Colloquium abstracts

Atul Dixit*
IIT Gandhinagar
July 2, 2020

Superimposing theta structure on a generalized modular relation:  By a modular relation for a certain function F , we mean a relation governed by the map $z o -1/z$ but not necessarily by $z o z+1$. Equivalently, the relation can be written in the form $F(alpha) = F(eta)$, where $alphaeta = 1$. There are many generalized modular relations in the literature such as the general theta transformation of the form $F(w, alpha) = F (iw, eta)$ or the Ramanujan-Guinand formula of the form $F (z, alpha) = F (z, eta)$ etc. The latter, equivalent to the functional equation of the non-holomorphic Eisenstein series on $SL_2(Z)$, admits a beautiful generalization of the form $F(z,w,alpha) = F(z,iw,eta)$ 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?