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 $α\in K^\times$. Under the Generalised Riemann Hypothesis, there is a density ${\mathrm{dens}}(α)$ counting the proportion of the primes of $K$ for which $α$ is a primitive root. This density ${\mathrm{dens}}(α)$ is a rational multiple of an Artin constant $A(τ)$ that depends on the largest integer $τ\geq 1$ such that $α\in (K^\times)^τ$. The aim of this paper is bounding the ratio ${\mathrm{dens}}(α)/A(τ)$, under the assumption that ${\mathrm{dens}}(α)\neq 0$. 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 $K$.

Theorems & Definitions (21)

• Theorem 2.1
• proof
• Example 2.2
• Definition 1
• Example 3.1
• Remark 4.1
• Proposition 4.2
• proof
• Proposition 4.3
• proof