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 $dens\left(\alpha \right)$ counting the proportion of the primes of $K$ for which $\alpha$ is a primitive root. This density $dens\left(\alpha \right)$ is a rational multiple of an Artin constant $A\left(\tau \right)$ that depends on the largest integer $\tau ⩾1$ such that $\alpha \in {\left(K^{\times }\right)}^{\tau }$. The aim of this paper is bounding the ratio $dens\left(\alpha \right)/A\left(\tau \right)$, under the assumption that $dens\left(\alpha \right)\ne 0$. Over $Q$, 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 $K$.