Some Series Related to Extended Riemann Hypothesis for Dedekind Zeta Functions

TL;DR

This paper derives closed-form expressions for series involving Dedekind zeta function zeros under the Extended Riemann Hypothesis and shows these forms are equivalent to the hypothesis itself.

Contribution

It provides new series representations linked to the zeros of Dedekind zeta functions and establishes their equivalence to the Extended Riemann Hypothesis.

Findings

Closed-form series involving Dedekind zeta zeros derived

Equivalence between series forms and the Extended Riemann Hypothesis proven

New criteria for the validity of the Riemann Hypothesis for Dedekind zeta functions

Abstract

We obtain closed form of some infinite series involving derivatives of an analogue of the Riemann xi function for Dedekind zeta function and nontrivial zeros of Dedekind zeta function assuming the Extended Riemann Hypothesis. Conversely, we prove that if this closed form holds, then all of the zeros of Dedekind zeta function beyond a certain height lie on the critical line. This yields a large number of equivalent statements of Riemann Hypothesis.