Ramanujan Special: Krishnaswami Alladi (Florida) - Thursday, Feb 6, 2025 - 6:30 PM (IST)

 Happy new year. 

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

Live LInk: https://youtube.com/live/nDmJ9IUM2xA?feature=share

Abstract

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

$$

\sum_{n\ge 2, \, p(n)\equiv\ell(mod\,k)}\frac{\mu(n)}{n}=-\frac{1}{\phi(k)}\tag1

$$ 

for all positive integers $k$, where $1\le\ell\le k$, $(\ell, k)=1$, $\mu(n)$ is

the Moebius function, $p(n)$ is the smallest prime factor of $n$, and $\phi(k)$

is the Euler function. In the last few years, this duality and identity (1) have

attracted considerable attention, and extended to the setting of algebraic

number fields by the use of the Chebotarev Density Theorem by several young researchers.

In the 1977 work, I established four general duality identities connecting the

smallest prime factor with the $k$-th largest prime factor, and the $k$-th

smallest prime factor with the largest prime factor, utilizing not just

$\mu(n)$, but also $\omega(n)$, the number the distinct prime factors of $n$, a function first systematically studied by Hardy and Ramanujan in 1917.

Recently, along with my PhD student Jason Johnson, I used the second order

duality together with the Prime Number Theorem for Arithmetic Progressions

to establish

$$

\sum_{n\ge 2, \, p(n)\equiv\ell(mod\,k)}\frac{\mu(n)\omega(n)}{n}=0.\tag2

$$

The proof involves a variety of elementary and analytic techniques, and a study

of the distribution of the second largest prime factor of $n$. Identity (2)

has more recently been extended to the setting of algebraic number fields

using the Chebotarev Density of Theorem by Sroyon Sengupta, another of my

PhD students. Starting from the 1977 results, I will

discuss all the recent developments pertaining to this problem.

Ramanujan Special: Frank Garvan (Florida) - Thursday, Jan 25, 2024 - 7:30 PM (IST)

 Happy new year. 

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.

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

Abstract

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.

Ramanujan Special: Shaun Cooper (Massey University) - Thursday, Feb 2, 2023 - 2:30 PM

 Happy new year. 

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.

Several conjectures will be presented.

Here are the slides of the talk.

Ramanujan Special: Alan Sokal (University College, London and New York University) Thursday, January 27, 2022, 4-5 PM (IST)

Dear all, 

Welcome to 2022. We begin the year with a Ramanujan Special talk by Alan Sokal. The talk announcement is below. 

We encourage you to distribute this announcement to friends and colleagues in your department or otherwise, so that they come to know of our seminar. 

Talk Announcement:

Title: Coefficientwise Hankel-total positivity

Speaker: Alan Sokal (University College London and New York)

 

When: Thursday, January 27,  2022 - 4:00 PM - 5:00 PM (IST) 

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

Live Link: https://youtu.be/wUo4QcAS3mQ

Click here for the presentation

Tea or Coffee: Please bring your own.

Abstract:  

  A matrix $M$ of real numbers is called totally positive
   if every minor of $M$ is nonnegative.  Gantmakher and Krein showed
   in 1937 that a Hankel matrix $H = (a_{i+j})_{i,j \ge 0}$
   of real numbers is totally positive if and only if the underlying
   sequence $(a_n)_{n \ge 0}$ is a Stieltjes moment sequence.
   Moreover, this holds if and only if the ordinary generating function
   $\sum_{n=0}^\infty a_n t^n$ can be expanded as a Stieltjes-type
   continued fraction with nonnegative coefficients:
$$
   \sum_{n=0}^{\infty} a_n t^n
   \;=\;
   \cfrac{\alpha_0}{1 - \cfrac{\alpha_1 t}{1 - \cfrac{\alpha_2 t}{1 -  \cfrac{\alpha_3 t}{1- \cdots}}}}
$$
   (in the sense of formal power series) with all $\alpha_i \ge 0$.
   So totally positive Hankel matrices are closely connected with
   the Stieltjes moment problem and with continued fractions.

   Here I will introduce a generalization:  a matrix $M$ of polynomials
   (in some set of indeterminates) will be called
   coefficientwise totally positive if every minor of $M$
   is a polynomial with nonnegative coefficients.   And a sequence
   $(a_n)_{n \ge 0}$ of polynomials will be called
   coefficientwise Hankel-totally positive if the Hankel matrix
   $H = (a_{i+j})_{i,j \ge 0}$  associated to $(a_n)$ is coefficientwise
   totally positive.  It turns out that many sequences of polynomials
   arising naturally in enumerative combinatorics are (empirically)
   coefficientwise Hankel-totally positive.  In some cases this can
   be proven using continued fractions, by either combinatorial or
   algebraic methods;  I will sketch how this is done.  In many other
   cases it remains an open problem.

   One of the more recent advances in this research is perhaps of
   independent interest to special-functions workers:
   we have found branched continued fractions for ratios of contiguous
   hypergeometric series ${}_r \! F_s$ for arbitrary $r$ and $s$,
   which generalize Gauss' continued fraction for ratios of contiguous
   ${}_2 \! F_1$.  For the cases $s=0$ we can use these to prove
   coefficientwise Hankel-total positivity.

   Reference: Mathias P\'etr\'eolle, Alan D.~Sokal and Bao-Xuan Zhu,
   arXiv:1807.03271

Ramanujan Special: Wadim Zudilin on Thursday, January 7, 2021, 3:55 pm-5 PM

A very happy new year to all. We have decided that the first talk of every year will be a  Ramanujan Special Talk. This year a colloquium talk will be given by Wadim Zudilin. The announcement is below.

We wish you many new theorems, ideas and papers in 2021. Please do send any ideas or suggestions you have for the organisers to make this seminar more successful and help serve the interests of this community. 

Talk Announcement

Title: 10 years of q-rious positivity. More needed!

Speaker: Wadim Zudilin (Radboud University, Nijmegen).

Date and Time: Thursday, January 7, 2021, 3:55 PM IST (GMT+5:30)

Tea or coffee: Bring your own.

Where: Zoom: Please write to sfandnt@gmail.com for a link at-least 24 hours before the talk.

Abstract:

 

The $q$-binomial coefficients \[ \prod_{i=1}^m(1-q^{n-m+i})/(1-q^i),\] for integers $0\le m\le n$, are known to be polynomials with non-negative integer coefficients. This readily follows from the $q$-binomial theorem, or the many combinatorial interpretations of them. Ten years ago, together with Ole Warnaar we observed that this non-negativity (aka positivity) property generalises to products of ratios of $q$-factorials that happen to be polynomials; we prove this observation for (very few) cases. During the last decade a resumed interest in study of generalised integer-valued factorial ratios, in connection with problems in analytic number theory and combinatorics, has brought to life new positive structures for their $q$-analogues. In my talk I will report on this "$q$-rious positivity" phenomenon, an ongoing project with Warnaar.