First, motivated by a theorem in Ramanujan's lost notebook, Martino Fassina, Sun Kim, Alexandru Zaharescu, and the speaker developed representations for finite sums of products of trig functions for which we provided theorems and several conjectures.

Second, a paper of Richard McIntosh served as motivation. First, he made a very interesting conjecture, which was recently proved by Likun Xie, Zaharescu, and the speaker. Second, he examined a particular trigonometric sum, which inspired Sun Kim, Zaharescu, and the speaker to evaluate in closed form several classes of trigonometric sums, and find reciprocity theorems for others.

New conference announcement

A new conference on algebraic combinatorics has been announced by Arvind Ayyer. It is called

In this talk, we present a simple extension of the Ramanujan-Serre derivative map and

describe how it can be used to derive a general method for explicit evaluation of convolution sums of the divisor functions. We provide explicit examples for four types of convolution sums.

This is a joint work with Brundaban Sahu and Anup Kumar Singh.

We are back to our usual time with a talk by Galina Filipuk all the way from Warsaw, Poland. Please note that we will be open to changing the time, since speakers from the US find this time to be very inconvenient, and we surely would like speakers from the US. The discussions in the previous talk went quite late into the night (for New Zealand) and we thank Shaun Cooper for a very nice talk.

Talk Announcement:

Title: (Quasi)-Painleve equations and Painleve equivalence problem

Speaker: Galina Filipuk (University of Warsaw, Poland) When: Feb 16, 2022, 4:00 PM- 5:00 PM IST (11:30 CET in Warsaw)

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

Abstract

Painleve equations are second order nonlinear differential equations solutions of which have no movable critical points (algebraic singularities). They appear in many applications (e.g., in the theory of orthogonal polynomials) but in disguise. How to find a transformation to the canonical form? This is known as the Painleve equivalence problem. The so-called geometric approach may help in many cases.

In this talk I shall present some recent results on the geometric approach for the Painleve and quasi-Painleve equations.

The first talk of the year (on Feb 2, 2023) is a "Ramanujan Special". This year's speaker is Shaun Cooper. Please note that the talk will be earlier than usual.

The last year was quite exciting for our group with many talks as well as a mini course. We hope this year is equally exciting. Please consider the seminar to present your latest preprint.

Talk Announcement: Ramanujan Special

Title: Apéry-like sequences defined by four-term recurrence relations: theorems and conjectures

Speaker: Shaun Cooper (Massey University, Auckland, New Zealand)

When: Feb 2, 2022, 2:30 PM- 3:30 PM IST (Note special time) (IST= GMT - 5:30)

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

Abstract

The Apéry numbers are famous for having been introduced and used by R. Apéry to prove that~$\zeta(3)$ is irrational. They may be defined by the recurrence relation $$ (n+1)^3A(n+1)=(2n+1)(17n^2+17n+5)A(n)-n^3A(n-1), $$ with the single initial condition $A(0)=1$ being enough to start the recurrence. The Apéry numbers are all integers, a fact not obvious from the recurrence relation, and they satisfy interesting congruence properties. The generating function $$ y=\sum_{n=0}^\infty A(n)w^n $$ has a splendid parameterisation given by $$ y = \prod_{j=1}^\infty \frac{(1-q^{2j})^7(1-q^{3j})^7}{(1-q^{j})^5(1-q^{6j})^5} \quad \mbox{and} \quad w=q\,\prod_{j=1}^\infty \frac{(1-q^{j})^{12}(1-q^{6j})^{12}}{(1-q^{2j})^{12}(1-q^{3j})^{12}}. $$ In this talk I will briefly survey other sequences defined by three-term recurrence relations that have properties similar to those satisfied by the Apéry numbers described above. I will also introduce some sequences defined by four-term recurrence relations and describe some of their properties.

Schur polynomials are the characters of irreducible representations of classical groups of type A parametrized by partitions. For a fixed integer $t \geq 2$ and a primitive $t$'th root of unity \omega, Schur polynomials evaluated at elements $\omega^{k} x_i$ for $0 \leq k \leq t-1$ and $1 \leq i \leq n$, were considered by D. J. Littlewood (AMS press, 1950)and independently by D. Prasad (Israel J. Math., 2016). They characterized partitions for which the specialized Schur polynomials are nonzero and showed that if the Schur polynomial is nonzero, it factorizes into characters of smaller classical groups of type A.

In this talk, I will present a generalization of the factorization result to the characters of classical groups of type B, C and D. We give a uniform approach for all cases. The proof uses Cauchy-type determinant formulas for these characters and involves a careful study of the beta sets of partitions. This is joint work with A. Ayyer and is available here. (Preprint: https://arxiv.org/abs/2109.11310)

The talk this week is by Nicholas Smoot from the Research Institute of Symbolic Computation (RISC) at Johannes Kepler University (JKU), Linz, Austria. The announcement is below.

Since Ramanujan's groundbreaking work, a large variety of infinite congruence families for partition functions modulo prime powers have been discovered. These families vary enormously with respect to the difficulty of proving them. We will discuss the application of the localization method to proving congruence families by walking through the proof of one recently discovered congruence family for the counting function for 5-elongated plane partitions. In particular, we will discuss a critical aspect of such proofs, in which the associated generating functions of a given congruence family are members of the kernel of a certain linear mapping to a vector space over a finite field. We believe that this approach holds the key to properly classifying congruence families.

The non-vanishing as well as primitivity of the values of the Fourier coefficients of non-CM Hecke eigen forms, in particular the Ramanujan $\tau$ function is a deep and mysterious theme in Number theory. In this talk, we will report on our recent work on the number of distinct prime factors of the values of the Fourier coefficients of non-CM Hecke eigen forms, in particular the Ramanujan $\tau$ function.

We are happy that Kaneenika Sinha (IISER, Pune) has consented to give a mini-course on Central limit theorems in number theory. The course will comprise two lectures. The announcement is below. Graduate students who are interested in number theory are especially welcome to hear Professor Sinha.

Mini-course announcement

Title: Central Limit theorems in Number Theory

Speaker: Kaneenika Sinha (IISER, Pune)

Abstract:

The goal of these lectures is to review a theme that binds the study of different types of arithmetic functions, namely central limit theorems. After reviewing the "prototype" theorem in this theme, namely the classical Erdos-Kac theorem about the prime-omega function, we will survey different types of central limit theorems in the context of zeroes of zeta functions, eigenvalues of Hecke operators acting on spaces of cusp forms and eigenvalues of regular graphs.

After a refreshing break in a summer with many opportunities to meet in person, we are back with a talk by Professor Ole Warnaar of the University of Queensland, Australia.

Talk Announcement:

Title: Cylindric partitions and Rogers--Ramanujan identities

Speaker: S. Ole Warnaar (University of Queensland, Australia)

Cylindric partitions, first introduced by Gessel and Krattenthaler in 1997, are an affine (or toroidal) analogue of ordinary plane partitions. Perhaps somewhat surprisingly, cylindric partitions have a close connection to Rogers--Ramanujan-type $q$-series identities. In this talk I will try to explain this connection, and report on some new Rogers--Ramanujan identities for the affine Lie algebra $\mathrm{A}_2^{(1)}$ that follow naturally from this connection.

It is an open question of Baker whether the numbers $L(1, \chi)$ for non-trivial Dirichlet characters $\chi$ with period $q$ are linearly independent over $\mathbb{Q}$. The best known result is due to Baker, Birch and Wirsing which affirms this when $q$ is co-prime to $\varphi(q)$. In this talk, we will discuss an extension of this result to any arbitrary family of moduli. The interplay between the resulting ambient number fields brings in new technical issues and complications hitherto absent in the context of a fixed modulus. In the process, we also extend a result of Okada about linear independence of the cotangent values over $\mathbb{Q}$ as well as a result of Murty-Murty about ${\overline{\mathbb{Q}}}$ linear independence of such $L(1, \chi)$ values.

If time permits, we will also discuss the interrelation between the non-vanishing of Dedekind zeta values and its derivatives.

Polya established that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for the Riemann zeta function. This led to the study of Jensen polynomials for large classes of functions. Recently, Jensen polynomials for various partition functions and more generally, the Jensen polynomials for weakly holomorphic forms are considered. Their hyperbolicity was established under various conditions. In this talk we will consider the Jensen polynomials for power partitions and discuss the hyperbolicity of these Jensen polynomials

Where:Zoom. Please write to the organizers for the link

Tea or Coffee: Please bring your own.

Abstract:

In this talk, we discuss the asymptotic behavior of the number of partitions into primes in arithmetic progressions. While the study for ordinary partitions goes back to Hardy and Ramanujan, partitions into primes were recently re-visited by Vaughan. Our error term is sharp and improves on previous known estimates in the special case of primes as parts of the partition. As an application, monotonicity of this partition function is established explicitly via an asymptotic formula.

Random walk on the $n$-cube gives rise to two matrices, the adjacency matrix and the Kac matrix, that have an elegant spectral theory with many applications.

We define a random walk on subspaces that yields q-analogs of the adjacency and Kac matrices with an equally elegant spectral theory (though more difficult to prove). We give an application to tree counting.

On the coming Thursday, Atul Dixit (IIT, Gandhinagar) and Gaurav Bhatnagar (Ashoka University) will present the talks they presented at the recently concluded JMM meeting held online. Each talk will be approximately 20 minutes followed by 10 minutes for questions. The titles and abstracts are below.

Here is the announcement for the second talk of the day.

Title: An easy proof of Ramanujan's famous mod $5$ congruences

We give an elementary proof of Ramanujan's famous congruences $p(5n+4) \equiv 0 \mod 5$ and $\tau(5n+5)\equiv 0 \mod 5$. The proof requires no more than Euler's techniques and a result of Jacobi. The proof extends to embed the congruences into 4 infinite families of congruences for rational powers of the eta function.

This is joint work with Hartosh Singh Bal.

The video below is the same as the one in Atul's talk. It just begins in the middle.

On the coming Thursday, Atul Dixit (IIT, Gandhinagar) and Gaurav Bhatnagar (Ashoka University) will present the talks they presented at the recently concluded JMM meeting held online. Each talk will be approximately 20 minutes followed by 10 minutes for questions. The titles and abstracts are below.

We should say that during the summer, the talks in the seminar may be less as we are finally able to catch up with colleagues in person.

Here is the announcement for Atul's talk.

Title:Combinatorial identities associated with a bivariate generating function for overpartition pairs

We obtain a three-parameter $q$-series identity that generalizes two results of Chan and Mao. By specializing our identity, we derive new results of combinatorial significance in connection with $N(r, s, m, n)$, a function counting certain overpartition pairs recently introduced by Bringmann, Lovejoy and Osburn. For example, one of our identities gives a closed-form evaluation of a double series in terms of Chebyshev polynomials of the second kind, thereby resulting in an analogue of Euler's pentagonal number theorem. Another of our results expresses a multi-sum involving $N(r, s, m, n)$ in terms of just the partition function $p(n)$. Using a result of Shimura we also relate a certain double series with a weight 7/2 theta series. This is joint work with Ankush Goswami.