Superimposing theta structure on a generalized modular relation

Abstract

A generalized modular relation of the form \(F(z, w, \alpha )=F(z, iw,\beta )\), where \(\alpha \beta =1\) and \(i=\sqrt{-1}\), is obtained in the course of evaluating an integral involving the Riemann \(\Xi \)-function. This modular relation involves a surprising new generalization of the Hurwitz zeta function \(\zeta (s, a)\), which we denote by \(\zeta _w(s, a)\). We show that \(\zeta _w(s, a)\) satisfies a beautiful theory generalizing that of \(\zeta (s, a)\). In particular, it is shown that for \(0<a<1\) and \(w\in \mathbb {C}\), \(\zeta _w(s, a)\) can be analytically continued to Re\((s)>-1\) except for a simple pole at \(s=1\). The theories of functions reciprocal in a kernel involving a combination of Bessel functions and of a new generalized modified Bessel function \({}_1K_{z,w}(x)\), which are also essential to obtain the generalized modular relation, are developed.