at noon
online

(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.

at noon
online

We describe a thin family of Dirichlet L-functions which have an irregular and perhaps unexpected behavior in their value distribution. This behavior has an arithmetic explanation and corresponds to the nonvanishing of a certain Gauss type sum. We give a complete classification of the characters for which these sums are nonzero and count the number of corresponding characters. It turns out that this Gauss type sum vanishes for 100% of primitive Dirichlet characters but there is an infinite (but zero density) subfamily of characters where the sum is nonzero.

Experimentally, this thin family of L-functions seems to have a significant and previously undetected bias in distribution of gaps between their zeros. After uncovering this bias, we re-examined the gaps between the zeros of the Riemann zeta-function and discovered an even more surprising phenomenon. If we list the gaps in increasing order and sum over arithmetic progressions of gaps, there seems to be a "Chebyshev-type" bias in the corresponding measures; the sum over certain arithmetic progressions of gaps are much larger than others! These observations seem to go well beyond the Random Matrix Theory model of L-functions.

This is joint work with Jonathan Bober and Zhenchao Ge.

at noon
online

(University of Lethbridge)

We obtain the asymptotic formula with an error term $O(X^{\frac{1}{2} + \varepsilon})$ for the smoothed first moment of quadratic twists of modular $L$-functions. The argument is largely based on a recursive method of Young (Acta Arith. 138(1):73–99, 2009 and Sel. Math. (N.S.) 19(2):509–543, 2013).

at noon
online

(University of Lethbridge)

A subset $X$ boundedly generates a group $G$ if every element $g$ of $G$ is the product of a bounded number of elements of $X$. This is a very powerful notion in abstract group theory, but geometric group theorists (and others) may also need a good bound on the norms of the elements of $X$ that are used. (We do not want to have to use large elements of $X$ to represent a small element of $G$.) In 1993, Lubotzky, Mozes, and Raghunathan proved an excellent result of this type for the case where $G$ is the group $\mathrm{SL}(n,\mathbb{Z})$ of $n$-by-$n$ matrices with integer entries and determinant one, and $X$ consists of the elements of the natural copies of $\mathrm{SL}(2,\mathbb{Z})$ in $G$. We will explain the proof of this result, and discuss a generalization to other arithmetic groups.

at noon
online

The ternary divisor function, denoted $\tau_3(n)$, counts the number of ways to write a natural number $n$ as an ordered product of three positive integers. Thus, $\sum_{n=1}^\infty \tau_3(n) n^{-s} = \zeta^3(s).$ Given two coprime positive integers $a$ and $q$, we study the distribution of $\tau_3$ in arithmetic progressions $n \equiv a \ (\text{mod } q).$ The distribution of $\tau_3$ in arithmetic progressions has a rich history and has applications to the distribution of prime numbers and moments of Dirichlet $L$-functions. We show that $\tau_3$ is equidistributed on average for moduli $q$ up to $X^{2/3}$, extending the individual estimate of Friedlander and Iwaniec (1985). We will also discuss an averaged variance of $\tau_3$ in arithmetic progressions related to a recent conjecture of Rodgers and Soundararajan (2018) about asymptotic of this variance. One of the key inputs to this asymptotic come from a modified additive correlation sum of $\tau_3$.

at noon
online

(University of Lethbridge)

In this talk I will report on some recent joint work with Cameron Franc (McMaster) where we investigate the structure of modular forms for the non-congruence groups of minimal index. In contrast to the case of congruence groups, little is known about the Fourier coefficients of even the Eisenstein series let alone more general forms. One way in which non-congruence forms differ from congruence forms is that when their Fourier coefficients are algebraic the denominators are conjectured to diverge to infinity. We will demonstrate that this conjecture holds for the index 7 subgroups of $\mathrm{PSL}_2(\mathbb{Z})$ while highlighting some questions which remain open about the coefficients of Eisenstein series.

at noon
online

(University of Lethbridge)

For non-negative integers $a$, $b$, and $n$, let $t(a,b;n)$ be the number of representations of $n$ as a sum of triangular numbers with coefficients $1$ and $3$ ($a$ of ones and $b$ of threes) and let $r(a, b; n)$ be the number of representations of $n$ as a sum of squares with coefficients $1$ and $3$ ($a$ of ones and $b$ of threes). It is known that for $a$ and $b$ satisfying $1\leq a+3b \leq 7$, we have $$ t(a,b;{n}) = \frac{2}{2+{a\choose4}+ab} r(a,b;8n+a+3b) $$ and for $a$ and $b$ satisfying $a+3b=8$, we have $$ t(a,b;{n}) = \frac{2}{2+{a\choose4}+ab} \left( r(a,b;8n+a+3b) - r(a,b; (8n+a+3b)/4) \right). $$ Such identities are not known for $a+3b>8$. We report on our joint work with Zafer Selcuk Aygin (University of Calgary) in which, for general $a$ and $b$ with $a+b$ even, we prove asymptotic equivalence of formulas similar to the above, as $n\rightarrow\infty$. One of our main results extends a theorem of Bateman, Datskovsky, and Knopp where the case $b=0$ and general $a$ was considered. Our approach is different from Bateman-Datskovsky-Knopp's proof where the circle method and singular series were used. We achieve our results by explicitly computing the Eisenstein components of the generating functions of $t(a,b;n)$ and $r(a,b;n)$.

at noon
online

The Titchmarsh divisor problem asks for an asymptotic evaluation of the average of the divisor function evaluated at shifted primes. We will discuss how strong error terms that are uniform in the shift parameters could be obtained using spectral theory of automorphic forms. We will also discuss the automorphic analogue of the Titchmarsh divisor problem. This is a joint work with E. Assing and V. Blomer.

at noon
online

In this talk, we discuss the asymptotic behavior of the number of ways to write a given positive integer as a sum of primes powers concerning a Chebotarev condition. In special cases, this reduces to the study of partitions into primes in arithmetic progressions. While this study for ordinary partitions goes back to Hardy and Ramanujan, partitions into primes were recently re-visited by Vaughan. Our error term is sharp and improves on previous known estimates in the special case of primes as parts of the partition. As an application, monotonicity of this partition function is established explicitly via an asymptotic formula in connection to a result of Bateman and Erdos.

at noon
online

(University of Lethbridge)

It turns out that homology of finite graphs is trivial, however when it comes to infinite graphs, it gets interesting. The problem of defining cycles in infinite graphs has attracted so many researchers. Diestel and Kühn have proposed viewing a graph as 1-complex, and defining a topology on the point set of the graph together with its ends. In this setting, a circle in the graph is a homeomorph of the unit circle $S^1$ in this topological space. For locally finite graphs this setting appears to be natural, as many classical theorems on cycles in finite graphs extend to the infinite setting. In this talk, we discuss this approach in details and also define a new notion for Hamiltonicity of infinite graphs. Depending on time constraint, we address the Lovász conjecture for infinite groups.