Department of

June 2020 July 2020August 2020Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su MoTuWe Th Fr Sa 1 2 3 4 5 6 1 2 3 4 1 7 8 9 10 11 12 13 5 6 7 8 9 10 11 2 3 4 5 6 7 8 14 15 16 17 18 19 20 12 13 14 15 16 17 18 9 101112 13 14 15 21 22 23 24 25 26 27 19 20 21 22 23 24 25 16 17 18 19 20 21 22 28 29 30 26 27 28 29 30 31 23 24 25 26 27 28 29 30 31

Tuesday, January 14, 2020

**Abstract:** By a modular relation for a certain function $F$, we mean that which is 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 $\mathrm{SL}_{2}(\mathbb{Z})$, admits a beautiful generalization of the form $F(z, w,\alpha)=F(z, iw, \beta)$, that is, one can superimpose theta structure on it.

Recently, a modular relation involving infinite series of the Hurwitz zeta function $\zeta(z, a)$ was obtained. It generalizes a result of Ramanujan from the Lost Notebook. Can one superimpose theta structure on it? 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 the whole complex plane except $z=1$. Hurwitz's formula for $\zeta(z, a)$ is also generalized in this setting. We also prove a generalized modular relation involving infinite series of $\zeta_w(z, a)$, which is of the form $F(z, w,\alpha)=F(z, iw, \beta)$. This is joint work with Rahul Kumar.

Friday, January 17, 2020

Tuesday, January 21, 2020

Wednesday, January 22, 2020

Thursday, January 23, 2020

Friday, January 24, 2020

Monday, January 27, 2020

Tuesday, January 28, 2020

Wednesday, January 29, 2020

Thursday, January 30, 2020

Friday, January 31, 2020

Monday, February 3, 2020

Tuesday, February 4, 2020

Wednesday, February 5, 2020

Thursday, February 6, 2020

Friday, February 7, 2020

Saturday, February 8, 2020

Monday, February 10, 2020

Tuesday, February 11, 2020

Wednesday, February 12, 2020

Thursday, February 13, 2020

Friday, February 14, 2020

Monday, February 17, 2020

Tuesday, February 18, 2020

Wednesday, February 19, 2020

Thursday, February 20, 2020

Friday, February 21, 2020

Monday, February 24, 2020

Tuesday, February 25, 2020

Wednesday, February 26, 2020

Thursday, February 27, 2020

Friday, February 28, 2020

Sunday, March 1, 2020

Monday, March 2, 2020

Tuesday, March 3, 2020

Wednesday, March 4, 2020

Thursday, March 5, 2020

Friday, March 6, 2020

Monday, March 9, 2020

Tuesday, March 10, 2020

Wednesday, March 11, 2020

Thursday, March 12, 2020

Friday, March 13, 2020

Monday, March 23, 2020

Tuesday, March 24, 2020

Wednesday, March 25, 2020

Thursday, March 26, 2020

Friday, March 27, 2020

Monday, March 30, 2020

Tuesday, March 31, 2020

Wednesday, April 1, 2020

Thursday, April 2, 2020

Friday, April 3, 2020

Monday, April 6, 2020

Tuesday, April 7, 2020

Wednesday, April 8, 2020

Thursday, April 9, 2020

Friday, April 10, 2020

Monday, April 13, 2020

Tuesday, April 14, 2020

Wednesday, April 15, 2020

Thursday, April 16, 2020

Friday, April 17, 2020

Monday, April 20, 2020

Tuesday, April 21, 2020

Wednesday, April 22, 2020

Thursday, April 23, 2020

Friday, April 24, 2020

Monday, April 27, 2020

Tuesday, April 28, 2020

Wednesday, April 29, 2020

Thursday, April 30, 2020

Friday, May 1, 2020

Monday, May 4, 2020

Tuesday, May 5, 2020

Wednesday, May 6, 2020

Thursday, May 7, 2020

Tuesday, May 12, 2020

Wednesday, May 13, 2020

Tuesday, May 19, 2020

Tuesday, May 26, 2020

Tuesday, June 2, 2020

Tuesday, June 9, 2020

Tuesday, June 16, 2020

Tuesday, June 23, 2020

Thursday, June 25, 2020

Tuesday, June 30, 2020

Tuesday, July 7, 2020

Tuesday, July 14, 2020

Tuesday, July 21, 2020

Tuesday, July 28, 2020

Tuesday, August 4, 2020

Thursday, August 6, 2020

Tuesday, August 11, 2020

Thursday, August 13, 2020

Thursday, August 20, 2020

Thursday, August 27, 2020

Thursday, September 3, 2020

Thursday, September 10, 2020

Thursday, September 24, 2020

Thursday, October 22, 2020

Thursday, October 29, 2020

Thursday, November 5, 2020