Refinements of Artin's primitive root conjecture

Let ord𝑝(𝑎)be the order of 𝑎in (ℤ/𝑝ℤ)∗. In 1927, Artin conjectured that the set of primes 𝑝for which an integer 𝑎≠−1,◻is a primitive root (i.e. ord𝑝(𝑎)=𝑝−1) has a positive asymptotic density among all primes. In 1967 Hooley proved this conjecture assuming the Generalized Riemann Hypothesis (GRH). In this talk, we will study the behaviour of ord𝑝(𝑎)as 𝑝varies over primes. In particular, we will show, under GRH, that the set of primes 𝑝for which ord𝑝(𝑎)is “𝑘prime factors away” from 𝑝−1− 1 has a positive asymptotic density among all primes, except for particular values of 𝑎and 𝑘. We will interpret being “𝑘prime factors away” in three different ways:
𝑘=𝜔(𝑝−1ord𝑝(𝑎)),𝑘=Ω(𝑝−1ord𝑝(𝑎)),𝑘=𝜔(𝑝−1)−𝜔(ord𝑝(𝑎)).

We will present conditional results analogous to Hooley’s in all three cases and for all integer 𝑘. From this, we will derive conditionally the expectation for these quantities.

Furthermore, we will provide partial unconditional answers to some of these questions.

This is joint work with Leo Goldmakher and Greg Martin.