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

TL;DR

This paper develops new analogues of Ramanujan's formula for the Riemann zeta function at odd integers within the context of number fields, extending classical identities and deriving applications to Eisenstein series, eta functions, and class numbers.

Contribution

It introduces a novel number field analogue of Ramanujan-Grosswald's formula for zeta functions at odd integers and extends related identities and applications.

Findings

Derived a formula for Dedekind zeta function at odd arguments.

Generalized transformation formulas for Eisenstein series and eta functions.

Established a new connection between class numbers and Dedekind zeta functions.

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.