Live LInk: https://youtube.com/live/eVgu2PvtG7U?feature=share
Abstract
We will recall earlier work where Apéry's proof of irrationality of ζ(3)was related to a continued fraction in Ramanujan's notebooks. We will then recall the method of Telescoping continued fractions from joint work with Bhatnagar (2023). We will then proceed to apply this method to certain series to prove several entries from Ramanujan's notebooks related toπ, ζ(2), ζ(3)andG,the Catalan's constant.
Best wishes,
Gaurav Bhatnagar, Atul Dixit and Krishnan Rajkumar (organisers)
This week we have a talk by Archna Kumari of IIT, Delhi. Here is the announcement.
Talk Announcement:
Title:Some results in weighted and elliptic enumeration
Speaker: Archna Kumari (IIT, Delhi) When: Mar 27, 2025, 4:00 PM- 5:00 PM IST (Our usual time)
Where: Zoom: Write to sf and nt at gmail dot com for the link
Live LInk: https://youtube.com/live/6U6er9-X7ZE?feature=share
Abstract
In the literature, there are a lot of q-identities. In this talk, we will talk about two types of q-identities and their extension to the elliptic case. First, we extend some Fibonacci identities using combinatorial methods. Since many of these identities can be derived through telescoping, we use this technique to find elliptic versions of basic elementary identities, such as the sum of the first odd or even numbers, the geometric series sum, and the sum of the first cubes. Along the way, we discover a multi-parameter identity that seems to be new, even in the q-setting. Second, wewill extend some q-hypergeometric identities to elliptic hypergeometric. We derive four expansion formulas and, as a result, some transformation formulas. In the q case, when the nome p=0, one of the formulas generalizes the basic hypergeometric transformation formula due to Liu, and Wang and Ma. The remaining three are equivalent to the well-poised Bailey lemma. Thus, we recover transformation formulas from Warnaar and Spiridonov.
This is joint work with Gaurav Bhatnagar and Michael Schlosser.
Next week's talk is by Atul Dixit of IIT, Gandhinagar. Here is the announcement.
Talk Announcement:
Title:The Rogers-Ramanujan dissection of a theta function
Speaker: Atul Dixit (IIT, Gandhinagar) When: Mar 6, 2025, 4:00 PM- 5:00 PM IST (Our usual time)
Where: Zoom: Write to the organisers for the link
Live Link: https://youtube.com/live/1v54wzJk-_o?feature=share
Abstract
Page 27 of Ramanujan's Lost Notebook contains a beautiful identity which, as shown by Andrews, not only gives a famous modular relation between the Rogers-Ramanujan functions G(q) and H(q) as a corollary but also a relation between two fifth order mock theta functions and G(q) and H(q). In this talk, we will discuss a generalization of Ramanujan's relation that we recently obtained which gives an infinite family of such identities. Our result shows that a theta function can always be ``dissected'' as a finite sum of products of generalized Rogers-Ramanujan functions.
Several well-known results are shown to be consequences of our theorem, for example, a generalization of the Jacobi triple product identity and Andrews' relation between two of his generalized third order mock theta functions. As will be shown, the identities resulting from our main theorem for s>2 transcend the modular world and hence look difficult to be written in the form of a modular relation. Using asymptotic analysis, we also offer compelling evidence that explains how Ramanujan may have arrived at his generalized modular relation. This is joint work with Gaurav Kumar.
The talk next week will be by James Sellers of the University of Minnesota, Duluth. We are back to our usual time, since the speaker is currently in Europe.
Talk Announcement:
Title:Partitions into Odd Parts with Designated Summands
Speaker: James Sellers (University of Minnesota, Duluth, USA) When: Feb 20, 2025, 4:30 PM- 5:30 PM IST (12 Noon CET) (Note the time).
In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects which they called partitions with designated summands. These are built by taking unrestricted integer partitions and designating exactly one of each occurrence of a part. In that same work, Andrews, Lewis, and Lovejoy also studied such partitions wherein all parts must be odd, and they denoted the number of such partitions of size n by the function PDO(n).
In this talk, I will report on recent proofs of infinite families of divisibility properties satisfied by PDO(n). Some of these proofs follow in elementary fashion while others rely on modular forms (in work completed jointly with Shane Chern).
I will then transition to very recent joint work with Shishuo Fu in which we consider a "refined" view of PDO(n) based on ideas which originated with P. A. MacMahon. This new approach allows for a more combinatorial view of the well-known identity that, for all n,
PDO(2n) = \sum_{0 \leq k \leq n} PDO(k) PDO(n-k),
a result which is (trivially) proven via generating functions but which has eluded combinatorial proof for many years.
The first talk of the year (on Feb 6, 2025) is a ``Ramanujan Special". This year's speaker is Krishnaswami Alladi. Among other things, he is the founding editor of the Ramanujan Journal. Please note that the talk will be later than usual.
Talk Announcement: The 2025 Ramanujan Special
Title:Speaker: Krishnaswami Alladi (University of Florida, USA)
When: Feb 6, 2025, 6:30 PM- 7:30 PM IST (8 AM EST) (Note special time)
(EST= IST - 10:30)
Where: Zoom: Write to sfandnt@gmail.com for a link
In 1977, I noticed two duality identities connecting the smallest and largest prime factors of integers, and vice-versa, the connection being provided by the Moebius function. Using this duality, I generalized the famous results of Edmund Landau on the Moebius function which are equivalent to the Prime Number Theorem. When this duality is combined with the Prime Number Theorem for Arithmetic Progressions, this leads to the striking result that
Our talk this week is by Jeremy Lovejoy of CNRS. Here is the announcement.
Talk Announcement:
Title:Bailey pairs, radial limits of q-hypergeometric false theta functions, and a conjecture of Hikami
Speaker: Jeremy Lovejoy (CNRS, Paris)
When: Nov 21, 2024, 4:00 PM- 5:00 PM IST (11:30 AM CET)
Where: Zoom: Please write to the organisers
Live LInk: https://youtube.com/live/PCvF5pykt9c?feature=share
Abstract:In the first part of this talk we describe a conjecture of Hikami on the values of the radial limits of a family of q-hypergeometric false theta functions. Hikami conjectured that the radial limits are obtained by evaluating a truncated version of the series. He proved a special case of his conjecture by computing the Kashaev invariant of certain torus links in two different ways. We show how to prove the full conjecture using Bailey pairs. In the second part ofthe talk we explain how the framework of Bailey pairs leads to further results of this type. The talkis based on joint work with Rishabh Sarma.
Best wishes,
Gaurav Bhatnagar, Atul Dixit and Krishnan Rajkumar (organisers)
Abstract:The Voronoi summation formulas for the divisor function and r_2(n) are well-known. Not only do these formulas have interesting structure, but they have also been used to improve the error term in the Dirichlet divisor problem and the Gauss circle problem respectively.
In this work we derive Voronoi summation formulas for some other functions related to the generalized divisor function-d^2(n) and Liouville Lambda function-\lambda(n) and Mobius function-\mu(n). We also make use of Vinogradov-Korobov zero free region for the Riemann zeta function to obtain the results.
We also derive beautiful analogues of Cohen's identity and the Ramanujan-Guinand formula associated to these functions.
We also derive certain Omega-bounds for the weighted sums of d^2(n), \lambda(n) and \mu(n) assuming the Linear Independence conjecture.
This is a joint work with Atul Dixit.
Best wishes,
Gaurav Bhatnagar, Atul Dixit and Krishnan Rajkumar (organisers)
Using physics methods, Saha and Sinha recently obtained many intriguing expansions of string amplitudes (see Sinha's talk in this seminar 3 October). From a mathematical perspective, they are very unusual deformations of classical hypergeometric identities. One very special case gives a deformation of Madhava's famous series for pi, which received a lot of media attention. I will discuss these identities from a mathematical perspective. They can be derived from partial fraction expansions for symmetric rational functions, which may have some independent interest.
We are back to our usual time with a talk by Aninda Sinha of IISc, Bangalore.
Next week (on Oct 10), we will have a follow-up talk by Hjalmar Rosengren on the topic: String amplitudes and partial fractions: A mathematician's perspective. The formal announcement will be made on the weekend. But please mark your calendars.
Talk Announcement:
Title:Field theory expansions of string theory amplitudes
Speaker:Aninda Sinha (Indian Institute of Science, Bangalore) When: Oct 3, 2024, 4:00 PM- 5:00 PM IST
This talk is based on: Phys. Rev. Lett.132 (2024) 22,221601 (e-print: 2401.05733 [hep-th])
I will explain the reasons (both physics and maths) for trying to find a new formula, satisfying certain physics inspired criteria, for the Euler-Beta and related functions. I will give a sketch of the derivation which relies on a novel dispersion relation. Towards the end, time permitting, new results from the Bootstrap will be presented.
Best wishes,
Gaurav Bhatnagar, Atul Dixit and Krishnan Rajkumar (organisers)
We are excited to begin after the summer break with a talk by Professor George Andrews, Atherton Professor of Mathematics, Penn State University.
Please note the unusual time for the seminar.
Talk Announcement:
Title:Extensions of MacMahon's Generalization of Sums of Divisors
Speaker: George Andrews (Penn State University) When: Sept 19, 2024, 5:30 PM- 6:30 PM IST (8:00 AM EDT)
Where: Zoom: Write to the organisers for the link.
Abstract:
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 theq-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.
