Number Field Analogue of Jacobi Theta Relation And Zeros of Dedekind zeta function on Re$(s)=1/2$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916861368598528 |
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| author | Bansal, Diksha Rani Maji, Bibekananda |
| author_facet | Bansal, Diksha Rani Maji, Bibekananda |
| contents | 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 $ζ_\mathbb{F}(s)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_18121 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Number Field Analogue of Jacobi Theta Relation And Zeros of Dedekind zeta function on Re$(s)=1/2$ Bansal, Diksha Rani Maji, Bibekananda Number Theory Primary 11M06, 11R42, Secondary 11M26 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 $ζ_\mathbb{F}(s)$ 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. |
| title | Number Field Analogue of Jacobi Theta Relation And Zeros of Dedekind zeta function on Re$(s)=1/2$ |
| topic | Number Theory Primary 11M06, 11R42, Secondary 11M26 |
| url | https://arxiv.org/abs/2507.18121 |