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

We give a number field analogue of a result of Ramanujan, Hardy and 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 ζ<inf>K</inf>(s). New elegant transformations are obtained when K is a quadratic extension, one of which involves the modified Bessel function of the second kind.