Department of Mathematics and Computer Science Number Theory and Combinatorics Seminar Spring 2020 Talks are at noon on Monday in W561 of University Hall For more information, contact Nathan Ng < ng AT cs DOT uleth DOT ca > or Dave Morris .
 The next talk: Mar 2 at noon in W561 Nathan Ng (University of Lethbridge) Moments of the Riemann zeta function and mean values of long Dirichlet polynomials The $2k$-th moments $I_k(T)$ of the Riemann zeta function have been studied extensively. In the late 90's, Keating-Snaith gave a conjecture for the size of $I_k(T)$. At the same time Conrey-Gonek connected $I_k(T)$ to mean values of long Dirichlet polynomials with divisor coefficients. Recently this has been further developed by Conrey-Keating in a series of 5 articles. I will discuss my work relating $I_3(T)$ to smooth shifted ternary additive divisor sums and also recent joint work with Alia Hamieh on mean values of long Dirichlet polynomials with higher divisor coefficients.
 Talks in the series this semester: (Click on any title for more info, including the abstract. Then click on it again to hide the info.)

 Date Speaker Title Jan 13 everyone Open problem session at noon in B543 (University of Lethbridge) Please bring your favourite (math) problems. Anyone with a problem to share will be given about 5 minutes to present it. We will also choose most of the speakers for the rest of the semester. Jan 20 Joy Morris Regular Representations of Groups at noon in C620 (University of Lethbridge) A natural way to understand groups visually is by examining objects on which the group has a natural permutation action. In fact, this is often the way we first show groups to undergraduate students: introducing the cyclic and dihedral groups as the groups of symmetries of polygons, logos, or designs. For example, the dihedral group $D_8$ of order $8$ is the group of symmetries of a square. However, there are some challenges with this particular example of visualisation, as many people struggle to understand how reflections and rotations interact as symmetries of a square. Every group $G$ admits a natural permutation action on the set of elements of $G$ (in fact, two): acting by right- (or left-) multiplication. (The action by right-multiplication is given by $\{\tau_g: g \in G\}$, where $\tau_g(h)=hg$ for every $h \in G$.) This action is called the right- (or left-) regular representation of $G$. It is straightforward to observe that this action is regular (that is, for any two elements of the underlying set, there is precisely one group element that maps one to the other). If it is possible to find an object that can be labelled with the elements of $G$ in such a way that the symmetries of the object are precisely the right-regular representation of $G$, then we call this object a regular representation of $G$. A Cayley (di)graph Cay$(G,S)$ on the group $G$ (with connection set $S \subset G$) is defined to have the set $G$ as its vertices, with an arc from $g$ to $sg$ for every $s \in S$. It is straightforward to see that the right-regular representation of $G$ is a subset of the automorphism group of this (di)graph. However, it is often not at all obvious whether or not Cay$(G,S)$ admits additional automorphisms. For example, Cay$(\mathbb Z_4, \{1,3\})$ is a square, and therefore has $D_8$ rather than $\mathbb Z_4$ as its full automorphism group, so is not a regular representation of $\mathbb Z_4$. Nonetheless, since a regular representation that is a (di)graph must always be a Cayley (di)graph, we study these to determine when regular representations of groups are possible. I will present results about which groups admit graphs, digraphs, and oriented graphs as regular representations, and how common it is for an arbitrary Cayley digraph to be a regular representation. Jan 27 Habiba Kadiri Explicit results about primes in Chebotarev's density theorem at noon in W561 (University of Lethbridge) Let $L/K$ be a Galois extension of number fields with Galois group $G$, and let $C \subset G$ be a conjugacy class. Attached to each unramified prime ideal $\mathfrak{p}$ in $\mathcal{O}_K$ is the Artin symbol $\sigma_{\mathfrak{p}}$, 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 $\sigma_{\mathfrak{p}}=C$ and norm $N_{\mathfrak{p}} \le x$ is asymptotically $\frac{|C|}{|G|}Li(x)$ as $x\to \infty$, where $Li(x)$ 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. Feb 3 Selcuk Aygin On the eta quotients whose derivatives are also eta quotients at noon in C620 (University of Lethbridge) In classical q-series studies there are examples of eta quotients whose derivatives are also eta quotients. The most famous examples can be found in works of S. Ramanujan and N. Fine. In 2019, in a joint work with P. C. Toh, we have given 203 pairs of such eta quotients, which we believe to be the complete list (see "When is the derivative of an eta quotient another eta quotient?", J. Math. Anal. Appl. 480 (2019) 123366). Recently, D. Choi, B. Kim and S. Lim have given a complete list of such eta quotients with squarefree levels (see "Pairs of eta-quotients with dual weights and their applications", Adv. Math. 355 (2019) 106779). Their findings support the idea that our list is complete. In this talk we introduce a beautiful interplay between eta quotients, their derivatives and Eisenstein series. Then we share our work in progress (joint with A. Akbary) in proving the completeness of our list beyond squarefree levels. Feb 10 Amir Akbary Reciprocity Laws at noon in W561 (University of Lethbridge) The Artin reciprocity law provides a solution to Hilbert's ninth problem (9. Proof of the Most General Law of Reciprocity in any Number Field). In this talk we provide an exposition of this theorem with emphasis on its relation with the classical law of quadratic reciprocity and describe its motivating role in the far reaching Langlands Reciprocity Conjecture. Feb 24 Peng-Jie Wong Cyclicity of CM Elliptic Curves Modulo $p$ at noon in W561 (University of Lethbridge) Let $E$ be a CM elliptic curve defined over $\Bbb{Q}$. In light of the Lang-Trotter conjecture, there is a question asking for an asymptotic formula for the number of primes $p\le x$ for which the reduction modulo $p$ of $E$ is cyclic. This has been studied by Akbary, Cojocaru, Gupta, M.R. Murty, V.K. Murty, and Serre. In this talk, we will discuss their work and some variants of the question. Mar 2 Nathan Ng Moments of the Riemann zeta function and mean values of long Dirichlet polynomials at noon in W561 (University of Lethbridge) The $2k$-th moments $I_k(T)$ of the Riemann zeta function have been studied extensively. In the late 90's, Keating-Snaith gave a conjecture for the size of $I_k(T)$. At the same time Conrey-Gonek connected $I_k(T)$ to mean values of long Dirichlet polynomials with divisor coefficients. Recently this has been further developed by Conrey-Keating in a series of 5 articles. I will discuss my work relating $I_3(T)$ to smooth shifted ternary additive divisor sums and also recent joint work with Alia Hamieh on mean values of long Dirichlet polynomials with higher divisor coefficients. Mar 9 Nathan Ng Title TBA at noon in W561 (University of Lethbridge) Abstract TBA Mar 16 Speaker TBA Title TBA at noon in W561 (University of Lethbridge) Abstract TBA Mar 23 Shabnam Ahktari Title TBA at noon in W561 (University of Oregon) Abstract TBA Mar 30 Sourabh Das Title TBA at noon in W561 (University of Lethbridge) Abstract TBA
 Past semesters: Fall F2007 F2008 F2009 F2010 F2011 F2012 F2013 F2014 F2015 F2016 F2017 F2018 F2019 Spring S2008 S2009 S2010 S2012 S2013 S2014 S2015 S2016 S2017 S2018 S2019