Counting primes with a given primitive root, uniformly

Abstract

The celebrated Artin conjecture on primitive roots asserts that given any integer $g$ which is neither $-1$ nor a perfect square, there is an explicit constant $A(g)>0$ such that the number $\Pi(x;g)$ of primes $p\le x$ for which $g$ is a primitive root is asymptotically $A(g)\pi(x)$ as $x\to\infty$, where $\pi(x)$ counts the number of primes not exceeding $x$. Artin's conjecture has remained unsolved since its formulation in 1927. Nevertheless, Hooley demonstrated in 1967 that Artin's conjecture is a consequence of the Generalized Riemann Hypothesis (GRH) for Dedekind zeta functions of certain cyclotomic-Kummer extensions over $\mathbb{Q}$. In this paper, we use GRH to establish a uniform version of the Artin--Hooley asymptotic formula. Specifically, we prove that $\Pi(x;g) \sim A(g) x/\log{x}$ whenever $\log{x}/\log\log{2|g|} \to \infty$, i.e., whenever $x$ tends to infinity faster than any power of $\log{(2|g|)}$. Under GRH, we also show that the least prime $p_g$ possessing $g$ as a primitive root satisfies the upper bound $p_g=O(\log^{19}(2|g|))$ uniformly for all non-square $g\ne-1$. We conclude with an application to the average value of $p_g$ and a discussion of an analogue concerning the least "almost-primitive'' root.