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

Diksha Rani Bansal, Bibekananda Maji

TL;DR

This paper develops a number-field analogue of Ramanujan's identity for odd zeta values by introducing a framework based on Dedekind zeta functions, a Steen-function–driven Lambert-type series $\mathfrak{F}_{\mathbb{F},k}(z)$, and a key Dirichlet-series $\Lambda_{\mathbb{F},k}(s)$. A central transformation theorem (Theorem DB) relates $\mathfrak{F}_{\mathbb{F},2k+1}(z)$ to its value at $-1/z$ with explicit residue contributions, and specialized results are obtained for totally real and purely imaginary fields. The authors derive number-field analogues of Ramanujan-Grosswald identities, a Dedekind eta analogue for $k=0$, and exact class-number-type evaluations that tie into Kronecker's limit formula. Together, these results extend Ramanujan-type phenomena to the Dedekind zeta setting, with applications to Eisenstein series transformations and eta-type identities in arithmetic geometry.

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.

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

TL;DR

This paper develops a number-field analogue of Ramanujan's identity for odd zeta values by introducing a framework based on Dedekind zeta functions, a Steen-function–driven Lambert-type series , and a key Dirichlet-series . A central transformation theorem (Theorem DB) relates to its value at with explicit residue contributions, and specialized results are obtained for totally real and purely imaginary fields. The authors derive number-field analogues of Ramanujan-Grosswald identities, a Dedekind eta analogue for , and exact class-number-type evaluations that tie into Kronecker's limit formula. Together, these results extend Ramanujan-type phenomena to the Dedekind zeta setting, with applications to Eisenstein series transformations and eta-type identities in arithmetic geometry.

Abstract

Ramanujan's famous formula for 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 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 and Dedekind eta function . 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.
Paper Structure (9 sections, 16 theorems, 157 equations)

This paper contains 9 sections, 16 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Dedekind zeta function
  3. Main Results
  4. Transformation formulae for totally real number fields
  5. Transformation formulae for imaginary number fields
  6. A number field analogue of transformation formula for Dedekind eta function
  7. Required Tools
  8. Proof of Main Results
  9. Concluding Remarks

Key Result

Theorem 2.1

Let $\mathbb{F}$ be a number field of degree $d=r_1+2r_2$ and $\zeta_{\mathbb{F}}(s)$ be the Dedekind zeta function defined in Dedekind. We consider $r =r_1+r_2-1$. For any non-zero integer $k$, we define Let $\mathfrak{F}_{\mathbb{F}, k}(z)$ be the infinite series defined as in Infinite series F. Then for any $z \in \mathbb{H}$ and $k >0$, we have where and the residual terms are defined as A

Theorems & Definitions (36)

  • Theorem 2.1
  • remark 1
  • Corollary 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Theorem 2.5
  • remark 2
  • Corollary 2.6
  • remark 3
  • Corollary 2.7
  • ...and 26 more