Number Theory Seminar

Paul Peringuey from the University of British Columbia

Date:  Monday, March 3

Time:  12:15 - 1:15 p.m.

Room:  M1060 (Markin Hall)

Refinements of Artin's primitive root conjecture

Let ord_p(a) be the order of a in (Z/pZ)*. In 1927, Artin conjectured that the set of primes p for which an integer a ≠ -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 (GRH). In this talk, we will study the behavior 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 = ω((p- 1)/ord_p(a)), k = Ω((p - 1)/ord_p(a)), and k = ω(p - 1) - ω(ord_p(a)), and 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.

EVERYONE IS WELCOME!

Room or Area: 
M1060

Contact:

Cherie Secrist | cherie.secrist@uleth.ca | (403) 329-2470