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arXiv · 2507.18121

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

Diksha Rani Bansal·Bibekananda Maji
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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.

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Diksha Rani Bansal, Bibekananda Maji. 2025-07-24. Number Field Analogue of Jacobi Theta Relation And Zeros of Dedekind zeta function on Re$(s)=1/2$. https://arxiv.org/abs/2507.18121

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