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.

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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.

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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