A modular relation involving non-trivial zeros of the dedekind zeta function, and the generalized Riemann hypothesis

Abstract

We give a number field analogue of a result of Ramanujan, Hardyand Littlewood,thereby obtaining a modular relation involving the non-trivial zeros of the Dedekind zeta function.We also provide a Riesz-type criterion for the Generalized Riemann Hypothesis for?K(s). Newelegant transformations are obtained whenKis a quadratic extension, one of which involves themodified Bessel function of the second kind.