Study on asymptotics of modified Bessel functions dates back to 18th century. In this talk, we will describe how from the study of asymptotics of modified Bessel function of first kind of non-negative order, one can comes up with a host of inequalities that finally leads to answer combinatorial properties, for example log-concavity, higher order Turán inequality of certain arithmetic sequences arising from Fourier coefficients of modular forms. In addition to that, we will discuss briefly on a result of Bringmann et al. and analyze with the work addressed above.

Where:Zoom(please write to the organizers for the link)

Live Link: https://youtu.be/nFWdn5rNMa4

Tea or Coffee: Please bring your own.

Abstract:

We will obtain large values of the argument of the Riemann zeta function using the resonance method. We will also apply the method to the iterated arguments. This is a joint work with A. Chirre.

It is well known that the prime numbers are equidistributed in arithmetic progressions. Such a phenomenon is also observed more generally for a class of arithmetic functions. A key result in this context is the Bombieri-Vinogradov theorem which establishes that the primes are equidistributed in arithmetic progressions ``on average" for moduli in the range for any . In , building on an idea of Maier, Friedlander and Granville showed that such equidistribution results fail if the range of the moduli is extended to for any . We discuss variants of this result and give some applications. This is joint work with Aditi Savalia.

In this talk, we construct elliptic analogues of the rook numbers and file numbers by attaching elliptic weights to the cells in a board. We show that our elliptic rook and file numbers satisfy elliptic extensions of corresponding factorization theorems which in the classical case was established by Goldman, Joichi and White and by Garsia and Remmel in the file number case. This factorization theorem can be used to define elliptic analogues of various kinds of Stirling numbers of the first and second kind, and Abel numbers.

We also give analogous results for matchings of graphs, elliptically extending the result of Haglund and Remmel.

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.

The
six Painlevé equations, whose solutions are called the Painlevé
transcendents, were derived by Painlevé and his colleagues in the
late 19th and early 20th centuries in a classification of second order
ordinary differential equations whose solutions have no movable critical
points. In the 18th and 19th centuries, the classical special
functions such as Bessel, Airy, Legendre and hypergeometric functions,
were recognized and developed in response to the problems of the day in
electromagnetism, acoustics, hydrodynamics, elasticity and many other
areas. Around the middle of the 20th century, as science and
engineering continued to expand in new directions, a new class of
functions, the Painlevé functions, started to appear in
applications. The list of problems now known to be described by the
Painlevé equations is large, varied and expanding rapidly. The list
includes, at one end, the scattering of neutrons off heavy nuclei, and
at the other, the distribution of the zeros of the Riemann-zeta function
on the critical line $\mbox{Re}(z) =\tfrac12$. Amongst many others,
there is random matrix theory, the asymptotic theory of orthogonal
polynomials, self-similar solutions of integrable equations,
combinatorial problems such as the longest increasing subsequence
problem, tiling problems, multivariate statistics in the important
asymptotic regime where the number of variables and the number of
samples are comparable and large, and also random growth problems.

The
Painlevé equations possess a plethora of interesting properties
including a Hamiltonian structure and associated isomonodromy problems,
which express the Painlevé equations as the compatibility condition
of two linear systems. Solutions of the Painlevé equations have some
interesting asymptotics which are useful in applications. They possess
hierarchies of rational solutions and one-parameter families of
solutions expressible in terms of the classical special functions, for
special values of the parameters. Further the Painlevé equations
admit symmetries under affine Weyl groups which are related to the
associated Bäcklund transformations.

In this talk I shall
discuss special polynomials associated with rational solutions of
Painlevé equations. Although the general solutions of the six
Painlevé equations are transcendental, all except the first
Painlevé equation possess rational solutions for certain values of
the parameters. These solutions are expressed in terms of special
polynomials. The roots of these special polynomials are highly symmetric
in the complex plane and speculated to be of interest to number
theorists. The polynomials arise in applications such as random matrix
theory, vortex dynamics, in supersymmetric quantum mechanics, as
coefficients of recurrence relations for semi-classical orthogonal
polynomials and are examples of exceptional orthogonal polynomials.

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.