Superimposing theta structure on a generalized modular relation

A generalized modular relation of the form F(z, w, α) = F(z, iw, β) , where αβ= 1 and i=-1, is obtained in the course of evaluating an integral involving the Riemann Ξ -function. This modular relation involves a surprising new generalization of the Hurwitz zeta function ζ(s, a) , which we denote by ζ<inf>w</inf>(s, a). We show that ζ<inf>w</inf>(s, a) satisfies a beautiful theory generalizing that of ζ(s, a). In particular, it is shown that for 0 < a< 1 and w∈ C, ζ<inf>w</inf>(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 <inf>1</inf>K<inf>z</inf><inf>,</inf><inf>w</inf>(x) , which are also essential to obtain the generalized modular relation, are developed.