{
  "id": "1508.04084",
  "version": "v1",
  "published": "2015-08-17T17:08:30.000Z",
  "updated": "2015-08-17T17:08:30.000Z",
  "title": "On some integrals involving the Hurwitz-type Euler zeta functions",
  "authors": [
    "Su Hu",
    "Daeyeoul Kim",
    "Min-Soo Kim"
  ],
  "comment": "18 pages",
  "categories": [
    "math.CA",
    "math-ph",
    "math.MP",
    "math.NT"
  ],
  "abstract": "The Hurwitz-type Euler zeta function is defined as a deformation of the Hurwitz zeta function: \\begin{equation*} \\zeta_E(s,x)=\\sum_{n=0}^\\infty\\frac{(-1)^n}{(n+x)^s}. \\end{equation*} In algebraic number theory, it represents a partial zeta function of cyclotomic fields in one version of Stark's conjectures (see [12, p.4249, (6.13)]). Its special case, the alternative series \\begin{equation*}~\\label{Riemann}\\zeta_{E}(s)=\\sum_{n=1}^{\\infty}\\frac{(-1)^{n-1}}{n^{s}},\\end{equation*} has also appeared as a particular situation of the well-known Witten zeta functions in mathematical physics (see [15, p.248, (3.14)]). In this paper, by using the method of Fourier expansions, we shall evaluate several integrals with integrands involving Hurwitz-type Euler zeta functions $\\zeta_E(s,x)$. There are the analogues of Espinosa and Moll's formulas in [7] for Hurwitz zeta functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-17T17:08:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33B15",
      "33E20",
      "11M35",
      "11B68"
    ],
    "keywords": [
      "hurwitz-type euler zeta function",
      "hurwitz zeta function",
      "well-known witten zeta functions",
      "algebraic number theory",
      "partial zeta function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150804084H"
    }
  }
}