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

In 1914, Hardy proved that there are infinitely many non-trivial zeros of the Riemann zeta function $ζ(s)$ 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 ζF(s) on Re$(s)=1/2$, for any number field F. Quite interestingly, we also prove that the Jacobi theta relation is equivalent to an intriguing identity of Hardy, Littlewood and Ramanujan.