Number Theory

The Riemann zeta function plays a central role in our understanding of the prime numbers. In this talk we will review some of its amazing properties as well as properties of other similar functions, the Dirichlet L-functions. We will then see how the method of moments can help us in the study of L-functions and some surprising properties of their values. This talk will be accessible to advanced undergraduate students and is part of the May12, Celebration of Women in Mathematics.

Over fifty years ago Richard Kenneth Guy joined the then Department of Mathematics, Statistics and Computer Science at the nascent University of Calgary. Although he retired from the University in 1982, he continued, even in his last year, to come in to the University every day and work on the mathematics that he loved. In this talk I will provide a glimpse into the life and research of this most remarkable man. In doing this, I will recount several of the important events of Richard’s life and briefly discuss some of his mathematical contributions.

About Dr. Williams

: Dr. Hugh Williams is internationally recognized as an expert in computational number theory and its applications to cryptography. Shortly after obtaining his Ph.D. in 1969 from the Department of Applied Analysis and Computer Science at the University of Waterloo, he joined the newly established Department of Computer Science at the University of Manitoba, where he was promoted to the rank of Full Professor in 1979. He also served there as Associate Dean of Science for Research Development for seven years (1994-2001). He moved to the University of Calgary in 2001 to take up the iCORE Chair for Algorithmic Number Theory and Cryptography (2001-2013) and retired as Emeritus Professor of Mathematics and Statistics in 2016. Dr. Williams has authored over 150 refereed journal papers, 30 refereed conference papers and 20 books or book chapters, and from 1983-85 held a national Killam Research Fellowship. In February 2009, Dr, Williams was selected for a six year term as the inaugural Director of the Tutte Institute for Mathematics and Computing (TIMC), a highly classified research facility established by the federal government. In 2016, he was appointed Professor Emeritus in Mathematics and Statistics at the University of Calgary.

TBA

Even in the title of one of his papers, Richard Guy called the elliptic curve with equation $y^2 = x^3 - 4x + 4$ his favorite. During the CNTA-XIV meeting in Calgary in 2016, I recalled some of his reasons for this (with Richard listening from the front row). The story as well as a few additional developments will also be the topic of the present lecture.

Habiba Kadiri (University of Lethbridge, Canada)

Let $L/K$ be a Galois extension of number fields with Galois group $G$, and let $C⊂G$ be a conjugacy class. Attached to each unramified prime ideal p in OK is the Artin symbol $\sigma p$, a conjugacy class in $G$. In 1922 Chebotarev established what is referred to his density theorem (CDT). It asserts that the number $\pi C(x)$ of such primes with $\sigma p=C$ and norm $Np≤x$ is asymptotically $\left|C\right|\left|G\right|\mathrm{Li} (x)$ as $x\rightarrow\infty$ where $\mathrm{Li} (x)$ is the usual logarithmic integral. As such, CDT is a generalisation of both the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions. In light of Linnik's result on the least prime in an arithmetic progression, one may ask for a bound for the least prime ideal whose Artin symbol equals C. In 1977 Lagarias and Odlyzko proved explicit versions of CDT and in 1979 Lagarias, Montgomery and Odlyzko gave bounds for the least prime ideal in the CDT. Since 2012 several unconditional explicit results of these theorems have appeared with contributions by Zaman, Zaman and Thorner, Ahn and Kwon, and Winckler. I will present several recent results we have proven with Das, Ng, and Wong.