Superimposing theta structure on a generalized modular relation

Abstract

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