Events

J.C. Saunders, Ben Gurion University of the Negev

We deal with various Diophantine equations involving the Euler totient function. In particular, for $a,b,c,m,n\in\mathbb{N}$ with $m\geq 2$ we study the equations $\varphi(ax^m)=\frac{b\cdot n!}{c}$ and $\varphi\left(\frac{b\cdot n!}{c}\right)=ax^m$ where $\varphi(x)$ is the Euler totient function. We also deal with similar equations involving Lucas sequences of the first kind and second kind, generalising the work of Luca and Stanica.

MC 5417

Ertan Elma, Department of Pure Mathematics, University of Waterloo

Let χ be a Dirichlet character modulo a prime number p ⩾ 3 and let \mathfrak a_χ:=(1-χ(-1))/2. Define the mean value
\begin{align*}
\mathcal{M}_{p}(s,\chi):=\frac{2}{p-1}\sum_{\substack{\psi \bmod p\\\psi(-1)=-1}}L(1,\psi)L(s,\chi\overline{\psi})
\end{align*}
for a complex number s such that s≠ 1 if \mathfrak a _χ=1.

Adrian Diaconu, University of Minnesota

I will begin by discussing some recent results (joint in part with Ian Whitehead) about moments of quadratic Dirichlet L-functions. While our understanding of these moments over number fields still remains largely elusive, their function field analogs are more tractable. The main focus will be to describe some partial results in the function field setting. (Joint work with Jonas Bergström, Dan Petersen and Craig Westerland.)

MC 5417

Arpita Kar, Queen's University

Kübra Benli, University of Georgia

Let $p$ be a prime number. For each positive integer $k\geq 2$, it is widely believed that the smallest prime that is a $k$th power residue modulo $p$ should be $O(p^{\epsilon})$, for any $\epsilon>0$. Elliott has proved that such a prime is at most $p^{\frac{k-1}{4}+\epsilon}$, for each $\epsilon>0$. In this talk we will discuss the distribution of the prime $k$th power residues modulo $p$ in the range $[1, p]$, with a more emphasis on the subrange $[1,p^{\frac{k-1}{4}+\epsilon}]$,  for $\epsilon>0$.

MC 5417

Anup Dixit, Queen's University

As a natural generalization of the Euler-Mascheroni constant $\gamma$, Ihara introduced the Euler-Kronecker constant $\gamma_K$ attached to a number field $K$. He conjectured that for cyclotomic fields $\mathbb{Q}(\zeta_n)$, this constant is always positive. In this talk, we will discuss a connection of this conjecture with the number of small prime factors of $p-1$, for primes $p$. Motivated by this, we also establish a localized Erdos-Kac type theorem in this context.

MC 5417

Jerry Wang, Department of Pure Mathematics, University of Waterloo

In this talk, we will discuss the squarefree sieve for discriminants of arbitrary polynomials. We will find that there are striking differences between odd and even degree polynomials, where additional techniques of counting on varieties will be required for the latter. This is joint work with Manjul Bhargava and Arul Shankar.

MC 5417

Angelos Koutsianas, University of British Columbia

In this talk, I will talk about Darmon's program and the resolution of the generalized Fermat equation of signature (p,p,5) using Frey hyperelliptic curves. This is joint work with Imin Chen (Simon Fraser University).

MC 5417

Sacha Mangerel, Centre de Recherches Mathématiques 

Siddhi Pathak, Penn State University