Department of Mathematics and Statistics

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

Number Theory - Siddhi Pathak (Queen's University)

Tuesday, October 16th, 2018

Time: 10:15 a.m.  Place: Jeffery Hall 422

Speaker: Siddhi Pathak (Queen's University)

Title: On the Euler-Kronecker constants.

Abstract: Values of the series $\sum_{n=1}^{\infty} A(n)/B(n)$ where A(X) and B(X) are polynomials with suitable conditions to ensure convergence, have been studied by several authors in the past. The evaluation of these series can be considered as a generalization of Euler's theorem evaluating $\zeta(2k)$. In this talk, we study elliptic analogues of these series.

Number Theory - Anup Dixit (Queen's University)

Tuesday, October 9th, 2018

Time: 10:00 a.m.  Place: Jeffery Hall 422

Speaker: Anup Dixit (Queen's University)

Title: On the Euler-Kronecker constants.

Abstract: In 2006, Y. Ihara introduced the Euler-Kronecker constant $\gamma_K$ attached to any number field $K$, which is a generalization of the Euler-Mascheroni constant. This constant surprisingly arises in several seemingly unrelated aspects of analytic number theory. Ihara studied this constant systematically and produced bounds on $gamma_K$ under GRH. In this talk, we prove unconditional bounds for $\gamma_K$ in some cases and discuss its connection to the Brauer-Siegel theorem.

Number Theory - Siddhi Pathak (Queen's University)

Tuesday, October 2nd, 2018

Time: 10:00 a.m.  Place: Jeffery Hall 422

Speaker: Siddhi Pathak (Queen's University)

Title: On the values of the Epstein zeta function.

Abstract: Given a positive definite binary quadratic form, $Q(X,Y)$, the Epstein zeta function attached to $Q$ is given by $Z_Q(s) = \sum_{m,n} Q(m,n)^{-s}$, where the sum is over all tuples $(m,n)$ in $\mathbb{Z} \times \mathbb{Z}$, excluding $(0,0)$. This series converges absolutely for $Re(s)>1$. In this talk, we will present a result by J. R. Smart that 'evaluates' $Z_Q(k)$ for positive integers $k > 1$.

Number Theory - Neha Prabhu (Queen's University)

Tuesday, September 25th, 2018

Time: 10:00 a.m.  Place: Jeffery Hall 422

Speaker: Neha Prabhu (Queen's University)

Title: The error term in the Sato-Tate theorem of Birch.

Abstract: In 1968, Birch proved a vertical analogue of the Sato-Tate conjecture for elliptic curves showing the asymptotic behaviour of $a_E(p)$, a quantity associated to an elliptic curve $E$ mod p. An error term for this result was obtained by Banks and Shparlinski in 2009 using the results of Katz and Neiderreiter. In this talk, we shall see that this error term can also be obtained in an elementary fashion using ideas in Birch's paper. This is joint work with Ram Murty.

Number Theory - Zhen Zhao (Queen's University)

Tuesday, September 18th, 2018

Time: 10:00 a.m.  Place: Jeffery Hall 422

Speaker: Zhen Zhao (Queen's University) 

Title: Asymptotics of the dimensions of irreducible representations of the symmetric group

Abstract: We will discuss the results of the 1985 paper of S. V. Kerov and A. M. Vershik giving the asymptotic rates growth for the dimensions of the largest and 'typical' dimensions of the irreducible representations of the symmetric group of order n.

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

Tuesday, September 11th, 2018

Time: 10:00 a.m.  Place: Jeffery Hall 422

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

Title: Some reflections on Hasse's inequality (Part 1)

Abstract: In 1935, Hasse proved the Riemann hypothesis for zeta functions attached to elliptic curves (mod p) which was originally conjectured in the 1922 doctoral thesis of Emil Artin. There are two papers, one by Davenport in 1931 and another by Mordell in 1933 that discuss elementary approaches to this conjecture that give suprisingly non-trivial estimates. Both papers make use of a clever averaging argument that later appears in Bombieri's proof of Weil's theorem on the Riemann hypothesis for curves. We will give a motivated (and somewhat leisurely) discourse on these developments.

Number Theory - Richard Gottesman (Queen's University)

Wednesday, August 29th, 2018

Time: 2:30-3:20p.m.  Place: Jeffery Hall 422

Speaker: Richard Gottesman (Queen's University)

Title: VECTOR VALUED MODULAR FORMS ON GAMMA_0 ( 2 )

Abstract: I will give an introduction to vector-valued modular forms and describe my research on the arithmetic of vector-valued modular forms with respect to a representation of Gamma_0(2). The collection of vector-valued modular forms form a graded module over the graded ring of modular forms. I will explain how understanding the structure of this module allows one to show that the component functions of vector-valued modular forms satisfy an ordinary differential equation whose coefficients are modular forms. In certain cases, we can use a Hauptmodul to transform such a differential equation into a Fuchsian differential equation on the project line minus three points. We then are able to use Gaussian hypergeometric series to explicitly solve this differential equation. Finally, we make use of these ideas together with some algebraic number theory to study the prime numbers that divide the denominators of the Fourier coefficients of the component functions of vector-valued modular forms.

Number Theory - Anup Dixit (Queen's University)

Wednesday, August 22nd, 2018

Time: 2:30-3:20p.m.  Place: Jeffery Hall 422

Speaker: Anup Dixit (Queen's University)

Title: ON THE UNIVERSALITY OF CERTAIN L-FUNCTIONS.

Abstract: In 1975, S. Voronin proved a fascinating result on Riemann zeta-function, which states that every non-vanishing holomorphic function on a compact set in the critical strip $1/2  \Re(s)  1$ is well approximated by vertical shifts of the zeta function, infinitely often. This is called the universality property of the Riemann zeta-function. This property can be shown for many familiar L-functions, for instance all L-functions in the Selberg class are known to be universal. Moreover, functions such as the Hurwitz zeta-function or Lerch zeta-function, which are not elements in the Selberg class also satisfy the universality property. This motivated Y. Linnik and I. Ibragimov to conjecture that every Dirichlet series, with has an analytic continuation and satisfies some "growth condition" must be universal. In this talk, we will formulate this conjecture more precisely and prove some partial results towards the conjecture.

Number Theory - Arpita Kar (Queen's University)

Tuesday, August 14th, 2018

Time: 2:30-3:20p.m.  Place: Jeffery Hall 422

Speaker: Arpita Kar (Queen's University)

Title: ON A THEOREM OF HARDY AND RAMANUJAN.

Abstract: In 1917, G.H Hardy and S. Ramanujan coined the phrase ``normal order" and proved that $\omega(n)$ has normal order $\log \log n$. (Here, $\omega(n)$ denotes the number of distinct prime factors of $n$.) In other words, they showed that $\omega(n) \approx \log \log n$ for all but $o(x)$ many integers $n \leq x$, as $x \to \infty$. In this talk, we will show that the size of this exceptional set is, in fact $O(\frac{x}{(\log x)^A)})$ for any $A>0$, improving upon the work of Hardy, Ramanujan and Tur\'an.

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