UBC Number Theory Seminar: Paul Péringuey

Let ordp(a) be the order of a in (ℤ/pℤ)∗. 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. ordp(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 ordp(a) as p varies over primes, in particular we will show, under GRH, that the set of primes p for which ordp(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=ω((p−1)/ordp(a)), k=Ω((p−1)/ordp(a)) and k=ω(p−1)−ω(ordp(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.