IMSc Webinar

Superimposing theta structure on a generalized modular relation

Atul Dixit

IIT Gandhinagar

By a modular relation for a certain function F, we mean a relation governed by the map z → −1/z but not necessarily by z → z+1. Equivalently, the relation can be written in the form F(α) = F(β), where αβ = 1. There are many generalized modular relations in the literature such as the general theta transformation of the form F(w, α) = F(iw, β) or the Ramanujan-Guinand formula of the form F(z, α) = F(z, β) etc. The latter, equivalent to the functional equation of the non-holomorphic Eisenstein series on SL2(Z), admits a beautiful generalization of the form F(z, w, α) = F(z, iw, β) 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 ζ(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 ζ_(z, a). We show that this new zeta function, ζ_w(z, a), satisfies a beautiful theory. In particular, it is shown that ζ_w(z, a) can be analytically continued to Re(z) > −1, z ≠ 1. We also prove a two-variable generalization of Ramanujan’s formula which involves infinite series of ζ_w(z, a) and which is of the form F(z, w, α) = F(z, iw, β). This is joint work with Rahul Kumar.
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