{
  "id": "2407.03660",
  "version": "v1",
  "published": "2024-07-04T06:02:49.000Z",
  "updated": "2024-07-04T06:02:49.000Z",
  "title": "A Number Field Analogue of Ramanujan's identity for $ζ(2m+1)$",
  "authors": [
    "Diksha Rani Bansal",
    "Bibekananda Maji"
  ],
  "comment": "33 pages, comments are welcome!",
  "categories": [
    "math.NT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-04T06:02:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11R42",
      "11R29"
    ],
    "keywords": [
      "number field analogue",
      "ramanujans identity",
      "dedekind zeta function",
      "transformation formula",
      "eisenstein series"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}