Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Number Theory Seminar

Number Theory - Chantal David (Concordia University)

Tuesday, April 23nrd, 2019

Time: 1:30-2:30 p.m.  Place: Jeffery Hall 319

Speaker: Chantal David (Concordia University)

Title: Moments of cubic Dirichlet twists over function fields (Joint work with A. Florea and M. Lalin.)

Abstract: We obtain an asymptotic formula for the mean value of $L$--functions associated to cubic characters over $\F_q[T]$. We solve this problem in the non-Kummer setting when $q \equiv 2 \pmod 3$ and in the Kummer case when $q \equiv 1 \pmod 3$. The proofs rely on evaluating averages of cubic Gauss sums over function fields, which can be done using the theory of metaplectic Eisenstein series. In the non-Kummer setting, we display some explicit cancellation between the main term and the dual term coming from the approximate functional equation of the $L$--functions.

Number Theory - Tariq Osman (Queen's University)

Tuesday, March 26th, 2019

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

Speaker: Tariq Osman (Queen's University)

Title: Counting Integer Points on Vinogradov's Quadric.

Abstract: Consider the variety defined by the pair of equations $x_1 + x_2 + x_3 = y_1 + y_2 + y_3$ and $ x_1^2 + x_2^2 + x_3^2 = y_1^2 + y_2^2 + y_3^2$, known as Vinogradov's quadric. Following a brief historical overview and a few motivational remarks, we will derive an asymptotic formula (due to Ragovskya) for the number of integer points on Vinogradov's quadric in a large box.

Number Theory - Allysa Lumley (York University)

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 $1/2<\sigma<1$.

Number Theory - Arpita Kar (Queen's University)

Tuesday, March 12th, 2019

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

Speaker: Arpita Kar (Queen's University)

Title: On the normal number of prime factors of Euler Phi function at shifts of prime arguments.

Abstract: Let $\omega(n)$ and $\Omega(n)$ denote the number of prime factors of a natural number $n$ counted without and with multiplicity respectively. Let $\phi(n)$ denote the Euler totient function. In 1984, R.Murty and K. Murty defined a certain class of multiplicative functions and computed the normal order of $\omega(f(p))$ and $\omega(f(n))$ for $f$ belonging in that class. An example of functions in this class is $\phi(n)$. In this talk, we will discuss the normal number of prime factors of $\phi(n)$ at shifts of prime arguments, that is, $\Omega(\phi(p+a))$, for primes $p$ and any non-zero integer $a$.
This is joint work with Prof. Ram Murty.

Number Theory - Brad Rodgers (Queen's University)

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.

Number Theory - Anup Dixit (Queen's University)

Tuesday, February 26th, 2019

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

Speaker: Anup Dixit (Queen's University)

Title: An extremal property of general Dirichlet series.

Abstract: A general Dirichlet series is given by $F(s)=\sum_{n=1}^{\infty} a_n/ \lambda_n^s$. Our goal is to realize $F(s)$ as a unique boundary element for a class of functions. In other words, is there a naturally occurring class of functions, for which $F(s)$ is the unique solution to an extremal problem? In this talk, we undertake this question and as a consequence show that an $L$-function satisfying a certain growth condition is uniquely determined by its degree, conductor, and its residue at $s=1$.

Number Theory - Siddhi Pathak (Queen's University)

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.

Number Theory - Ahmet Muhtar Guloglu (Bilkent University, Turkey)

Tuesday, February 5th, 2019

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

Speaker: Ahmet Muhtar Güloğlu (Bilkent University, Turkey)

Title: Cyclicity of elliptic curves modulo primes in arithmetic progressions (joint work with Yildirim Akbal).

Abstract: Let E be an elliptic curve defined over the rational numbers. We investigate the cyclicity of the group of F_p-rational points of the reduction of E modulo primes in a given arithmetic progression as a special case of Serre's Cyclicity Conjecture.

Number Theory - Neha Prabhu (Queen's University)

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.