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)

 

Live Link: https://youtu.be/_GSUW18gSo4

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.

 

 

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)

 

Live link: https://youtu.be/zWksTp6g0rw

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.

 

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: https://youtu.be/ONLnzklDDeA

Tea or Coffee: Please bring your own.

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

Ankush Goswami (IIT, Gandhinagar) - May 27, 2021 - 3:55 -5:00 PM (IST)

 

The next speaker in our series is Ankush Goswami of IIT, Gandhinagar. Until recently, Ankush was a post-doc in Linz (Austria).

Talk Announcement:

Title: Partial theta series with periodic coefficients and quantum modular forms

Speaker: Ankush Goswami (IIT, Gandhinagar)

 

When: May 27, 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/W5av3JiovnI

Tea or Coffee: Please bring your own.

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.

Siddhi Pathak - May 13, 2021 - 3:55 -5:00 PM (IST)

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.

R Balasubramanian (I.M.Sc., Chennai) April 29, 2021 - 3:55 PM - 5:00 PM (IST)

 

Talk Announcement:

Title: Hardy's approximation to the Riemann Zeta Function

Speaker: Professor R Balasubramanian (IMSc, Chennai)

When: April 29, 2021 - 3:55 PM - 5:00 PM (IST)

Where: Zoom. 

 

Live link: https://youtu.be/53c6ow5IQTk 

Tea or Coffee: Please bring your own.

Link to slides
 

 

Nayandeep Deka Baruah (Tezpur University) - April 15, 2021 - 3:55 PM-5:00 PM (IST)

 

Our next speaker is Nayandeep Deka Baruah of Tezpur University, Assam, India.

Talk Announcement:

Title: Matching coefficients in the series expansions of certain $q$-products and their inverses.

Speaker: Nayandeep Deka Baruah (Tezpur University)

When: April 15, 2021 - 3:55 PM - 5:00 PM (IST)

Where: Zoom: Please write to sf-and-nt@gmail.com for the link.

Tea or Coffee: Please bring your own.

Abstract:We show that the series expansions of certain  $q$-products have \textit{matching coefficients} with their inverses. Several of the results are associated to Ramanujan's continued fractions. For example, let $R(q)$ denote the Rogers-Ramanujan continued fraction having the well-known $q$-product repesentation $R(q)=\left(q,q^4;q^5\right)_{\infty}/\left(q^2,q^3;q^5\right)_{\infty}$. If
\begin{align*}
\sum_{n=0}^{\infty}\alpha(n)q^n=\dfrac{1}{R^5\left(q\right)}=\left(\sum_{n=0}^{\infty}\alpha^{\prime}(n)q^n\right)^{-1},\\
\sum_{n=0}^{\infty}\beta(n)q^n=\dfrac{R(q)}{R\left(q^{16}\right)}=\left(\sum_{n=0}^{\infty}\beta^{\prime}(n)q^n\right)^{-1},
\end{align*}
then
\begin{align*}
\alpha(5n+r)&=-\alpha^{\prime}(5n+r-2), \quad r\in\{3,4\}; \\ \text{and} & \\
\beta(10n+r)&=-\beta^{\prime}(10n+r-6), \quad r\in\{7,9\}.
\end{align*}
This is a joint work with Hirakjyoti Das.

Shishuo Fu (Chongqing University, PRC) - April 1, 2021 - 3:55 PM - 5:00 PM (IST)

The next speaker in our seminar is Shishuo Fu of Chongqing University, PRC. It may be Fool's day, but we're not kidding. It really is Shishuo who has consented to give a talk all the way from China!

The live broadcast did not work as anticipated in the previous talk; I hope it works this time. At any rate, its best to try and come for the zoom session. 

Talk Announcement

Title: Bijective recurrences for Schroeder triangles and Comtet statistics

Speaker: Shishuo Fu (Chongqing University, PRC)

 

When: April 1, 2021 - 3:55 PM - 5:00 PM (IST)

 

Where: Zoom:  Please write to sfandnt@gmail.com for a link 

 

Tea or Coffee: Please bring your own.

Abstract:

In this talk, we bijectively establish recurrence relations for two triangular arrays, relying on their interpretations in terms of Schroeder paths (resp. little Schroeder paths) with given length and number of hills. The row sums of these two triangles produce the large (resp. little) Schroeder numbers. On the other hand, it is well-known that the large Schroeder numbers also enumerate separable permutations. This propelled us to reveal the connection with a lesser-known permutation statistic, called initial ascending run (iar), whose distribution on separable permutations is shown to be given by the first triangle as well. A by-product of this result is that "iar" is equidistributed over separable permutations with "comp", the number of components of a permutation. We call such statistics Comtet and we briefly mention further work concerning Comtet statistics on various classes of pattern avoiding permutations. The talk is based on joint work with Zhicong Lin and Yaling Wang.

 

Christian Krattenthaler (University of Vienna) March 18, 2021 - 3:55 PM-5:00 PM (IST)

The next talk is by Christian Krattenthaler. I hope this time the live broadcast works. Here is the announcement.

Talk announcement

Title: Determinant identities for moments of orthogonal polynomials

Speaker: Christian Krattenthaler (University of Vienna, Austria)

When: March 18, 2021 - 3:55 PM - 5:00 PM (IST)

 

Where: Zoom:  Please write to sfandnt@gmail.com for a link

Tea or Coffee: Please bring your own.

Abstract: We present a formula that expresses the Hankel determinants of a linear combination of length d+1 of moments of orthogonal polynomials in terms of a d x d determinant of the orthogonal polynomials. As a literature search revealed, this formula exists somehow hidden in the folklore of the theory of orthogonal polynomials as it is related to "Christoffel's theorem". In any case, it deserves to be better known and be presented correctly and with full proof. (During the talk I will explain the meaning of these somewhat cryptic formulations.) Subsequently, I will show an application of the formula. I will close the talk by presenting a generalisation that is inspired by Uvarov's formula for the orthogonal polynomials of rationally related densities.

Announcement: Atul Dixit (IIT, Gandhinagar) awarded the 2021 Gabor Szego award by OPSF SIAM

Dear all,

We are happy to report that Atul Dixit, one of the co-organizers of this seminar, has been awarded the 2021 Gábor Szegö Prize. This prize is awarded every two years by the SIAM Activity Group on Orthogonal Polynomials and Special Functions (SIAG/OPSF). It is awarded to an early-career researcher for outstanding research contributions within 10 years of obtaining a Ph.D.

 

 

The selection committee for the 2021 award consists of Peter Clarkson (Chair), University of Kent; Kerstin Jordaan, University of South Africa; Adri Olde Daalhuis, The University of Edinburgh; Sarah Post, University of Hawaii; and Yuan Xu, University of Oregon.

 

The selection committee in its letter to him cited his “impressive scientific work solving problems related to number theory using special functions, in particular related to the work of Ramanujan.”

Atul obtained his Ph.D. under the direction of Bruce Berndt in 2012 from the University of Illinois at Urbana-Champaign. Subsequently he did a post-doc at Tulane with Victor Moll as his mentor. Currently, he is in IIT, Gandhinagar and has quickly developed a reputation among young and upcoming mathematicians in this country that has attracted a bright set of Ph.D. students and post-docs to his team. 

We wish Atul continued success, both personally and for the group he is leading. 

Gaurav Bhatnagar and Krishnan Rajkumar (co-organizers with Atul of this seminar). 

Links.

1. Atul's talk in this seminar earlier.
   
2. About the prize. 
3. Atul Dixit's home page
   

Liuquan Wang (Wuhan University) March 4, 2021 - 3:55 PM (IST)

The next talk is by Liuquan Wang of Wuhan University.

Talk announcement

Title: Parity of coefficients of mock theta functions

Speaker: Liuquan Wang (Wuhan University, PRC)

When: March 4, 2021 - 3:55 PM - 5:00 PM (IST)

 

Where: Zoom:  Please write to sfandnt@gmail.com for a link or watch here:

https://youtu.be/w6sS7n3gEXI

Tea or Coffee: Please bring your own.

Abstract: We study the parity of coefficients of classical mock theta functions. Suppose $g$ is a formal power series with integer coefficients, and let $c(g;n)$ be the coefficient of $q^n$ in its series expansion. We say that $g$ is of parity type $(a,1-a)$ if $c(g;n)$ takes even values with probability $a$ for $n\geq 0$. We show that among the 44 classical mock theta functions, 21 of them are of parity type $(1,0)$. We further conjecture that 19 mock theta functions are of parity type $(\frac{1}{2},\frac{1}{2})$ and 4 functions are of parity type $(\frac{3}{4},\frac{1}{4})$. We also give characterizations of $n$ such that $c(g;n)$ is odd for the mock theta functions of parity type $(1,0)$.

 

 

Gaurav Bhatnagar (Ashoka) February 18, 2021 - 3:55 PM-5:00 PM

 

Talk announcement

Title: The Partition-Frequency Enumeration Matrix

Speaker: Gaurav Bhatnagar (Ashoka University)

When: February 18, 2021 - 3:55 PM - 5:00 PM (IST)

 

Where: Google Meet:  Please write to sfandnt@gmail.com for a link.

Tea or Coffee: Please bring your own. 

ABSTRACT

We develop a calculus that gives an elementary approach to enumerate partition-like objects using an infinite upper-triangular number-theoretic matrix. We call this matrix the Partition-Frequency Enumeration (PFE) matrix. This matrix unifies a large number of results connecting number-theoretic functions to partition-type functions. The calculus is extended to arbitrary generating functions, and functions with Weierstrass products. As a by-product, we recover (and extend) some well-known recurrence relations for many number-theoretic functions, including the sum of divisors function, Ramanujan's $\tau$ function, sums of squares and triangular numbers, and for $\zeta(2n)$, where $n$ is a positive integer. These include classical results due to Euler, Ramanujan, and others.  As one application, we embed Ramanujan's famous congruences $p(5n+4)\equiv 0\;$ (mod $5)$ and $\tau(5n+5)\equiv 0\; $ (mod $5)$ into an infinite family of such congruences.

This is joint work with Hartosh Singh Bal. 

Victor Moll (Tulane) - February 4, 2021 -3:55 PM - 5:00 PM (IST)

The next talk will be by Victor Moll of Tulane University. This will be the first mathematician from the US giving a talk in our seminar. Victor has kindly consented to stay awake to make his talk more suitable for Indian timings. But in future, we do expect speakers from the US will speak at times later at night (IST). 

At any rate, we hope more speakers from the US will give talks in our seminar. As we have mentioned earlier, our website now contains video recordings of the presentations. This makes it more convenient for the US participants to view the talks.

Talk Announcement

Title:  Valuations of interesting sequences

Speaker: Victor Moll (Tulane)

When: February 4, 2021 - 3:55 PM - 5:00 PM (IST)

Where: Google Meet; Please write to sfandnt@gmail.com if you want a link.

Tea or Coffee: Please bring your own. 

ABSTRACT

Given a sequence ${ a_{n} }$ of integers and a prime $p$, the sequence of

valuation $\nu_{p}(a_{n})$ presents interesting  challenges. This talk will discuss a

variety of examples in order to illustrate these challenges and present our approach

to this problem.

 

Josef Küstner (University of Vienna) January 21, 2021 - 3:55 - 5:00 PM (IST)

Our next talk is on January 21, 2021. Please do feel free to volunteer to speak, if you have submitted something recently and the topic is suitable for this group. 

We have now upgraded the sf-and-nt website a bit. In case a video recording is available, we have embedded it, so you can view the recording at leisure. I wish to thank Shivam Sahu, a recent graduate of the math department of Ashoka University, for help in this activity. In particular, it will be of help when you are reading the associated papers in more detail. Any suggestions for the website are welcome.

The talk for the week is as follows.

Talk Announcement

Title:  Elliptic and q-analogs of the Fibonomial numbers

Speaker: Josef Küstner (University of Vienna)

When: January 21, 2021 - 3:55 PM - 5:00 PM (IST)

Where: Google Meet; Please write to sfandnt@gmail.com if you want a link.

Tea or Coffee: Please bring your own. 

ABSTRACT

The Fibonomial numbers are integer numbers obtained from the binomial coefficients by replacing each term by its corresponding Fibonacci number. In 2009, Sagan and Savage introduced a simple combinatorial model for the Fibonomial numbers.
 

In this talk, I will present a combinatorial description for a q-analog and an elliptic analog of the Fibonomial numbers which is achieved by introducing certain q- and elliptic weights to the model of Sagan and Savage.
 

This is joint work with Nantel Bergeron and Cesar Ceballos.

Ramanujan Special: Wadim Zudilin on Thursday, January 7, 2021, 3:55 pm-5 PM

A very happy new year to all. We have decided that the first talk of every year will be a  Ramanujan Special Talk. This year a colloquium talk will be given by Wadim Zudilin. The announcement is below.

We wish you many new theorems, ideas and papers in 2021. Please do send any ideas or suggestions you have for the organisers to make this seminar more successful and help serve the interests of this community. 

Talk Announcement

Title: 10 years of q-rious positivity. More needed!

Speaker: Wadim Zudilin (Radboud University, Nijmegen).

Date and Time: Thursday, January 7, 2021, 3:55 PM IST (GMT+5:30)

Tea or coffee: Bring your own.

Where: Zoom: Please write to sfandnt@gmail.com for a link at-least 24 hours before the talk.

Abstract:

 

The $q$-binomial coefficients \[ \prod_{i=1}^m(1-q^{n-m+i})/(1-q^i),\] for integers $0\le m\le n$, are known to be polynomials with non-negative integer coefficients. This readily follows from the $q$-binomial theorem, or the many combinatorial interpretations of them. Ten years ago, together with Ole Warnaar we observed that this non-negativity (aka positivity) property generalises to products of ratios of $q$-factorials that happen to be polynomials; we prove this observation for (very few) cases. During the last decade a resumed interest in study of generalised integer-valued factorial ratios, in connection with problems in analytic number theory and combinatorics, has brought to life new positive structures for their $q$-analogues. In my talk I will report on this "$q$-rious positivity" phenomenon, an ongoing project with Warnaar.