标题： 数域中Jacobi Theta关系及Dedekind zeta函数在Re$(s)=1/2$上的零点 显示英文标题

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

摘要： 1914年，哈代证明了黎曼zeta函数$\zeta(s)$在临界线 Re$(s)=1/2$上存在无限多个非平凡零点，这是利用雅可比theta关系得出的。 在本文中，我们首先建立了雅可比theta关系的数域类比，并作为应用，我们展示了对于任何数域$\mathbb{F}$，戴德金zeta函数$\zeta_\mathbb{F}(s)$在 Re$(s)=1/2$上存在无限多个非平凡零点。 有趣的是，我们还证明了雅可比theta关系与哈代、李特尔伍德和拉马努金的一个引人入胜的恒等式是等价的。 显示英文摘要

摘要： 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.