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

TL;DR

This paper establishes a modular relation involving the non-trivial zeros of the Dedekind zeta function for number fields, and introduces a Riesz-type criterion for the Generalized Riemann Hypothesis, with special cases for quadratic extensions.

Contribution

It extends classical results to the setting of number fields and provides new criteria and transformations related to the Galois zeros of Dedekind zeta functions.

Findings

Derived a number field analogue of Ramanujan, Hardy, and Littlewood's result.

Established a Riesz-type criterion for the Generalized Riemann Hypothesis.

Obtained new transformations involving the modified Bessel function for quadratic extensions.

Abstract

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 $\zeta_K(s)$. New elegant transformations are obtained when $K$ is a quadratic extension, one of which involves the modified Bessel function of the second kind.