Findings

g(p)< sqrt(p) - 2 for all p>409

hat{g}(p)< sqrt(p) - 2 for all p>2791

Under GRH, bounds hold for large primes

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 $g(p)< \sqrt{p} - 2$ for all $p>409$. Our method also shows that under GRH we have $\hat{g}(p)< \sqrt{p}-2$ for all $p>2791$, where $\hat{g}(p)$ is the least prime primitive root modulo $p$.