An extension of Ramanujan-Guinand identity for the Dedekind zeta function and a new formula for ${\zeta }_{\mathbb{F}}(2)$ and ${\zeta }_{\mathbb{F}}(2m+1)$

Abstract

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 ${\zeta }_{\mathbb{F}}(2)$ for any number field $\mathbb{F}$. Quite surprisingly, as an application of our main theorem, we also obtain a new formula for ${\zeta }_{\mathbb{F}}(2)$ for arbitrary number field $\mathbb{F}$. Moreover, we derive an elegant identity for ${\zeta }_{\mathbb{F}}(2m+1)$ for any positive integer $m$ and any number field $\mathbb{F}$.

23 pages, comments are welcome!