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arXiv · 2407.03660

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

Diksha Rani Bansal·Bibekananda Maji
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Abstract

Ramanujan's famous formula for $ζ(2m+1)$ 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 $ζ(2m+1)$ 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 $η(z)$. 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.

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Diksha Rani Bansal, Bibekananda Maji. 2024-07-04. A Number Field Analogue of Ramanujan's identity for $ζ(2m+1)$. https://doi.org/10.1016/j.jmaa.2025.129538

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