Department of

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

Thursday, January 23, 2020

Thursday, January 30, 2020

Thursday, February 6, 2020

Thursday, February 13, 2020

Thursday, February 20, 2020

Thursday, February 27, 2020

Thursday, March 5, 2020

Thursday, April 2, 2020

Tuesday, April 7, 2020

Thursday, April 16, 2020

Thursday, April 30, 2020