## Thursday, June 4, 2020

### Atul Dixit

We are back in business, after a short break due to the lockdown. We decided to begin the seminar online. There is a great advantage to this. First of all, many people in Delhi who were not in a position to travel to JNU or ISI can now attend from home. Secondly we have greatly expanded the ability to get good speakers for the seminar. Finally, people working in related areas around the country, and perhaps in Europe and Singapore can now contribute. We wish at present to keep the timings as per our original calendar (suitable to our schedule in India) but if we get a great speaker, we may modify as per their convenience. We will continue to meet once every two weeks.

As our first talk of the new season, we are delighted to have as a speaker Atul Dixit from IIT, Gandhinagar. Atul has developed a small group of students and post-docs in Gandhinagar, doing interesting mathematics related to the themes covered in our seminar, and we hope they will join this group.

Title: Superimposing theta structure on a generalized modular relation\\

Speaker: Atul Dixit, IIT, Gandhinagar

When: Thursday, June 4, 2020; 3:55-5:00 pm IST (GMT+5:30)

Abstract: By a modular relation for a certain function $F$, we mean a relation governed by the map $z\to -1/z$ but not necessarily by $z\to z+1$. Equivalently, the relation can be written in the form $F(\alpha)=F(\beta)$, where $\alpha\beta=1$. There are many generalized modular relations in the literature such as the general theta transformation $F(w,\alpha)=F(iw, \beta)$ or the Ramanujan-Guinand formula $F(z, \alpha)=F(z, \beta)$ etc. The latter, equivalent to the functional equation of the non-holomorphic Eisenstein series on $\operatorname{SL}_{2}(\mathbb{Z})$, admits a beautiful generalization of the form $F(z, w,\alpha)=F(z, iw, \beta)$ obtained by Kesarwani, Moll and the speaker, that is, one can superimpose theta structure on it.
In 2011, the speaker obtained a generalized modular relation involving infinite series of the Hurwitz zeta function $\zeta(z, a)$. It generalizes a result of Ramanujan from the Lost Notebook. Can one superimpose theta structure on the generalized modular relation? While answering this question affirmatively, we were led to a surprising new generalization of $\zeta(z, a)$. We show that this new zeta function, $\zeta_w(z, a)$, satisfies a beautiful theory. In particular, it is shown that $\zeta_w(z, a)$ can be analytically continued to Re$(z)>-1, z\neq1$. We also prove a two-variable generalization of Ramanujan's formula which involves infinite series of $\zeta_w(z, a)$ and which is of the form $F(z, w,\alpha)=F(z, iw, \beta)$. This is joint work with Rahul Kumar.