AI 中文总结
本文为戴德金ζ函数建立拉马努金-吉南恒等式的数域类似物，得到ζ_F(2)的新公式及ζ_F(2m+1)的恒等式，还获戴德金η函数对数的变换公式。
AI 中文摘要
在拉马努金的《失落笔记》第253页，他记录了一个关联广义除数函数与修正贝塞尔函数的有趣恒等式，该恒等式后被吉南重新发现，现称为拉马努金-吉南恒等式。本文中，我们为戴德金ζ函数建立了该恒等式的数域类似物，特别地，我们得到了拉马努金-吉南恒等式、拉马努金-科什利亚科夫恒等式，还获得了戴德金η函数对数的变换公式。1986年扎吉尔得到了任意数域F的ζ_F(2)公式，令人惊讶的是，作为我们主定理的应用，我们还得到了任意数域F的ζ_F(2)的新公式，此外，我们推导出了对任意正整数m和任意数域F的ζ_F(2m+1)的优美恒等式。
英文摘要
On page 253 of his Lost Notebook, Ramanujan recorded an intriguing identity relating a generalized divisor function and a modified Bessel function, which was later rediscovered by Guinand and is now known as the Ramanujan-Guinand identity. In this paper, we establish a number field analogue of this identity for the Dedekind zeta function. In particular, we recover Ramanujan-Guinand identity, Ramanujan-Koshliakov identity, and also obtain the transformation formula for the logarithm of the Dedekind eta function. In 1986, Zagier obtained a formula for $ζ_\mathbb{F}(2)$ for any number field $\mathbb{F}$. Quite surprisingly, as an application of our main theorem, we also obtain a new formula for $ζ_\mathbb{F}(2)$ for arbitrary number field $\mathbb{F}$. Moreover, we derive an elegant identity for $ζ_\mathbb{F}(2m+1)$ for any positive integer $m$ and any number field $\mathbb{F}$.
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