{
  "id": "1901.10373",
  "version": "v1",
  "published": "2019-01-29T16:37:47.000Z",
  "updated": "2019-01-29T16:37:47.000Z",
  "title": "A Ramanujan-type formula for $ζ^{2}(2m+1)$ and its generalizations",
  "authors": [
    "Atul Dixit",
    "Rajat Gupta"
  ],
  "comment": "29 pages, submitted for publication",
  "categories": [
    "math.NT",
    "math.CA"
  ],
  "abstract": "A Ramanujan-type formula involving the squares of odd zeta values is obtained. The crucial part in obtaining such a result is to conceive the correct analogue of the Eisenstein series involved in Ramanujan's formula for $\\zeta(2m+1)$. The formula for $\\zeta^{2}(2m+1)$ is then generalized in two different directions, one, by considering the generalized divisor function $\\sigma_z(n)$, and the other, by studying a more general analogue of the aforementioned Eisenstein series, consisting of one more parameter $N$. A number of important special cases are derived from the first generalization. For example, we obtain a series representation for $\\zeta(1+\\omega)\\zeta(-1-\\omega)$, where $\\omega$ is a non-trivial zero of $\\zeta(z)$. We also evaluate a series involving the modified Bessel function of the second kind in the form of a rational linear combination of $\\zeta(4k-1)$ and $\\zeta(4k+1)$ for $k\\in\\mathbb{N}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-29T16:37:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11J81"
    ],
    "keywords": [
      "ramanujan-type formula",
      "odd zeta values",
      "important special cases",
      "rational linear combination",
      "ramanujans formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}