A Number Field Analogue of Ramanujan's identity for $\zeta (2m+1)$

Abstract

Ramanujan's famous formula for $\zeta (2m+1)$ has captivated the attention of numerous mathematicians over the years. Grosswald, in 1972, found a simple extension of Ramanujan's formula which in turn gives transformation formula for Eisenstein series over the full modular group. Recently, Banerjee, Gupta and Kumar found a number field analogue of Ramanujan's formula. In this paper, we present a new number field analogue of the Ramanujan-Grosswald formula for $\zeta (2m+1)$ by obtaining a formula for Dedekind zeta function at odd arguments. We also obtain a number field analogue of an identity of Chandrasekharan and Narasimhan, which played a crucial role in proving our main identity. As an application, we generalize transformation formula for Eisenstein series $G_{2k}(z)$ and Dedekind eta function $\eta (z)$. A new formula for the class number of a totally real number field is also obtained, which provides a connection with the Kronceker's limit formula for the Dedekind zeta function.

33 pages, comments are welcome!