Department of Mathematics and Statistics

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

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.

Number Theory - Siddhi Pathak (Queen's University)

Wednesday, August 1st, 2018

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

Speaker: Siddhi Pathak (Queen's University)

Title: Adjoint of the Serre derivative, Poincaré series and special values of shifted Dirichlet series.

Abstract: The Serre derivative is a linear differential operator from the space of modular forms of weight k to the space of modular forms of weight k+2. In this talk, we compute the adjoint of the Serre derivative with respect to the Petersson inner product, as done by A. Kumar. Using this as a pretext, we will highlight the connection between the Serre derivative and special values of shifted Dirichlet series via the tool of Poincaré series.

Number Theory - Anup Dixit (Queen's University)

Tuesday, July 24th, 2018

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

Speaker: Anup Dixit (Queen's University)

Title: ON THE GENERALIZED BRAUER-SIEGEL THEOREM.

Abstract: For a number field $K/ \mathbb{Q}$, the class number $h_K$ captures how far the ring of integers of $K$ is from being a PID. The study of class numbers is an important theme in number theory. In order to understand how the class number varies upon varying the number field, Siegel showed that the class number times the regulator tends to infinity in any family of quadratic number fields. Brauer extended this result to family of Galois extensions over $\mathbb{Q}$. This is the Brauer-Siegel theorem. Recently, Tsfasman and Vladut conjectured a Brauer-Siegel statement for an asymptotically exact family of number fields. In this talk, we prove the classical Brauer-Siegel and the generalized version in several unknown cases. We also discuss some effective versions of Brauer-Siegel in the classical setting.

Number Theory - Kalyan Chakraborty (Harish-Chandra Research Institute)

Wednesday, July 18th, 2018

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

Speaker: Kalyan Chakraborty (Harish-Chandra Research Institute, Allahabad, India)

Title: ON CLASS NUMBER OF CERTAIN QUADRATIC FIELDS.

Abstract: Class number of a number field and in particular that of a quadratic number field, is one of the fundamental and mysterious objects in algebraic number theory and related topics. I will discuss some results concerning the divisibility of the class numbers of certain families of real (respectively, imaginary) quadratic fields in both qualitative and quantitative aspects. I will also look at the 3-rank of the ideal class groups of certain imaginary quadratic fields. The talk will be based on some recent works done along with my collaborators.

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

Wednesday, July 11th, 2018

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

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

Title: DIOPHANTINE M-TUPLES.

Abstract: Fix an integer n. A Diophantine m-tuple for n is a collection of integers a(1), ..., a(m) such that a(i)a(j)+n is a perfect square whenever i is unequal to j. The study of such tuples has a long and glorious history reaching as far back as the time of Diophantus in the second century C.E. I will survey this topic and report on some recent work with my NSERC USRA student, Riley Becker.

Number Theory - Siddhi Pathak (Queen's University)

Wednesday, July 4th, 2018

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

Speaker: Siddhi Pathak (Queen's University)

Title: THE SELBERG TRACE FORMULA.

Abstract: Let $G$ be a locally compact group and $\Gamma$ be a discrete subgroup of $G$ such that $\Gamma \backslash G$ is compact. In 1956, A. Selberg introduced the trace formula relating the traces of irreducible unitary representations of G on $L^2(\Gamma \backslash G)$ to orbital integrals.

In this talk, we will give a brief exposition on Tamagawa's formulation of the Selberg trace formula. Further, we will emphasize that the Poisson summation formula as well as the Frobenius reciprocity for finite groups are special cases of the Selberg trace formula.

Number Theory - Atul Dixit (IIT Gandhinagar, India)

Wednesday, June 27th, 2018

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

Speaker: Atul Dixit (IIT Gandhinagar, India)

Title: RAMANUJAN'S FORMULA FOR ODD ZETA VALUES AND SUBSEQUENT DEVELOPMENTS.

Abstract: While the Riemann zeta function $\zeta(s)$ at even positive integers is known to be always transcendental, the arithmetic nature of $\zeta(s)$ at odd positive integers remains mysterious with only $\zeta(3)$ known to be irrational, thanks to Ap\'{e}ry. In 1901, Lerch obtained a beautiful result involving $\zeta(2m+1), m$ odd, and an Eisenstein series, which implies that either $\zeta(2m+1)$ or the Eisenstein series is transcendental. In his famous notebooks, Srinivasa Ramanujan obtained a beautiful result generalizing that of Lerch. This result has had a tremendous impact on Mathematics with its applications in modular forms, computer science, to name a few. A brief historical survey on Ramanujan's formula will be given. We will then concentrate on the generalized Lambert series $\sum_{n=1}^{\infty}\frac{n^{N-2h}}{e^{n^{N}x}-1}$ studied by Kanemitsu, Tanigawa and Yoshimoto, which we recently found to be located on page $332$ of Ramanujan's Lost Notebook in a slightly more general form. In a joint work with Bibekananda Maji, we have extended a transformation of this series given by the above authors. The novel feature of this extension is that it not only gives Ramanujan's formula for $\zeta(2m+1)$ but also its beautiful new generalization linking $\zeta(2m+1)$ and $\zeta(2Nm+1), N\in\mathbb{N}$. We then discuss some results associated with a more general Lambert series $\displaystyle\sum_{n=1}^{\infty}\frac{n^{N-2h}\exp{(-an^{N}x)}}{1- \exp{(-n^{N}x)}}$ that we have obtained in a joint work with Bibekananda Maji, Rahul Kumar and Rajat Gupta. Applications of many of these formulas towards new transcendence results of Zudilin- and Rivoal-type are given.

Number Theory - Neha Prabhu (Queen's University)

Wednesday, June 20th, 2018

Time: 2:00-3:20p.m.  Place: Jeffery Hall 422

Speaker: Neha Prabhu (Queen's University)

Title: EXTREMAL PRIMES FOR NON-CM ELLIPTIC CURVES.

Abstract: For an elliptic curve E over the field of rationals, the distribution of the number of points on E modulo a prime p has been well-studied over the last few decades. A relatively recent study is that of extremal primes for a given curve E. These are the primes p of good reduction for which the number of points on E mod p is either maximal or minimal. If E is a curve with CM, an asymptotic for the number of extremal primes was determined by James and Pollack. The talk will discuss the non-CM case and focus on obtaining upper bounds. This is joint work with C. David, A. Gafni, A. Malik and C. Turnage-Butterbaugh

Number Theory - Francois Seguin

Wednesday, May 30th, 2018

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

Speaker: Francois Seguin

Title: The complementary two-variable problem.

Abstract: In light of our previous talks concerning the two-variable Artin conjecture, one could ask about a "reverse" or "complementary" analogue. During this talk, we will describe this analogue as well as present a proof due to Schinzel from 1960. We will then discuss how much the proof can be modified to obtain a more precise result.

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