Department of Mathematics and Computer Science Number Theory and Combinatorics Seminar Fall 2017 Talks are at noon on Monday in C630 of University Hall For more information, or to receive an email announcement of each week's seminar, contact Nathan Ng < ng AT cs DOT uleth DOT ca > or Dave Morris .
 The next talk: Nov 20 at noon in C630 Forrest Francis (University of Lethbridge) Euler's Function on Products of Primes in Progressions Let $\phi(n)$ be Euler's totient function and let $q$ and $a$ be fixed coprime natural numbers. Denote by $S_{q,a}$ the set of natural numbers whose prime divisors are all congruent to $a$ modulo $q$. We can establish $\limsup_{n \in S_{q,a}} \frac{n}{\phi(n) (\log(\phi(q)\log{n}))^{1/\phi(q)}} = \frac{1}{C(q,a)},$ where $C(q,a)$ is a constant associated with a theorem of Mertens. We may then wish to know whether there are infinitely many $n$ in $S_{q,a}$ for which $\frac{n}{\phi(n)(\log\phi(q)\log{n})^{1/\phi(q)}} > \frac{1}{C(q,a)} \qquad (*)$ is true. In the case $q=a=1$, Nicolas (1983) established that if the Riemann hypothesis is true, then ($*$) holds for all primorials (products of the form $\prod_{p \leq x} p$), but if the Riemann hypothesis is false then there are infinitely many primorials for which ($*$) is true and infinitely many primorials for which ($*$) is false. In this talk we will show that, for some $q>1$, the work of Nicolas can be generalized by replacing the Riemann hypothesis with analogous conjectures for Dirichlet $L$-functions and replacing the primorials with products of the form $\prod_{\substack{p \leq x \\ p \,\equiv \,a \,(\mathrm{mod}\,q)}} p.$
 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 Sep 11 everyone Open problem session at noon in C630 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. Sep 18 Peng-Jie Wong Nearly supersolvable groups and Artin's conjecture at noon in C630 (University of Lethbridge) Let $K/k$ be a Galois extension of number fields with Galois group $G$, and let $\rho$ be a non-trivial irreducible representation of $G$ of dimension $n$. The Artin holomorphy conjecture asserts that the Artin $L$-function attached to $\rho$ extends to an entire function. It is well-known that when $n=1$, this conjecture follows from Artin reciprocity. Also, by the works of Langlands and many others, we know that this conjecture is valid for $n=2$ under certain conditions. However, in general, the Artin holomorphy conjecture is wildly open. In this talk, we will discuss how elementary group theory plays a role in studying the Artin holomorphy conjecture and introduce the notion of "nearly supersolvable groups". If time allows, we will explain how such groups lead to a proof of the Artin holomorphy conjecture for Galois extensions of degree less than 60. Sep 25 Muhammad Khan The contact graphs of totally separable packings at noon in C630 (University of Lethbridge) Contact graphs have emerged as an important tool in the study of translative packings of convex bodies and have found numerous applications in materials science. The contact number of a packing of translates of a convex body is the number of edges in the contact graph of the packing, while the Hadwiger number of a convex body is the maximum vertex degree over all such contact graphs. In this talk, we investigate the Hadwiger and contact numbers of totally separable packings of convex bodies, known as the separable Hadwiger number and the separable contact number, respectively. We show that the separable Hadwiger number of any smooth strictly convex body in dimensions $d = 2, 3, 4$ is $2d$ and the maximum separable contact number of any packing of $n$ translates of a smooth strictly convex domain is $\lfloor 2n - 2\sqrt{n} \rfloor$. Our proofs employ a characterization of total separability in terms of hemispherical caps on the boundary of a smooth convex body, Auerbach bases of finite dimensional real normed spaces, angle measures in real normed planes, minimal perimeter polyominoes and an approximation of smooth $o$-symmetric strictly convex domains by, what we call, Auerbach domains. This is joint work with K. Bezdek (Calgary) and M. Oliwa (Calgary). Oct 2 Andrew Fiori The average number of quadratic Frobenius pseudoprimes at noon in C630 (University of Lethbridge) Primality testing has a number of important applications. In particular in cryptographic applications the complexity of existing deterministic algorithms causes increasing latency as the size of numbers we must test grow and the number of tests we must run before finding a prime grows aswell. These observations lead one to consider potentially non-deterministic algorithms which are faster, and consequently leads one to consider the false positives these algorithms yield, which we call pseudoprimes. In this talk I will discuss my recent work with Andrew Shallue where we study Quadratic Frobenius Pseudoprimes. I shall describe our results on an asymptotic lower bounds on the number of false positives. These results represent a generalization of those Erdos-Pomerance concerning similar problems for (Fermat) pseudoprimes. Oct 16 Lee Troupe Normally Distributed Arithmetic Functions at noon in C630 (University of British Columbia) In the late 1930s, Paul Erdős attended a seminar at Cornell University given by Mark Kac, who suspected that divisibility by primes satisfies a certain "statistical independence" condition. If this were true, the central limit theorem could be used to show that the number of distinct prime factors of $n$, as $n$ varies over the natural numbers, is normally distributed, with mean $\log\log n$ and standard deviation $\sqrt{\log\log n}$. Erdős used sieve methods to confirm Kac's intuition, and the resulting Erdős-Kac theorem is a foundational result in the field of probabilistic number theory. Many different proofs of and variations on the Erdős-Kac theorem have been given in the intervening decades. This talk will highlight some of these results and the techniques used to obtain them, including recent work of the speaker and Greg Martin (UBC). Oct 23 Sam Broadbent, Habiba Kadiri, and Kirsten Wilk Sharper bounds for Chebyshev functions $\boldsymbol{\theta(x)}$ and $\boldsymbol{\psi(x)}$ at noon in C630 (University of Lethbridge) In this talk we report on some research projects from summer 2017 supported by NSERC-USRA. In the first part of the project, we surveyed all existing explicit results from the past 60 years on prime counting functions, with a special focus on $\theta(x)$ (counting $\log p$ for each prime $p \le x$). In the second part, we provided new bounds for the Chebyshev function $\psi(x)$ based on a recent zero density result for the zeros of the Riemann zeta function (due to Kadiri-Lumley-Ng). Finally, we have established the current best results for the prime counting function $\theta(x)$ for various ranges of $x$.       (Joint work with Noah Christensen, Allysa Lumley, and Nathan Ng) Oct 30 Akshaa Vatwani Variants of equidistribution in arithmetic progression at noon in C630 (University of Waterloo) It is well known that the prime numbers are equidistributed in arithmetic progression. Such a phenomenon is also observed more generally for a class of multiplicative functions. We derive some variants of such results and give an application to tuples of squarefree integers in arithmetic progression. We also discuss an interesting application that relates to the Chowla conjecture on correlations of the Möbius function, and show its relevance to the twin prime conjecture. Nov 6 Károly Bezdek Bounds for totally separable translative packings in the plane at noon in C630 (University of Calgary) A packing of translates of a convex domain in the Euclidean plane is said to be totally separable if any two packing elements can be separated by a line disjoint from the interior of every packing element. This notion was introduced by G. Fejes Toth and L. Fejes Toth (1973) and has attracted significant attention. In this lecture I will discuss the separable analogue of the classical inequality of N. Oler (from geometry of numbers) for totally separable translative packings of convex domains and then derive from it some new results. This includes finding the largest density of totally separable translative packings of an arbitrary convex domain and finding the smallest area convex hull of totally separable packings (resp., totally separable soft packings) generated by given number of translates of a convex domain (resp., soft convex domain). Last but not least, we determine the largest covering ratio (that is, the largest fraction of the plane covered) of totally separable soft circle packings with given soft parameter. This is a joint work with Zsolt Langi (Univ. of Technology, Budapest, Hungary). Nov 20 Forrest Francis Euler's Function on Products of Primes in Progressions at noon in C630 (University of Lethbridge) Let $\phi(n)$ be Euler's totient function and let $q$ and $a$ be fixed coprime natural numbers. Denote by $S_{q,a}$ the set of natural numbers whose prime divisors are all congruent to $a$ modulo $q$. We can establish $\limsup_{n \in S_{q,a}} \frac{n}{\phi(n) (\log(\phi(q)\log{n}))^{1/\phi(q)}} = \frac{1}{C(q,a)},$ where $C(q,a)$ is a constant associated with a theorem of Mertens. We may then wish to know whether there are infinitely many $n$ in $S_{q,a}$ for which $\frac{n}{\phi(n)(\log\phi(q)\log{n})^{1/\phi(q)}} > \frac{1}{C(q,a)} \qquad (*)$ is true. In the case $q=a=1$, Nicolas (1983) established that if the Riemann hypothesis is true, then ($*$) holds for all primorials (products of the form $\prod_{p \leq x} p$), but if the Riemann hypothesis is false then there are infinitely many primorials for which ($*$) is true and infinitely many primorials for which ($*$) is false. In this talk we will show that, for some $q>1$, the work of Nicolas can be generalized by replacing the Riemann hypothesis with analogous conjectures for Dirichlet $L$-functions and replacing the primorials with products of the form $\prod_{\substack{p \leq x \\ p \,\equiv \,a \,(\mathrm{mod}\,q)}} p.$ Nov 27 Sara Sasani A Strongly Regular Decomposition of the Complete Graph and its Association Scheme at noon in C630 (University of Lethbridge) A Strongly Regular Graph, SRG$({\color{blue}\nu,\color{blue}k,\color{blue}\lambda,\color{blue}\mu})$, is a $\color{blue}k$-regular graph with $\color{blue}\nu$ vertices such that every two adjacent vertices have $\color{blue}\lambda$ common neighbors, and every two non-adjacent vertices have $\color{blue}\mu$ common neighbors. For each positive integer $m$, a construction for $\color{red}{2^m}$ disjoint SRG$({\color{blue}{2^{2m}(2^m+2)}},\color{blue}{2^{2m}+2^m}, \color{blue}{2^m},\color{blue}{2^m})$ will be shown to form a decomposition of the complete graph with $\color{blue}{2^{2m}(2^m+2)}$ vertices, if the cliques of size $\color{red}{2^{2m}}$ is considered as a strongly regular graph with parameter $(\color{blue}{2^{2m}(2^m+2)},\color{blue}{2^{2m}-1}, \color{blue}{2^{2m}-2},\color{blue}{0})$. By decomposing the cliques and the strongly regular graphs further, we show the existence of a symmetric association scheme with $\color{red}{2^{m+2}-2}$ classes and explain, by an example, how to find the first and second eigenmatrices of the scheme. Dec 4 Clifton Cunningham Title TBA at noon in C630 (University of Calgary) Abstract TBA
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