## Monday, December 14, 2020

### Rahul Kumar (IIT, Gandhinagar), December 17, 2020, 3:55 PM - 5:00 PM IST

For the last seminar of the year, we are proud to host Rahul Kumar. He has recently defended his thesis in IIT, Gandhinagar under the direction of Atul Dixit, one of the co-organizers of this seminar. The announcement appears below.

Talk Announcement

Title:  A generalized modified Bessel function and explicit transformations of certain Lambert series

Speaker: Rahul Kumar (IIT, Gandhinagar)

When: December 17, 2020 - 3:55 PM - 5:00 PM (IST)

ABSTRACT:

An exact transformation, which we call a master identity, is obtained for the series $\sum_{n=1}^{\infty}\sigma_{a}(n)e^{-ny}$ for $a\in\mathbb{C}$ and Re$(y)>0$. As corollaries when $a$ is an odd integer, we derive the well-known transformations of the Eisenstein series on $\text{SL}_{2}\left(\mathbb{Z}\right)$, that of the Dedekind eta function as well as Ramanujan's famous formula for $\zeta(2m+1)$. Corresponding new transformations when $a$ is a non-zero even integer are also obtained as special cases of the master identity. These include a novel companion to Ramanujan's formula for $\zeta(2m+1)$. Although not modular, it is surprising that such explicit transformations exist. The Wigert-Bellman identity arising from the $a=0$ case of the master identity is derived too. The latter identity itself is derived using Guinand's version of the Vorono\"{\dotlessi} summation formula and an integral evaluation of N.~S.~Koshliakov involving a generalization of the modified Bessel function $K_{\nu}(z)$. Koshliakov's integral evaluation is proved for the first time. It is then generalized using a well-known kernel of Watson to obtain an interesting two-variable generalization of the modified Bessel function. This generalization allows us to obtain a new transformation involving the sums-of-squares function $r_k(n)$. This is joint work with Atul Dixit and Aashita Kesarwani.

## Sunday, December 13, 2020

### Conferences in December (of interest to the group)

Dear all,

The following are the conferences happening all over the country in December (mostly in honor of Ramanujan's birthday). You may forward this email to everyone in our group.

1. RMS & Rajagiri school of engineering and Technology: (Dec. 11-14, 2020)

2. 86th Annual conference of the Indian Mathematical Society (Dec. 17-20, 2020)

2. Recent Advances in Mathematics and Related Areas: (Dec. 18-22, 2020)

3. International Conference on Special Functions and Applications (Dec. 22-23, 2020)

(Deadline for contributed talks: Dec. 15, 2020)

4. International Conference on Number Theory and Algebra (Dec. 22-23, 2020)

(Contributed talks are invited. Deadline for registration: Dec. 18, 2020)

5. 35th Annual Conference of the Ramanujan Mathematical Society  (Dec. 28-30, 2020)

Best wishes,

Atul Dixit

## Saturday, December 5, 2020

### Sneha Chaubey (IIITD) December 10, 2020, 3:55 PM - 5:00 PM IST

The next talk in the SF and NT seminar is by Sneha Chaubey. There will be one further talk in the Seminar this year, next week, and then we will take a short break.

Here is the announcement for Sneha's talk.

Talk announcement

Title:  Generalized visible subsets of two dimensional integer lattice

Speaker: Sneha Chaubey (IIIT, Delhi)

When: December 10, 2020 - 3:55 PM - 5:00 PM (IST)