This talk is on joint work with Tewodros Amdeberhan and Roberto Tauraso. In our joint paper, Extensions of MacMahon’s Sums of Divisors (Research in the Math. Sciences, 11,8 (2024)), we examined the arithmetic properties of MacMahon’s original functions and their generalizations. In this talk, we consider further extensions of MacMahon’s work, and we look at both arithmetic and combinatorial aspects including a stunning identity for the generating function of the sum of the divisors of n.

Best wishes,

Gaurav Bhatnagar, Atul Dixit and Krishnan Rajkumar (organisers)

We have launched a course under the title of Ramanujan Explained. There will be a series of lectures, all given by Gaurav Bhatnagar, with accompanying notes and exercises. The goal is to cover (a large number of) Ramanujan's identities. The first talk in this series is in our next seminar slot. Kindly do share this announcement with students who may be interested in Ramanujan and his mathematics. The first few lectures will target $q$-hypergeometric series and special cases, and can serve as an introduction to basic hypergeometric series. We hope these lectures will serve as a useful supplement to the monumental work of Bruce Berndt (Ramanujan's Notebooks I-V) and George Andrews and Bruce Berndt (Ramanujan's Lost Notebook I-V).

About the Rogers-Ramanujan identities, Hardy famously remarked: "It would be difficult to find more beautiful formulae than the "Rogers-Ramanujan" identities... " In the first introductory lecture in the Ramanujan Explained course, we explain Askey's idea on how Ramanujan may have come across these identities. Continued fractions played an important part of Ramanujan's work, and Askey's explanation is all about the simplest $q$-continued fraction and how it naturally leads to the Rogers-Ramanujan identities.

The dispersion method has found an impressive number of applications in analytic number theory, from bounded gaps between primes to the greatest prime factors of quadratic polynomials. The method requires bounding certain exponential sums, using deep inputs from algebraic geometry, the spectral theory of GL2 automorphic forms, and GLn automorphic L-functions. We'll give a broad outline of this process, which combines various types of number theory; time permitting, we'll also discuss the key ideas behind some new results.

The next talk is by Gaurav Bhatnagar of Ashoka University. The announcement is as follows.

Talk Announcement:

Title: Elliptic enumeration and identities

Speaker: Gaurav Bhatnagar (Ashoka University) When: Mar 21, 2024, 4:00 PM- 5:00 PM IST

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

Live LInk: https://youtube.com/live/cpqHK-R2oXg?feature=share

Abstract

Many of the ideas of $q$-counting and $q$-hypergeometrics are now being extended to the elliptic case. The approach is not very far from the $q$-case. In this talk, we show several examples to illustrate this idea. First we extend some Fibonacci identities using combinatorial methods. Many such identities can be found by telescoping, so we next use telescoping to find elliptic extensions of elementary identities such as the sum of the firstodd or even numbers, the geometric sum and the sum of the firstcubes. In the course of our study, we obtained an identity with many parameters, which appears to be new even in the$q$-case. Finally, we introduce elliptic hypergeometric series and give an extension of some important identities of Liu. As applications, we find 5 double summations and 4 new elliptic transformation formulas. Again, these are new in the $q$-hypergeometric case, where the nome $p$ is 0.

This is a report of joint work with Archna Kumari and Michael Schlosser.

The next talk is by Shivani Goel, of the Indraprastha Institute of Information Technology (IIIT), Delhi. The announcement is as follows.

Talk Announcement:

Title:Distribution and applications of Ramanujan sums

Speaker: Shivani Goel (IIIT, Delhi) When: Mar 7, 2024, 4:00 PM- 5:00 PM IST

Where: Zoom: Ask the organisers for the link.

Live LInk: https://youtube.com/live/mQ9EiVeqimI?feature=share

Abstract

While studying the trigonometric series expansion of certain arithmetic functions, Ramanujan, in 1918, defined a sum of the $n^{th}$ power of the primitive $q^{th}$ roots of unity and denoted it as $c_q(n)$. These sums are now known as Ramanujan sums.

Our focus lies in the distribution of Ramanujan sums. One way to study distribution is via moments of averages. Chan and Kumchev initially considered this problem. They estimated the first and second moments of Ramanujan sums. Building upon their work, we extend the estimation of the moments of Ramanujan sums for cases where $k\ge 3$. Apart from this, We derive a limit formula for higher convolutions of Ramanujan sums to give a heuristic derivation of the Hardy-Littlewood formula for the number of prime $k$-tuplets less than $x$.

In his lost notebook, Ramanujan recorded beautiful identities. These include earlier versions of Guinand's formula for the divisor function and the transformation formula for the logarithm of Dedekind's -function.

In our presentation we will describe some generalizations of these formulas using a beautiful theory due to the forgotten mathematician N. S. Koshliakov. Our work will be presented under the point of view initiated by A.Dixitand R. Gupta, the first mathematicians of our century who have extended Koshliakov's theory in several directions.

This talk is based on joint work with Semyon Yakubovich.

The next talk is by Arvind Ayyer of the Indian Institute of Science, Bangalore, India. The talk announcement follows. Please note that the talk is half an hour later than our usual meeting time.

Talk Announcement:

Title:A new combinatorial formula for the modified Macdonald polynomials

Speaker: Arvind Ayyer (IISc, Bangalore, India) When: Feb 8, 2024, 4:30 PM- 5:30 PM IST (Note special time)

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

Macdonald polynomials are a remarkable family of symmetric functions that are known to have connections to combinatorics, algebraic geometry and representation theory. The modified Macdonald polynomials are obtained from the Macdonald polynomials using an operation called plethysm. A combinatorial formula for the latter was given by Haglund, Haiman and Loehr in a celebrated work (JAMS, 2004). We will give a new combinatorial formula (ALCO 2023).

Recently, a formula for the symmetric Macdonald polynomials was given by Corteel, Mandelshtam and Williams in terms of objects called multiline queues, which also compute probabilities of a statistical mechanics model called the multispecies ASEP on a ring. It is natural to ask whether the modified Macdonald polynomials can be obtained using a combinatorial gadget for some other statistical mechanics model. We answer this question in the affirmative via a multispecies totally asymmetric zero-range process (TAZRP) in (arXiv:2209.09859).

These are joint works with J. Martin and O. Mandelshtam.

The first talk of the year (on January 25, 2023) is a ``Ramanujan Special". This year's speaker is Frank Garvan. Please note that the talk will be later than usual. A report on the activities of this seminar in 2023 appears in the SIAM newsletter OPSFNET. We hope this year is equally exciting for our group. Please consider the seminar to present your latest preprint.

Talk Announcement: The 2024 Ramanujan Special

Title: Identities for Ramanujan's Mock Theta Functions and Dyson's Rank

Function

Speaker: Frank Garvan (University of Florida, USA) When: Jan 25, 2024, 7:30 PM- 8:30 PM IST (9 AM EST) (Note special time)

(EST= IST - 10:30)

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

In Ramanujan's Lost Notebook there are identities connecting Ramanujan's fifth order mock theta functions and Dyson's rank mod 5. We extend these connections to Zagier's higher order mock theta functions. We consider Dyson's problem of giving a group-theoretic structure to

the mock theta functions analogous to Hecke's theory of modular forms.From this much surprising symmetry and q-series identities arise in joint work with Rishabh Sarma and Connor Morrow.

The talk this week is by David Bradley of the University of Maine. Please note the special time. Since Professor Bradley is located in the US, we are starting later than usual.

This will be the final talk of the year. We will come back next year with a Ramanujan special, and hope that we get an opportunity to meet in person in the upcoming conference and holiday season. We wish you happy holidays and a great new year.

The announcement is as follows.

Talk Announcement:

Title:On Fractal Subsets of Pascal's "Pyramid" and the Number of Multinomial Coefficients Congruent to a Given Residue Modulo a Prime

We obtain an explicit formula and an asymptotic formula for the number of multinomial coefficients which are congruent to a given residue modulo a prime, and which arise in the expansion of a multinomial raised to any power less than a given power of that prime. Each such multinomial coefficient can be associated with a certain Cartesian product of intervals contained in the unit cube. For a fixed prime, the union of these products forms a set which depends on both the residue and the power of the prime. In the limit as the power of the prime increases to infinity, the sequence of unions converges in the Hausdorff metric to a non-empty compact set which is independent of the residue. We calculate the fractal dimension of this limiting set, and consider its monotonicity properties as a function of the prime. To study the relative frequency of various residue classes in the sequence of approximating sets, it would be desirable to have a closed-form formula for the number of entries in the first prows of Pascal’s "pyramid" which are congruent to a given nonzero residuermodulo the primep. Unfortunately, numerical computations with large prime moduli suggests that if there is such a formula, it is extremely complicated. Nevertheless, the evidence indicates that for sufficiently large primes p, the number ofbinomialcoefficients in Pascal'striangle which are congruent to rmod pfor r= 1, 1 <r<p−1, andr= p−1 is well approximated by the respective linear functions ofpgiven

by 3p, p/2, andp. In particular, for large primes pthere are approximately six times as many occurrences of the residue 1 in the first prows of Pascal’s triangle reduced modulo pthan there are of any other residue r in the range 1 <r<p− 1, and three times as many asr= p− 1. On the other hand, if we let the nonnegative integer kvary while keeping the prime pfixed, and look at the relative frequency of various residue classes that occur in the first pkrows, the seemingly substantial differences in frequency between r= 1, 1 <r<p−1, and r = p−1 when k= 1 are increasingly dissipated as kgrows without bound. We show that in the limit as ktends to infinity, all nonzero residues are equally represented with asymptotic proportion 1/(p− 1).

Let $\psi(x)$ denote the digamma function that is the logarithmic derivative of $\Gamma$ function. In 2007, Murty and Saradha studied the linear independence of special values of digamma function $\psi(a/q)+\gamma$ over some specific numbers fields which also imply the non-vanishing of $L(1,f)$ for any rational-valued Dirichlet type function $f$. In 2009, Gun, Murty and Rath studied the non-vanishing of $L'(0,f)$ for even Dirichlet-type periodic $f$ in terms of $L(1,\hat{f})$ and established that this is related to the linear independence of logarithm of gamma values. In this direction, they made a conjecture which they call it as a variant of Rohrlich conjecture concerning the linear independence of logarithm of gamma values. In this talk, first we will discuss the linear independence of digamma values over the field of algebraic numbers. Later, we provide counterexamples to this variant of Rohrlich conjecture.

Branching rules/laws (restriction of representations) also known as symmetry breaking in physics has been an active area of research since the onset of the topic by Herman Weyl in 1950. In this talk, I would give a brief description of Highest weight theory and the determinantal form of the Weyl character formula. I would proceed to talk about branching from $GL_n$ to $GL_{n-1}$ and give an idea about the other classical groups.

Selberg's multivariate extension of the beta integral appears all over mathematics: in random matrix theory, analytic number theory,multivariate orthogonal polynomials and conformal field theory. The goalof my talk will be to explain a recent unification of two importantgeneralisations of the Selberg integral, namely the Selberg integralassociated with the root system of type A_n due to Warnaar and theelliptic Selberg integral conjectured by van Diejen and Spiridonov andproved by Rains. The key tool in our approach is the ellipticinterpolation kernel, also due to Rains. This is based on joint work withEric Rains and Ole Warnaar.

The next talk is by David Wahiche of the University of Tours, France. The title and abstract is below.

Talk Announcement:

Title:From Macdonald identities to Nekrasov--Okounkov type formulas Speaker: David Wahiche (Universite' de Tours, France) When: Thursday, Oct 12, 2023, 4:00 PM- 5:00 PM IST

Between 2006 and 2008, using various methods coming from representation theory (Westbury), gauge theory (Nekrasov--Okounkov) and combinatorics (Han), several authors proved the so-called Nekrasov–Okounkov formula which involves hook lengths of integer partitions.

This formula does not only cover the generating series for P, but more generally gives a connection between powers of the Dedekind η function and integer partitions. Among the generalizations of the Nekrasov--Okounkov formula, a (q, t)-extension was proved by Rains and Warnaar, by using refined skew Cauchy-type identities for Macdonald polynomials. The same result was also obtained independently by Carlsson–Rodriguez-Villegas by means of vertex operators and the plethystic exponential. As mentioned in both of these papers, the special case q=t of their formula correspond to a q version of the Nekrasov--Okounkov formula, which was already obtained by Dehaye and Han (2011) and Iqbal et al. (2012).

Motivated by the work of Han et al. around the generalizations of the Nekrasov--Okounkov formula, one way of deriving Nekrasov--Okounkov formula is by using the Macdonald identities for infinite affine root systems (Macdonald 1972), which can be thought as extension of the classical Weyl denominator formula.

In this talk, I will try to explain how some reformulations of the Macdonald identities (Macdonald 1972, Stanton 1989, Rosengren and Schlosser 2006) can be decomposed in the basis of characters for each infinite of the 7 infinite affine root systems by the Littlewood decomposition. This echoes a representation theoretic interpretation of the Macdonald identities (see the book of Carter for instance) and an ongoing project with Cédric Lecouvey, I will mention some partial results we get.

At last, I will briefly explain how to go from these reformulations of Macdonald identities to q Nekrasov--Okounkov type formulas.

Dear all, sorry for the late announcement. The next talk is by Seema Kushwaha of IIIT, Allahabad. We are back to our usual time now. Hope to see you later this week.

Talk Announcement:

Title:Farey-subgraphs and Continued Fractions Speaker: Seema Kushwaha (IIIT, Allahabad) When: Thursday, Sept 28, 2023, 4:00 PM- 5:00 PM IST

Where: Zoom: Please send email to the organisers for a link.

Let $p$ be a prime and $l\in\mathbb{N}$. Let \begin{equation*}\label{X_n} \mathcal{X}_{p^l}=\left\{\frac{x}{y}:~x,y\in\mathbb{Z},~ y>0,~\mathrm{gcd}(x,y)=1~\textnormal{and}~{p^l}|y\right\}\cup\{\infty\}. \end{equation*} The set $\mX_{p^l}$ is the vertex set of a connected graph where vertices $x/y$ and $u/v$ are adjacent if and only if $ xv-uy=\pm p^l.$ These graphs give rise to a family of continued fraction, namely, $\f_{p^l}$-continued fractions \cite{seema_fareysubgraphs}.

Let $\mathcal{X}$ be a subset of the extended set of rational numbers. A {\it best $\mathcal{X}$-approximation} of a real number is a notion which is analogous to best rational approximation. An element $u/v$ of $\mX$ is called a \textit{best $\mX$-approximation} of $x\in\R$, if for every $u'/v'\in\mX$ different from $u/v$ with $0< v' \le v$, we have $|vx-u|<|v'x-u'|$.

In this talk, we will discuss the existence and uniqueness of $\f_{p^l}$-continued fractions and their approximation properties.