Refinements of Artin's primitive root conjecture

Let $ord_p(a)$ be the order of $a$ in $(\mathbb{Z}/p\mathbb{Z})^*$. In 1927 Artin conjectured that the set of primes p for which a given integer a (that is neither a square number nor −1) is a primitive root (i.e. $ord_p(a)=p-1$) has a positive asymptotic density among all primes. In 1967 Hooley proved this conjecture assuming the Generalized Riemann Hypothesis.

In this talk we will study the behaviour of $ord_p(a)$ as p varies over primes, in particular we will show, under GRH, that the set of primes p for which $ord_p(a)$ is ``k prime factors away'' from p−1 has a positive asymptotic density among all primes except for particular values of a and k. We will interpret being ``k prime factors away'' in three different ways, namely $k=\omega((p−1)/ord_p(a))$, $k=\Omega((p−1)/ord_p(a))$ and $k=\omega(p−1)-\omega(ord_p(a))$, and present conditional results analogous to Hooley's in all three cases and for all integer k. This is joint work with Leo Goldmakher and Greg Martin.