Where:Zoom: (Please write to sfandnt@gmail.com for the link)
Live Link: https://youtu.be/vYs5YGuS_L4
Tea or Coffee: Please bring your own.
Abstract:
In this talk we shall discuss about modular forms and certain types of congruences among the Fourier coefficients of modular forms. We shall also discuss about the non-existence of Ramanujan-type congruences for certain modular forms.
Gaurav Bhatnagar (Ashoka), Atul Dixit (IIT, Gandhinagar) and Krishnan Rajkumar (JNU)
Where:Zoom: Please write to sfandnt@gmail.com for a link.
Live Link: https://youtu.be/o7qNW8BhgJI
Tea or Coffee: Please bring your own.
Abstract:
In the 1980's, Greene defined hypergeometric functions over finite fields using Jacobi sums. These functions possess many properties that are analogous to those of the classical hypergeometric series studied by Gauss, Kummer and others. These functions have played important roles in the study of supercongruences, the Eichler-Selberg trace formula, and zeta-functions of arithmetic varieties. We study the distribution (over large finite fields) of the values of certain families of these functions. For the $_2F_1$ functions, the limiting distribution is semicircular, whereas the distribution for the $_3F_2$ functions is the more exotic \it{Batman distribution.}
Gaurav Bhatnagar (Ashoka), Atul Dixit (IIT, Gandhinagar) and Krishnan Rajkumar (JNU)
The
next talk is by Howard Cohl of NIST. Please note the special time.
Howard is zooming in from California, and we are grateful to him to be
able to speak at a time suitable to us.
Talk Announcement:
Title:The utility of integral representations for the Askey-Wilson polynomials and their symmetric sub-families
Speaker: Howard Cohl (NIST)
When: Thursday, September 2, 2021 - 6:30 PM - 7:30 PM (IST) (6 am Pacific Day Time (PDT))
Where:Zoom: Please write to us for the link.
Live Link: https://youtu.be/0hPgarkEXdc
Tea or Coffee: Please bring your own.
Abstract:
The
Askey-Wilson polynomials are a class of orthogonal polynomials which
are symmetric in four free parameters which lie at the very top of the q-Askey
scheme of basic hypergeometric orthogonal polynomials. These
polynomials, and the polynomials in their subfamilies, are usually
defined in terms of their finite series representations which are given
in terms of terminating basic hypergeometric series. However, they also
have nonterminating, q-integral,
and integral representations. In this talk, we will explore some of
what is known about the symmetry of these representations and how they
have been used to compute their important properties such as generating
functions. This study led to an extension of interesting contour
integral representations of sums of nonterminating basic hypergeometric
functions initially studied by Bailey, Slater, Askey, Roy, Gasper and
Rahman. We will also discuss how these contour integrals are deeply
connected to the properties of the symmetric basic hypergeometric
orthogonal polynomials.
Gaurav Bhatnagar (Ashoka), Atul Dixit (IIT, Gandhinagar) and Krishnan Rajkumar (JNU)
Nikolai Sergeevich Koshliakov was an outstanding Russian mathematician who made phenomenal contributions to number theory and differential equations. In the aftermath of World War II, he was one among the many scientists who were arrested on fabricated charges and incarcerated. Under extreme hardships while still in prison, Koshliakov (under a different name `N. S. Sergeev') wrote two manuscripts out of which one was lost. Fortunately the second one was published in 1949 although, to the best of our knowledge, no one studied it until the last year when Prof. Atul Dixit and I started examining it in detail. This manuscript contains a complete theory of two interesting generalizations of the Riemann zeta function having their genesis in heat conduction and is truly a masterpiece! In this talk, we will discuss some of the contents of this manuscript and then proceed to give some new results (modular relations) that we have obtained in this theory. This is joint work with Prof. Atul Dixit.
As a natural generalization of the Euler's constant $\gamma$, Y. Ihara introduced the Euler-Kronecker constants attached to any number field. In this talk, we will discuss the connection between these constants and certain arithmetic properties of number fields.
Euler's
remarkable formula for $\zeta(2m)$ immediately tells us that even zeta
values are transcendental. However, the algebraic nature of odd zeta
values is yet to be determined. Page 320 and 332 of Ramanujan's
Lost Notebook contains an intriguing identity for $\zeta(2m+1)$ and
$\zeta(1/2)$, respectively. Many mathematicians have studied these
identities over the years.
In
this talk, we shall discuss transformation formulas for a certain
infinite series, which will enable us to derive Ramanujan's formula for
$\zeta(1/2),$ Wigert's formula for $\zeta(1/k)$, as well as Ramanujan's
formula for $\zeta(2m+1)$. We also obtain a new identity for
$\zeta(-1/2)$ in the spirit of Ramanujan.
Abstract: Finding
solutions of differential equations has been a problem in pure
mathematics since the invention of calculus by Newton and Leibniz in the
17th century. Bessel functions are solutions of a particular
differential equation, called Bessel’s equation. In classical analytic
number theory, there are several summation formulas or trace formulas
involving Bessel functions. Two prominent such are the Kuznetsov trace
formula and the Voronoi summation formula. In this talk, I will present some Voronoi type summation formulas and its application to Number theory.
Abstract:Theta series first appeared in Euler’s work on partitions, but was systematically studied later by Jacobi. In his Lost Notebook, Ramanujan wrote down many identities (without proof) involving the so-called partial theta series. Unlike the theta series which are modular forms, the theory of partial theta series is not well understood. In this talk, I will consider a family of partial theta series and show their “quantum modular” behaviour. This is based on my recent joint work with Robert Osburn (UCD).
The talk should be accessible to graduate and advanced undergraduate students.
The next speaker in our series is Siddhi Pathak, S. Chowla Research Assistant Professor, Penn State University. The talk announcement is below.
Talk Announcement:
Title:Special values of L-functions
Speaker: Siddhi Pathak (Penn State)
When: May 13, 2021 - 3:55 PM - 5:00 PM (IST)
Where:Zoom (the link will be sent by email to our list)
Live link: https://youtu.be/SXl9IPgE2aI
Tea or Coffee: Please bring your own.
Abstract:In 1730s, Euler resolved the famous Basel problem by evaluating values of the Riemann zeta-function at even positive integers as rational multiples of powers of pi. Thus, we recognize that the values \zeta(2k) are transcendental and algebraically dependent. The situation is drastically different for odd zeta-values, that are not only expected to be transcendental, but also algebraically independent. Although we are far from proving this, there has been striking progress in the work of Apery, and more recently by Ball-Rivoal, Zudilin and others. In this talk, we discuss the analogous problem for Dirichlet L-functions, more generally, Dirichlet series with periodic coefficients.
This talk will be accessible to graduate students.
