{
  "id": "1602.06813",
  "version": "v1",
  "published": "2016-02-22T15:21:08.000Z",
  "updated": "2016-02-22T15:21:08.000Z",
  "title": "On generalized Eisenstein series and Ramanujan's formula for periodic zeta-functions",
  "authors": [
    "M. Cihat Dağlı",
    "Mümün Can"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper, transformation formulas for a large class of Eisenstein series defined by \\[ G(z,s;A_{\\alpha},B_{\\beta};r_{1},r_{2})=\\sum\\limits_{m,n=-\\infty}^{\\infty }\\ \\hspace{-0.19in}^{^{\\prime}}\\frac{f(\\alpha m)f^{\\ast}(\\beta n)} {((m+r_{1})z+n+r_{2})^{s}},\\text{ }\\operatorname{Re}(s)>2,\\text{ }\\operatorname{Im}(z)>0 \\] are investigated for $s=1-r$, $r\\in\\mathbb{N}$. Here $\\left\\{ f(n)\\right\\}$ and $\\left\\{ f^{\\ast}(n)\\right\\}$, $-\\infty<n<\\infty$ are sequences of complex numbers with period $k>0$, and $A_{\\alpha}=\\left\\{ f(\\alpha n)\\right\\} $ and $B_{\\beta}=\\left\\{ f^{\\ast}(\\beta n)\\right\\}$, $\\alpha,\\beta\\in\\mathbb{Z}$. Appearing in the transformation formulas are generalizations of Dedekind sums involving the periodic Bernoulli function. Reciprocity law is proved for periodic Apostol-Dedekind sum outside of the context of the transformation formulas. Furthermore, transformation formulas are presented for $G(z,s;A_{\\alpha},I;r_{1},r_{2})$ and $G(z,s;I,A_{\\alpha };r_{1},r_{2})$, where $I=\\left\\{ 1\\right\\}$. As an application of these formulas, analogues of Ramanujan's formula for periodic zeta-functions are derived.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-22T15:21:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M36",
      "11M41",
      "11F20",
      "11B68"
    ],
    "keywords": [
      "generalized eisenstein series",
      "ramanujans formula",
      "periodic zeta-functions",
      "transformation formulas",
      "periodic apostol-dedekind sum outside"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160206813C"
    }
  }
}