Number Field Analogue of Jacobi Theta Relation And Zeros of Dedekind zeta function on Re$(s)=1/2$

TL;DR

This paper extends Hardy's classical result on the zeros of the Riemann zeta function to Dedekind zeta functions of number fields by establishing a number field analogue of the Jacobi theta relation, linking theta identities to zero distributions.

Contribution

It introduces a novel number field analogue of the Jacobi theta relation and proves the existence of infinitely many zeros on the critical line for Dedekind zeta functions.

Findings

Infinitely many zeros of Dedekind zeta functions on Re(s)=1/2

Number field analogue of Jacobi theta relation established

Jacobi theta relation equivalent to classical identities of Hardy, Littlewood, Ramanujan

Abstract

In 1914, Hardy proved that there are infinitely many non-trivial zeros of the Riemann zeta function $\zeta(s)$ on the critical line Re$(s)=1/2$ using the Jacobi theta relation. In this paper, we first establish a number field analogue of the Jacobi theta relation and as an application, we show the existence of infinitely many non-trivial zeros of the Dedekind zeta function $\zeta_\mathbb{F}(s)$ on Re$(s)=1/2$, for any number field $\mathbb{F}$. Quite interestingly, we also prove that the Jacobi theta relation is equivalent to an intriguing identity of Hardy, Littlewood and Ramanujan.