Tuesday, April 2nd, 2019

Time: 1:00-2:00 p.m.  Place: Jeffery Hall 422

Speaker: M. Ram Murty (Queen's University)

Title: Some reflections on the Riemann Hypothesis.

Abstract: We will present some musings relating the Riemann hypothesis to probability theory.

Tuesday, March 19th, 2019

Time: 1:00-2:00 p.m.  Place: Jeffery Hall 422

Speaker: Allysa Lumley (York University)

Title: Complex moments and the distribution of values of $L(1, \chi_D)$ over function fields with applications to class numbers.

Abstract: In 1992, Hoffstein and Rosen proved a function field analogue to Gau\ss' conjecture (proven by Siegel) regarding the class number, $h_D$, of a discriminant $D$ by averaging over all polynomials with a fixed degree. In this case $h_D=|\text{Pic}(\mathcal{O}_D)|$, where $\text{Pic}(\mathcal{O}_D)$ is the Picard group of $\mathcal{O}_D$. Andrade later considered the average value of $h_D$, where $D$ is monic, squarefree and its degree $2g+1$ varies. He achieved these results by calculating the first moment of $L(1,\chi_D)$ in combination with Artin's formula relating $L(1,\chi_D)$ and $h_D$. Later, Jung averaged $L(1,\chi_D)$ over monic, squarefree polynomials with degree $2g+2$ varying. Making use of the second case of Artin's formula he gives results about $h_DR_D$, where $R_D$ is the regulator of $\mathcal{O}_D$.

For this talk we discuss the complex moments of $L(1,\chi_D)$, with $D$ monic, squarefree and degree $n$ varying. Using this information we can describe the distribution of values of $L(1,\chi_D)$ and after specializing to $n=2g+1$ we give results about $h_D$ and specializing to $n=2g+2$ we give results about $h_DR_D$.

If time permits, we will discuss similar results for $L(\sigma,\chi_D)$ with $1/2<\sigma<1$.

Tuesday, March 5th, 2019

Time: 1:00-2:00 p.m.  Place: Jeffery Hall 422

Speaker: Brad Rodgers (Queen's University)

Title: The distribution of traces of powers of matrices over finite fields.

Abstract: Consider a random N by N unitary matrix chosen according to Haar measure. A classical result of Diaconis and Shashahani shows that traces of low powers of this matrix tend in distribution to independent centered gaussians as N grows. A result of Johansson shows that this convergence is very fast -- superexponential in fact. Similar results hold for other classical compact groups. This talk will discuss analogues of these results for N by N matrices taken from a classical group over a finite field, showing that as N grows traces of powers of these matrices equidistribute superexponentially. A little surprisingly, the proof is connected to the distribution in short intervals of certain arithmetic functions in F_q[T]. This is joint work with O. Gorodetsky.

Tuesday, February 12th, 2019

Time: 1:00-2:00 p.m.  Place: Jeffery Hall 422

Speaker: Siddhi Pathak (Queen's University)

Title: Distribution of special values of $L$-series attached to Erdos functions.

Abstract: Inspired by Dirichlet's theorem that $L(1,\chi) \neq 0$ for a non-principal Dirichlet character $\chi$, S. Chowla initiated the study of non-vanishing of $L(1,f)$ for a rational valued, $q$-periodic arithmetical function $f$. In this context, Erdos conjectured that $L(1,f) \neq 0$ when $f$ takes values in $\{ -1, 1\}$. This conjecture remains open in the case $q \equiv 1 \bmod 4$ or when $q > 2 \phi(q) + 1$. In this talk, we discuss a density theoretic approach towards this conjecture and the distribution of these values.

Tuesday, January 29th, 2019

Time: 1:00-2:00 p.m.  Place: Jeffery Hall 422

Speaker: Neha Prabhu (Queen's University)

Title: A probabilistic approach to analytic number theory.

Abstract: In 2005, Allan Gut showed that the distribution of a certain random variable, constructed using a zeta-distributed random variable, is compound Poisson. Exploiting this property, he reproved some well-known facts about the Riemann zeta function, and Selberg’s identity, using probability theory. In this talk, I present these results.