Some series equivalent to the extended Riemann hypothesis for Dedekind zeta functions

Abstract

The extended Riemann hypothesis (ERH) for Dedekind zeta functions remains one of the most elusive open problems in number theory. Over the last century, many equivalent statements to the classical Riemann hypothesis alone have been discovered. We prove that the closed form of some infinite series over the non-trivial zeros of Dedekind zeta functions holds if and only if ERH is true.