Uniform bounds for the density in Artin's conjecture on primitive roots

Abstract

We consider Artin's conjecture on primitive roots over a number field $K$, reducing an algebraic number $\alpha\in K^\times$. Under the Generalised Riemann Hypothesis, there is a density ${\mathrm{dens}}(\alpha)$ counting the proportion of the primes of $K$ for which $\alpha$ is a primitive root. This density ${\mathrm{dens}}(\alpha)$ is a rational multiple of an Artin constant $A(\tau)$ that depends on the largest integer $\tau\geq 1$ such that $\alpha\in (K^\times)^\tau$. The aim of this paper is bounding the ratio ${\mathrm{dens}}(\alpha)/A(\tau)$, under the assumption that ${\mathrm{dens}}(\alpha)\neq 0$. Over $\mathbb Q$, this ratio is between $2/3$ and $2$, these bounds being optimal. For a general number field $K$ we provide upper and lower bounds that only depend on $K$.