James Sellers (Minnesota, Duluth) - Thursday, Feb 20, 2025 - 4:30 PM (IST)

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).

Where: Zoom:

Live Link: https://youtube.com/live/yks1v1Tq68I?feature=share 

Abstract

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.

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.