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

Sep 18 Peng-Jie Wong Nearly supersolvable groups and Artin's conjecture

Sep 25 Muhammad Khan The contact graphs of totally separable packings

Oct 2 Andrew Fiori The average number of quadratic Frobenius pseudoprimes

Oct 16 Lee Troupe Normally Distributed Arithmetic Functions
(University of British Columbia)

Oct 23 Sam Broadbent,
Habiba Kadiri,
and Kirsten Wilk
Sharper bounds for Chebyshev functions $\boldsymbol{\theta(x)}$ and $\boldsymbol{\psi(x)}$

Oct 30 Akshaa Vatwani Variants of equidistribution in arithmetic progression
(University of Waterloo)

Nov 6 Károly Bezdek Bounds for totally separable translative packings in the plane
(University of Calgary)

Nov 20 Forrest Francis Euler's Function on Products of Primes in Progressions

Nov 27 Sara Sasani A Strongly Regular Decomposition of the Complete Graph and its Association Scheme

Dec 4 Clifton Cunningham Title TBA
(University of Calgary)

Past semesters: Fall F2007 F2008 F2009 F2010 F2011 F2012 F2013 F2014 F2015 F2016
Spring S2008 S2009 S2010 S2012 S2013 S2014 S2015 S2016 S2017
Click here for a PDF file of abstracts all talks back to 2007 (approx 280K).