Department of Mathematics and Statistics

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

Number Theory Seminar - Brad Rodgers (Queen's University)

Thursday, October 17th, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 422

Speaker: Brad Rodgers (Queen's University)

Title: Moments and pseudomoments of the Riemann zeta-function, pt. 2.

Abstract: In a previous talk we discussed moments of the Riemann zeta function and "pseudomoments", in which the zeta-function is replaced by a finite Dirichlet polynomial. In this second talk we will further discuss random multiplicative functions and a variant of conjecture of Helson, proved with averaging weights by Bondarenko, Heap, and Seip. I hope to reintroduce the fundamental notions, so that this talk can be followed by audience members who missed the previous talk.

Number Theory Seminar - M. Ram Murty (Queen's University)

Thursday, October 10th, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 422

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

Title: THE PALEY GRAPH CONJECTURE AND DIOPHANTINE TUPLES.

Abstract: Let $n$ be a fixed natural number. An $m$-tuple $(a(1), ..., a(m))$ is said to be a Diophantine $m$-tuple with property $D(n)$ if $a(i)a(j)+n$ is a perfect square for $i, j$ distinct and less than or equal to $m$. We will show that the Paley graph conjecture in graph theory implies that the number of such tuples is $O((log n)^c)$ for any $c>0$. This is joint work with Ahmet Guloglu.

Number Theory Seminar - Brad Rodgers (Queen's University)

Tuesday, September 24th, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Brad Rodgers (Queen's University)

Title: Moments and pseudomoments of the Riemann zeta-function.

Abstract: In this talk I will discuss moments of the Riemann zeta-function and "pseudomoments", in which the zeta-function is replaced by a finite Dirichlet polynomial. I hope to explain the connection to random multiplicative functions and if there is sufficient time discuss a conjecture of Helson (recently proved by Harper) along with a weighted version previously proved by Bondarenko, Heap, and Seip.

Number Theory Seminar - Anup Dixit (Queen's University)

Tuesday, September 17th, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Anup Dixit (Queen's University)

Title: On Euler-Kronecker constants and the prime number theorem.

Abstract: As a generalization of the Euler-Mascheroni constant $\gamma$, Y. Ihara defined the the Euler-Kronecker constant $\gamma_K$ attached to a number field $K/\mathbb{Q}$. Ihara conjectured that for a cyclotomic fields $K$, $\gamma_K >0$. This initiated the study of the bounds on $\gamma_K$ for cyclotomic fields. In this talk, we describe an application of these bounds to the error term in the prime number theorem for certain arithmetic progressions. This is joint work with Prof. M. Ram Murty.

Number Theory Seminar - Seoyoung Kim (Queen's University)

Tuesday, September 10th, 2019

Time: 4:30-5:30 p.m.  Place: Jeffery Hall 319

Speaker: Seoyoung Kim (Queen's University)

Title: The Sato-Tate conjecture and Nagao’s conjecture.

Abstract: Nagao’s conjecture relates the rank of an elliptic surface to a limit formula arising from a weighted average of fibral Frobenius traces, and it is further generalized for smooth irreducible projective surfaces by Hindry and Pacheco. We show that the Sato-Tate conjecture for abelian surfaces studied by Fit\'{e}, Kedlaya, Rotger, Sutherland implies Nagao’s conjecture for certain twist families hyperelliptic curves of genus 2. Moreover, one can relate analogous discussions for higher genus g to the non-vanishing result on the symmetric power $L$-functions, from which analogous proof will hold for certain cases.

Number Theory Seminar - M. Ram Murty (Queen's University)

Wednesday, August 7th, 2019

Time: 3:30-4:30 p.m.  Place: Jeffery Hall 319

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

Title: AN INTRODUCTION TO THE CIRCLE METHOD.

Abstract: In 1916, in their ground-breaking paper deriving an asymptotic formula for the partition function, Hardy and Ramanujan introduced a powerful method in analytic number theory called the circle method. In a series of subsequent papers, Hardy and Littlewood developed the method to study Waring's problem, Goldbach's conjecture and other additive questions in number theory. I will present a gentle introduction to this method and indicate how it can be used to solve Waring's problem following an arrangement due to Linnik.

Number Theory Seminar - Akshaa Vatwani (IIT Gandhinagar)

Wednesday, July 31st, 2019

Time: 3:30-4:30 p.m.  Place: Jeffery Hall 319

Speaker: Akshaa Vatwani (IIT Gandhinagar)

Title: The Voronoi Summation Formula for a generalized divisor function.

Abstract: In 1904, Vorono\" {\i} formulated a conjecture for arithmetical functions, which is a generalization of the Poisson summation formula. He was able to establish this conjecture for the divisor function $d(n)$, giving a relation between Bessel functions and weighted sums of the divisor function. Such formulas have since been generalized to various arithmetical functions. We report on joint work with Atul Dixit, Bibekananda Maji and A. Sankaranarayanan, giving such a formula for the generalized divisor function:
$$
\sigma_{k,z} (n) = \sum_{d^k |n }d^z, \qquad \qquad k\in \mathbb N, \, z\in \mathbb{C}.
$$

Number Theory Seminar - Francesco Cellarosi (Queen's University)

Wednesday, July 24th, 2019

Time: 3:30-4:30 p.m.  Place: Jeffery Hall 319

Speaker: Francesco Cellarosi (Queen's University)

Title: COMPUTING KRONECKER SYMBOLS IN THE UNIVERSAL COVER OF SL_2(R).

Abstract: The Kronecker symbol generalizes Jacobi and Legendre symbols, and is intimately connected to the theory of Dirichlet characters. For co-prime integers $c$ and $d$ such that $c \equiv 0 \bmod 4$ and $d \equiv 1 \bmod 4$, I will explain an algorithm to compute the Kronecker symbol $(c/d)$ using a free group inside the universal cover of SL_2(R). Joint work with J. Griffin.

Number Theory Seminar - Molly Liu (Queen's University)

Wednesday, July 17th, 2019

Time: 3:30-4:30 p.m.  Place: Jeffery Hall 319

Speaker: Molly Liu (Queen's University)

Title: Eigenvalues and diameter of regular graphs.

Abstract: In graph theory, a regular graph is a graph where each vertex has the same number of neighbours. In this talk, we will discuss eigenvalues and diameter of regular graphs.

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