Mithun Kumar Das (NISER) - Thursday, August 6, 2026 - 4:00 PM (IST)

The talk today will be delivered by Mithun Das, an Inspire Faculty of the Department of Mathematics at NISER, Bhubaneswar. 

Talk Announcement:
Title: Equidistribution: where Number Theory meets Analysis

Speaker: Mithun Kumar Das, NISER.
When: August 6, 2026, 4:00 PM- 5:00 PM IST 

Where: Zoom: Write to the organisers for the link.

Live Link:https://youtube.com/live/TWTLG8z4nRU?feature=share

Abstract: The theory of equidistribution has a long and rich history. Although its study originated in number theory, it soon became clear that equidistribution problems extend far beyond their number-theoretic roots. In this talk, we will explore how the equidistribution of algebraic numbers can be understood through the lens of Fourier analysis. 

This talk is based on an AIM group project (2021) and a recent joint work with Emanuel Carneiro (ICTP).

Bittu Chahal (IIT Indore) - Thursday, July 23, 2026 - 4:00 PM (IST)

 Dear All,

The talk of this week will be delivered by Bittu Chahal, a post-doctoral fellow of the Department of Mathematics at IIT Indore. 

Talk Announcement:
Title: Spacing statistics of the Farey sequence

Speaker: Bittu Chahal, IIT Indore.
When: July 23, 2026, 4:00 PM- 5:00 PM IST 

Where: Zoom:

Live Link: https://youtube.com/live/ttGMGYzpdOs?feature=share

Abstract:

The Farey sequence $\mathcal{F}_Q$ of order $Q$ is the ascending sequence of fractions $a/b$ in the unit interval $(0,1]$ such that $\gcd(a,b)=1$ and $0<a\leq b\leq Q$. The study of the Farey sequence is of independent interest because of its role in the Diophantine approximation, the circle method, and its connection to the Riemann Hypothesis, as established by the classical work of Franel and Landau. In this talk, we discuss the spacing statistics of the Farey sequence by studying equidistribution and correlation measure. In particular, we establish an estimate for the corresponding Weyl sum and explore whether correlation measure is Poissonian or non-Poissonian.

Thank you.

With kind regards,

--

Gaurav Bhatnagar, Atul Dixit, Bibekananda Maji, Krishnan Rajkumar, and Manjil Saikia

Special Functions and Number Theory Seminar

A. Satyanarayana Reddy (Shiv Nadar University) - Thursday, July 2, 2026 - 4:00 PM (IST)

The talk this week is by A. Satyanarayana Reddy, Associate Professor of the Department of Mathematics at Shiv Nadar University. 

Talk Announcement:
Title: A Gateway to the Circle Method: Ramanujan Sums and the Ternary Goldbach Problem

Speaker: A. Satyanarayana Reddy, Shiv Nadar University.
When: July 2, 2026, 4:00 PM- 5:00 PM IST 

Where: Zoom

Live Link: https://youtube.com/live/MSXP_YYzNWA?feature=share

Abstract

The Ramanujan sum is a fundamental exponential sum possessing deep arithmetic and orthogonality properties. In this talk, we first establish these classical properties, then use them as a gateway to understand the Hardy-Littlewood Circle Method. We will see how additive problems like the Ternary Goldbach Conjecture are translated into integrals of exponential sums. 

--

Gaurav Bhatnagar, Atul Dixit, Bibekananda Maji, Krishnan Rajkumar, and Manjil Saikia

Special Functions and Number Theory Seminar

Rishabh Sarma (Penn State University) - Thursday, June 18, 2026 - 6:00 PM (IST)

 Dear All,

The talk in the coming week will be delivered by Rishabh Sarma, S. Chowla Research Assistant Professor of the Department of Mathematics at Pennsylvania State University. 

Talk Announcement:
Title: Modular symmetries for rank and crank statistics

Speaker: Rishabh Sarma, Pennsylvania State University.
When: June 18, 2026, 6:00 PM- 7:00 PM IST 

Where: Zoom: Write to the SFNT organizers.

Live Link: https://youtube.com/live/6F-TG4LKkkg?feature=share

Abstract

Let R(z,q) be the two-variable generating function for Dyson’s rank function. In his lost notebook Ramanujan gives the p-dissection of R(ζ_p,q) where ζ_p is a primitive p-th root of unity and p=5. This result is related to Dyson’s famous rank conjecture which was proved by Atkin and Swinnerton-Dyer. In this talk, we explore the modular structure of this function, the partition crank and their overpartition counterparts, leading to a symmetry phenomenon among the elements of the p-dissection of these functions.

--

Gaurav Bhatnagar, Atul Dixit, Bibekananda Maji, Krishnan Rajkumar, and Manjil Saikia

Special Functions and Number Theory Seminar

SF and NT Playlist

Ramanujan Explained Playlist

Yashovardhan Singh Gautam (IIT, Roorkee) - May 28, 2026, 4:00 PM- 5:00 PM IST

 Dear all,

The next talk is tomorrow, by Yashovardhan Singh Gautam of IIT, Roorkee. I apologize for the late notification.

There is also an announcement. 

Bibekananda Maji (IIT, Indore) and Manjil Saikia (Ahmedabad University) have joined us as co-organisers of the SF and NT Seminar. Their interests are in Combinatorics, Number Theory and Partitions. Both have spoken in the seminar previously. 

Talk Announcement: 

Title: On special values of Koshliakov zeta functions

Speaker: Yashovardhan Singh Gautam, Indian Institute of Technology, Roorkee, India

When: May 28, 2026, 4:00 PM- 5:00 PM IST 

Where: Zoom. Write to sfandnt@gmail.com for the link

Live Link: https://youtube.com/live/ubCNCfyP1VI?feature=share

Abstract

In this talk, we study the Koshliakov zeta function η_p(s), whose theory appears to be more involved than that of its counterpart ζ_p(s), owing to the fact that its defining series is not of Dirichlet type. We derive formulas for η_p(s) at both even and odd values of s. In the limiting case p → ∞, our results yield the celebrated formulas of Euler and Ramanujan for the Riemann zeta function. Moreover, our results lead to several consequences concerning closed-form expressions for Lambert series and their arithmetic properties, recovering results due to Berndt, Cauchy, Ramanujan, and others. We also propose p-analogues of the transformation formula for the classical Eisenstein series. Moreover, we introduce two families of p-analogues of Ramanujan polynomials and establish functional equations satisfied by them.

--

Gaurav Bhatnagar, Atul Dixit, Bibekananda Maji, Krishnan Rajkumar, and Manjil Saikia

Special Functions and Number Theory Seminar

SF and NT Playlist

Ramanujan Explained Playlist

Nargish Punia (IIT, Roorkee) - Thursday, May 14, 2026 - 4:00 PM (IST)

 Dear all,

The talk this week is by Nargish Punia of IIT, Roorkee. 

Talk Announcement: 

Title: On Partition classes arising from parity, differences and repeated smallest parts

Speaker: Nargish Punia, Indian Institute of Technology, Roorkee, India
When: May 14, 2026, 4:00 PM- 5:00 PM IST 

Where: Zoom: Write to the organisers at sf and nt at gmail.com for the link

Live Link: https://youtube.com/live/BMrnfrIzjxs?feature=share

Abstract

In this talk, we present various classes of partition functions such as those related to the parity of the number of parts, to differences of partition numbers, and to partitions with a repeated smallest part. We establish identities connecting these various classes of partitions. Moreover, we will discuss how these identities help us to extend the Euler’s partition theorem. If time permits, we will also present an analogue of Legendre’s theorem of the partition-theoretic interpretation of Euler’s pentagonal number theorem. This is joint work with Rahul Kumar.

--

Gaurav Bhatnagar, Atul Dixit, and Krishnan Rajkumar,

Special Functions and Number Theory Seminar

SF and NT Playlist

Ramanujan Explained Playlist

Nikita Kalinin (Guangdong Technion, China) -- Thursday February 19, 2026 - 4:00 PM (IST)

 Dear all,

The talk this week is by Nikita Kalinin, Guangdong Technion Israel Institute of Technology, Shantou, China.

Talk Announcement: 

Title: Evaluation of lattice sums via telescoping over topographs

Speaker: Nikita Kalinin, Guangdong Technion Israel Institute of Technology, Shantou, China.
When: Feb 19, 2026, 4:00 PM- 5:00 PM IST (1830 Beijing Time)

Where: Zoom: write to the organsers for the link

Live Link: https://youtube.com/live/iHy9UHCdBV8?feature=share

Abstract 

Conway’s topograph provides a geometric way to organize values of binary quadratic forms on an infinite trivalent planar tree. In this talk, I explain how summation over vertices and edges of a topograph leads to a systematic method for evaluating a broad class of lattice sums by telescoping.

The main idea is that many arithmetic sums indexed by primitive lattice points or Farey-type data admit a natural decomposition along oriented edges of the topograph. When written in this form, local identities at each vertex produce global cancellations, yielding exact closed formulas depending only on the root configuration and the discriminant. This viewpoint unifies and simplifies several known identities, including sums related to reciprocal products of quadratic form values, Mordell–Tornheim–type series, and formulas connected to Euler-type constants.

The talk will be largely elementary and geometric. I will introduce the necessary background on topographs, explain the telescoping mechanism in a concrete way, and illustrate how classical analytic identities emerge from purely combinatorial summation over the tree.

The talk is based on  https://arxiv.org/abs/2510.02082 and  https://arxiv.org/abs/2510.00012 

Gaurav Bhatnagar, Atul Dixit, and Krishnan Rajkumar,

Special Functions and Number Theory Seminar

SF and NT Playlist

Ramanujan Explained Playlist

Sudhir Kumar Pujahari (NISER) - Thursday, February 12, 2026 - 4:00 PM (IST)

 Dear all,

The next talk is by Sudhir Kumar Pujahari of the National Institute of Science Education and Research (NISER), Bhubaneswar, India. 

Talk Announcement: 

Title: Moments of the Trace of Frobenius in Arithmetic Progressions.

Speaker: Sudhir Kumar Pujahari, NISER (Bhubaneshwar, India)
When: Feb 12, 2026, 4:00 PM- 5:00 PM IST 

Where: Zoom: Write to the organisers for the link.

Live Link: https://youtube.com/live/dgnGN7RX9tA?feature=share

Abstract 

The statistical distribution of the trace of Frobenius for elliptic curves is a central theme in arithmetic geometry, famously encapsulated by the Sato–Tate conjecture. In this talk, we investigate the moments of the trace of Frobenius for elliptic curves over finite fields when the traces are constrained to a fixed arithmetic progression. We establish the asymptotic behavior of the moment as the size of the finite field tends to infinity. In the process, we will see a bridge between moments of traces of Frobenius and the theory of binary quadratic forms; specifically, we will derive these results from new asymptotic formulas for sums of Hurwitz class numbers restricted to arithmetic progressions. This talk is based on joint work with Ben Kane and Kathrin Bringmann.

Gaurav Bhatnagar, Atul Dixit, and Krishnan Rajkumar,

Special Functions and Number Theory Seminar

SF and NT Playlist

Ramanujan Explained Playlist

Ramanujan Special: Dennis Stanton (Minnesota) - Thursday, January 22, 2026 - 7:30 PM- 8:30 PM IST (8 AM CST)

Happy new year. 

The first talk of the year (on Jan 22, 2026) is a ``Ramanujan Special". This year's speaker is Dennis Stanton, who has decided to gift us a very special talk, containing several interesting open problems. Please note that the talk will be later than usual. 

Talk Announcement: The 2026 Ramanujan Special

Title: Some open problems 

Speaker: Dennis Stanton, University of Minnesota
When: Jan 22, 2026, 7:30 PM- 8:30 PM IST (8 AM CST) (Note special time) 

(IST= CST + 11:30)

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

Live Link: https://youtube.com/live/6ui6EMf4qJs?feature=share

Download presentation. 

Abstract 

I will discuss some open problems in q-series, partitions,

orthogonal polynomials, basic hypergeometric functions, and combinatorics, including the Rogers-Ramanujan identities.

Gaurav Bhatnagar, Atul Dixit, and Krishnan Rajkumar,

Special Functions and Number Theory Seminar

SF and NT Playlist

Ramanujan Explained Playlist

Aritram Dhar (Florida) -- Thursday December 18, 2025 - 4:00 PM (IST)

 Dear all,

We have a talk this week by Aritram Dhar, of the University of Florida. The title and abstract are below. 

Talk Announcement: 

Title: Extension of Bressoud's generalization of Borwein's conjecture, some exact results, and applications

Speaker: Aritram Dhar (University of Florida) 

When: December 18, 2025, 4:00 PM- 5:00 PM IST

Where: Zoom. Write to the organisers for a link

Live Link: https://youtube.com/live/1-GzIoCJtBM?feature=share

Abstract:

In this talk, we conjecture an extension of Bressoud's 1996 generalization of Borwein's famous 1990 conjecture. We then state a few infinite hierarchies of non-negative q-series identities which are interesting examples of our proposed conjecture and Bressoud's generalized conjecture. Using certain positivity-preserving transformations for q-binomial coefficients due to Berkovich and Warnaar, we prove the non-negativity of the infinite families. Time permitting, we present some applications of cubic positivity-preserving transformations where we establish new identities analogous to Andrews' representations of the Borwein polynomials. This is based on recent joint works with Alexander Berkovich.

Gaurav Bhatnagar, Atul Dixit, and Krishnan Rajkumar,

Special Functions and Number Theory Seminar

SF and NT Playlist

Ramanujan Explained Playlist

Hussein Mourtada (Paris-Diderot) - Thursday, October 23 , 2025 - 4:00 PM IST (12:30 PM CEST)

Dear all,

We have a talk this week by  Hussein Mourtada of the Université de Paris (Campus Paris-Diderot). The title and abstract are below. 

Talk Announcement: 

Title: Integer partitions, (hyper)graphs and monomial ideals

Speaker: Hussein Mourtada (Université de Paris (Campus Paris-Diderot))

When: October 23, 2025, 4:00 PM- 5:00 PM IST (12:30 PM CEST)

Where: Zoom: Please ask the organisers for a link

Live Link: https://youtube.com/live/iSTlKqaEC44?feature=share

Abstract:

We will present new partition identities that are, in a certain sense, dual to Gordon’s identities. These results arise from a correspondence between three classes of objects: a new family of partitions (called neighborly partitions), monomial ideals, and certain infinite (hyper)graphs. This talk is based on joint works, one with Zahraa Mohsen and another with Pooneh Afsharijoo.

Gaurav Bhatnagar, Atul Dixit, and Krishnan Rajkumar,

Special Functions and Number Theory Seminar

SF and NT Playlist

Ramanujan Explained Playlist

Jehanne Dousse (Geneva) - Thursday, October 16 , 2025 - 4:00 PM (IST)

 Dear all,

The talk next week is by Jehanne Dousse of the University of Geneva. The title and abstract are below.

There was a snafu in the previous talk organisation. Some announcements/reminders were inadvertently not sent. The video of the talk by Ritwik Pal (IIIT, Delhi) has been uploaded on sfandnt website. 

Talk Announcement: 

Title: Andrews-Gordon-Bressoud type identities and particle motion

Speaker: Jehanne Dousse  (University of Geneva)

When: October 16, 2025, 4:00 PM- 5:00 PM IST (12:30 PM CEST)

Where: Zoom: 

Live Link: https://youtube.com/live/AqUobI1aa30?feature=share

Abstract:

The Andrews-Gordon identities are among the most important q-series and partition identities, and generalise the famous Rogers-Ramanujan identities. Interestingly, while the product side of these identities clearly corresponds to partitions with congruence conditions, it is not obvious that the sum side of the q-series version is the generating function for the partitions with frequency conditions that appear in the combinatorial version. It was originally proved by George Andrews using recurrences, and then bijectively by Ole Warnaar using particle motion.

In this talk, we will explain and generalise the particle motion approach. We will show that the generalised version can be applied to the sum side of Bressoud's identity and that, using the Andrews-Gordon and Bressoud identities as starting points, it can prove many known and new identities.

This is based on joint work with Jihyeug Jang, Frédéric Jouhet and Isaac Konan.

Gaurav Bhatnagar, Atul Dixit, and Krishnan Rajkumar,

Special Functions and Number Theory Seminar

SF and NT Playlist

Ramanujan Explained Playlist

Ritwik Pal (IIIT, Delhi) - Sept 25, 2025 - 4:00-5:00 PM IST

 Dear all,

This week's talk is by Ritwik Pal of the Indian Institute of Information Technology (IIIT), Delhi. The announcement is below.

Talk Announcement: 

Title:   On shifted convolution sum of Fourier coefficients of $SL(3,\mathbb{Z})$ Hecke–Maass forms.

Speaker: Ritwik Pal (IIIT, Delhi)

When: Sept 25, 2025, 4:00-5:00 PM IST

Where: Zoom: Write to the organisers for a link

Live Link:  https://youtube.com/live/3QsUvbHNLGo?feature=share

Abstract:

We will present a brief overview of the shifted convolution sum problems, especially those related to the coefficients of automorphic L-functions. Then we will present our recent article on establishing a non-trivial upper bound for the Fourier coefficients of $SL(3,\mathbb{Z})$ Hecke–Maass forms. As a consequence, it gives a significant improvement over the previously known range of shifts for which a non-trivial upper bound of shifted convolution sum holds. This is a joint work with Sampurna Pal.

Gaurav Bhatnagar, Atul Dixit, and Krishnan Rajkumar,

Special Functions and Number Theory Seminar

SF and NT Playlist

Ramanujan Explained Playlist

Michael Schlosser (University of Vienna) - Thursday, Sept 11 , 2025 - 4:00 PM (IST)

 Dear all,

Welcome back from the summer break. We hope you had an exciting summer, filled with new theorems and fresh collaborations.

The first talk of the season is by Michael Schlosser on Sept 11. The announcement is below.

Talk Announcement: 

Title: Asymptotic formulas for the Fourier coefficients
of infinite $q$-products

Speaker: Michael Schlosser (University of Vienna)

When: Sept 11, 2025, 4:00 PM- 5:00 PM IST (12:30 PM CEST)

Where: Zoom: (Write to the organisers for a link)

Live Link: https://youtube.com/live/jpaIdBpfx_s?feature=share

Abstract:

We derive asymptotic expansions for weighted partition
numbers satisfying certain conditions. As applications we
partially settle some conjectures by Berkovic and Garvan,
and by Seo and Yee, on the nonnegativity of the coefficients
of certain infinite products, and a conjecture by Chan and
Yesilyurt on the periodicity of the signs of the coefficients of
a non-theta product. This is joint work with Nian Hong Zhou.

Pranjal Talukdar (Tezpur University) - Thursday, May 22, 2025 - 4:00 PM (IST)

The talk this week is by Pranjal Talukdar of Tezpur university. Here is the announcement of the talk. 

Talk Announcement: 

Title: On the least $r$-gaps in partitions and identities for the Rogers-Ramanujan Continued fraction

Speaker: Pranjal Talukdar (Tezpur University)

When: May 22, 2025, 4:00 PM- 5:00 PM IST

Where: Zoom: Please write to the organisers for a link

Live Link: https://youtube.com/live/ZCoyMURaMKQ?feature=share

Best wishes,

Gaurav Bhatnagar, Atul Dixit and Krishnan Rajkumar (organisers)

Abstract:

The minimal excludant of a partition $\pi=(\pi_1,\pi_2,\ldots,\pi_k)$ of $n$ is the smallest positive integer that is not present in $\pi$ and is denoted by $\textup{mex}(\pi)$. The least $r$-gap of $\pi$ is the least positive integer that does not appear in the partition at least $r$ times. In the first half of the talk, we derive some arithmetic functions related to the sum of least $r$-gaps. Using a Tauberian theorem due to Ingham, we obtain Hardy-Ramanujan-type asymptotic formula for two such functions. We also briefly mention some arithmetic properties for these functions.

In the second half, we prove some new identities for the Rogers\textendash Ramanujan continued fraction. For example, if $R(q)$ denotes the Rogers\textendash Ramanujan continued fraction, then

\begin{align*}&R(q)R(q^4)=\dfrac{R(q^5)+R(q^{20})-R(q^5)R(q^{20})}{1+R(q^{5})+R(q^{20})},\\

&\dfrac{1}{R(q^{2})R(q^{3})}+R(q^{2})R(q^{3})= 1+\dfrac{R(q)}{R(q^{6})}+\dfrac{R(q^{6})}{R(q)}.

\end{align*}

The contents of the talk is taken from two chapters of the speaker's Ph.D. thesis completed under the supervision of Prof. Nayandeep Deka Baruah at Tezpur University, India.