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

We consider Artin’s conjecture on primitive roots over a number field $K$, reducing an algebraic number $\alpha\in K^\times$. Under GRH, there is a density $dens(\alpha)$ counting the proportion of the primes of $K$ for which $\alpha$ is a primitive root.

This density $dens(\alpha)$ is a rational multiple of an Artin constant $A(\tau)$ that depends on the largest integer $\tau\geq 1$ such that

Supposing that $dens(\alpha)\neq 0$, we provide uniform bounds for the ratio $dens(\alpha)/A(\tau)$. This is joint work with Igor Shparlinski. We also present heuristics obtained with Mia Tholl.