UCalgary Algebra and Number Theory Seminar: Paul Péringuey

Let ${\text{ord}}_p(a)$be the order of $a$in ${\left(\mathbb{Z}/p\mathbb{Z}\right)}^{\ast <!--\; \ast \; -->}$. In 1927, Artin conjectured that the set of primes $p$for which an integer $a\ne <!--\; \ne \; -->-<!--\; -\; -->1,\mathrm{◻<!--\; ◻\; -->}$is a primitive root (i.e. ${\text{ord}}_p(a)=p-<!--\; -\; -->1$) 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 ${\text{ord}}_p(a)$as $p$varies over primes. In particular, we will show, under GRH, that the set of primes $p$for which ${\text{ord}}_p(a)$is “$k$prime factors away” from $p-<!--\; -\; -->1$− 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: $$k=\mathrm{\omega <!--\; \omega \; -->}\left(\frac{p-<!--\; -\; -->1}{{\text{ord}}_p(a)}\right),k=\mathrm{\Omega <!--\; \Omega \; -->}\left(\frac{p-<!--\; -\; -->1}{{\text{ord}}_p(a)}\right),k=\mathrm{\omega <!--\; \omega \; -->}(p-<!--\; -\; -->1)-<!--\; -\; -->\mathrm{\omega <!--\; \omega \; -->}({\text{ord}}_p(a)).$$

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.