## Saturday, November 20, 2021

### Koustav Banerjee (RISC, JKU, Linz) - Thursday Nov 25, 2021 - 3:55 PM (IST)

The final talk of the year will be by Koustav Banerjee of RISC. After this we will take a break and return next year.

Talk Announcement:

Title: Inequalities for the modified Bessel function of first kind and its consequences.

Speaker: Koustav Banerjee (RISC, Johann Kepler University, Linz, Austria)

When: Thursday, Nov 25, 2021 - 4:00 PM - 5:00 PM (IST)

Where: Zoom:

Tea or Coffee: Please bring your own.

Abstract:

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.

## Friday, November 5, 2021

### Kamalakshya Mahatab (CMI, Chennai) - Thursday, Nov 11, 2021 - 3:55 - 5:00 PM (IST)

Dear all,

The next speaker in our seminar is Kamalakshya Mahatab of CMI, Chennai

Talk Announcement:

Title: Large oscillations of the argument of the Riemann zeta function.

Speaker: Kamalakshya Mahatab (CMI, Chennai)

When: Thursday, Nov 11, 2021 - 4:00 PM - 5:00 PM (IST)

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.

## Sunday, October 24, 2021

### Akshaa Vatwani (IIT, Gandhinagar) - Thursday October 28, 3:55 PM - 5:00 PM (IST)

Dear all,

The next speaker in our seminar is Akshaa Vatwani of IIT, Gandhinagar

Talk Announcement:

Title: Limitations to equidistribution in arithmetic progressions

Speaker: Akshaa Vatwani (IIT, Gandhinagar)

When: Thursday, October 28, 2021 - 4:00 PM - 5:00 PM (IST)

Where: Zoom:

Tea or Coffee: Please bring your own.

Abstract:

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 $q$ in the range $q \le x^{1/2 -\epsilon }$ for any $\epsilon>0$. In $1989$, building on an idea of Maier, Friedlander and Granville showed that such equidistribution results fail if the range of the moduli $q$ is extended to $q \le x/ (\log x)^B$  for any $B>1$. We discuss variants of this result and give some applications. This is joint work with Aditi Savalia.

## Sunday, October 10, 2021

### Meesue Yoo (Chungbuk National University, Korea) - Thursday, October 14, 2021 - 3:55 PM (IST)

The next speaker in our seminar is Meesue Yoo of Chungbuk National University, Korea.

Talk Announcement:

Title: Elliptic rook and file numbers

Speaker: Meesue Yoo (Chungbuk National University, Korea)

When: Thursday, October 14, 2021 - 4:00 PM - 5:00 PM (IST)  (7:30 PM in S. Korea)

Where: Zoom: Please write to the organizers if you don't have the link.

Tea or Coffee: Please bring your own.

Abstract:

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.

This is joint work with Michael Schlosser.

## Monday, September 27, 2021

### Jaban Meher (NISER, Bhubaneswar) - Thursday Sept 30, 2021; 3:55-5:00 PM

Dear all,

The next talk is by Jaban Meher of NISER, Bhubaneswar. The announcement is as follows.

Talk Announcement:

Title: Modular forms and certain congruences

Speaker: Jaban Meher (NISER, Bhubaneswar)

When: Thursday, September 30, 2021 - 4:00 PM - 5:00 PM (IST)

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)

sfandnt@gmail.com

## Thursday, September 16, 2021

### Neelam Saikia (University of Virginia), Thursday, September, 16, 2021 - 3:55 PM (IST)

Dear all,

The next talk is by Neelam Saikia who is currently a post-doc with Ken Ono at the University of Virginia.

Talk Announcement:

Title: Frobenius trace distributions for Gaussian hypergeometric functions

Speaker: Neelam Saikia (University of Virginia)

When: Thursday, September 16, 2021 - 4:00 PM - 5:00 PM (IST)

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)

sfandnt@gmail.com

## Saturday, August 28, 2021

### Howard Cohl (NIST) Thursday, Sept 2, 2021, 6:30 PM IST

Dear all,

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)

sfandnt@gmail.com

## Thursday, August 19, 2021

### Rajat Gupta (IIT, Gandhinagar) - Thursday, Aug 19, 2021 - 3:55 PM - 5:00 PM (IST)

The next talk is by Rajat Gupta -- or shall we say Dr. Rajat Gupta!

Here is the announcement.

Talk Announcement:

Title: Koshliakov zeta functions and modular relations
Speaker: Rajat Gupta (IIT, Gandhinagar)

When: Thursday, Aug 19, 2021 - 3:55 PM - 5:00 PM (IST)

Where: Zoom  (Please write at sfandnt@gmail.com for a link)

Tea or Coffee: Please bring your own.

Abstract:

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.

## Thursday, August 5, 2021

### Peter A. Clarkson (University of Kent, UK) - Thursday, Aug 5, 2021 - 3:55 PM - 5:00 PM (IST)

The next speaker in our seminar is Professor Peter Clarkson of the University of Kent, Canterbury, UK.

Here is the announcement.

Talk Announcement:

Title: Special polynomials associated with the Painlevé equations
Speaker: Peter A. Clarkson (University of Kent, UK)

When: Thursday, Aug 5, 2021 - 3:55 PM - 5:00 PM (IST)

Where: Zoom  (Please write at sfandnt@gmail.com for a link)

Tea or Coffee: Please bring your own.

Abstract:

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.

## Saturday, July 17, 2021

### Anup Dixit (IMSc, Chennai) - July 22, 2021 - 3:55 PM - 5:00 PM (IST)

The next speaker in the SF and NT Seminar is Anup Biswanath Dixit of IMSc. (Chennai). Here is the announcement.

Talk Announcement:

Title: On Euler-Kronecker constants and the class number problem
Speaker: Anup Dixit (IMSc, Chennai)

When: Thursday, July 22, 2021 - 3:55 PM - 5:00 PM (IST)

Where: Zoom (please send email to sfandnt@gmail.com for a link)

Tea or Coffee: Please bring your own.

Abstract:

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.

## Thursday, July 8, 2021

### Bibekananda Maji (IIT, Indore) - July 8, 2021 - 3:55 PM - 5:00 PM (IST)

Dear all,

The next talk is by Bibekananda Maji of IIT, Indore. Here is the announcement.

Talk Announcement:

Title: On Ramanujan's formula for $\zeta(1/2)$ and $\zeta(2m+1)$

Speaker: Bibekananda Maji (IIT, Indore)

When: Thursday, July 8, 2021 - 3:55 PM - 5:00 PM (IST)

Where: Zoom (please send email to sfandnt@gmail.com for a link)

Tea or Coffee: Please bring your own.

Abstract:
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.

This is joint work with Anushree Gupta.

## Saturday, June 19, 2021

### Debika Banerjee (IIIT, Delhi) 24 June 2021 - 3:55 PM - 5:00 PM (IST)

Our next speaker is Debika Banerjee from IIIT, Delhi. Her talk is on a special function and its application to number theory. Really!

Talk Announcement:

Title: Bessel functions and their application to classical number theory

Speaker: Debika Banerjee (Indraprastha Institute of Information Technology (IIIT), Delhi)

When: June 24, 2021 - 3:55 PM - 5:00 PM (IST)

Where: Zoom (Please write to sfandnt@gmail.com for the link)

Tea or Coffee: Please bring your own.

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.

## Thursday, June 10, 2021

### Ritabrata Munshi (ISI, Kolkata) - 10 June 2021 - 3:55 - 5:00 PM (IST)

The next speaker in our Seminar is Professor Ritabrata Munshi of ISI, Kolkata. The talk announcement is as follows.

Talk Announcement:

Title: 100 years of sub-convexity

Speaker: Ritabrata Munshi (ISI, Kolkata)

When: June 10, 2021 - 3:55 PM - 5:00 PM (IST)

Where: Zoom (the link will be sent by email to our list)

Live link:

Tea or Coffee: Please bring your own.

Abstract:  I will present a historical survey of the sub-convexity problem.