2025/05/08 by Fan, Steve, Pollack, Paul
#11N36 #FOS: Mathematics #Number Theory (math.NT) #Primary 11N05 #Secondary 11A07
paper · doi:10.48550/arxiv.2505.05601
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 Π(x;g) of primes p≤ x for which g is a primitive root is asymptotically A(g)π(x) as x→∞, where π(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 ℚ. In this paper, we use GRH to establish a uniform version of the Artin--Hooley asymptotic formula. Specifically, we prove that Π(x;g) ∼ A(g) x/logx whenever logx/loglog2|g| → ∞, i.e., whenever x tends to infinity faster than any power of log(2|g|). Under GRH, we also show that the least prime pg possessing g as a primitive root satisfies the upper bound pg=O(log19(2|g|)) uniformly for all non-square g≠-1. We conclude with an application to the average value of pg and a discussion of an analogue concerning the least "almost-primitive'' root.