{
  "id": "1405.6270",
  "version": "v2",
  "published": "2014-05-24T05:37:48.000Z",
  "updated": "2014-12-08T08:38:49.000Z",
  "title": "A few remarks on values of Hurwitz Zeta function at natural and rational arguments",
  "authors": [
    "Paweł J. Szabłowski"
  ],
  "comment": "16 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We exploit some properties of the Hurwitz zeta function $\\zeta (n,x)$ in order to study sums of the form $\\frac{1}{\\pi ^{n}}\\sum_{j=-\\infty}^{\\infty}1/(jk+l)^{n}$ and $\\frac{1}{\\pi ^{n}}\\sum_{j=-\\infty}^{\\infty}(-1)^{j}/(jk+l)^{n}$ for $% 2\\leq n,k\\in \\mathbb{N},$ and integer $l\\leq k/2$. We show that these sums are algebraic numbers. We also show that $1<n\\in \\mathbb{N}$ and $p\\in \\mathbb{Q\\cap (}0,1\\mathbb{)}$ $:$ the numbers $(\\zeta (n,p)+(-1)^{n}\\zeta (n,1-p))/\\pi ^{n}$ are algebraic. On the way we find polynomials $s_{m}$ and $c_{m}$ of order respectively $2m+1$ and $2m+2$ such that their $n-$th coefficients of sine and cosine Fourier transforms are equal to $% (-1)^{n}/n^{2m+1}$ and $(-1)^{n}/n^{2m+2}$ respectively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-24T05:37:48.000Z",
      "title": "A few remarks on the relationship between Riemann and Hurwitz Zeta functions",
      "abstract": "We expose and exploit relationship between Riemann and Hurwitz zeta functions basically to study sums of the form $\\frac{1}{\\pi ^{n}}% \\sum_{j=-\\infty }^{\\infty }1/(jk+l)^{n}$ and $\\frac{1}{\\pi ^{n}}% \\sum_{j=-\\infty }^{\\infty }(-1)^{j}/(jk+l)^{n}$ for integer $n,k>1,$ and $% l\\leq k/2$. We show that for odd $k$ or $k\\allowbreak =\\allowbreak 2^{m}$ for some $m$ natural first sums while for even $k$ second sums are algebraic numbers. Hence we generalize known result for $k\\allowbreak =\\allowbreak 4$ and $l\\allowbreak =\\allowbreak 1$ dating back to L. Euler. Besides we recall and prove in believed to be new way known relationships between Hurwitz and Riemann Zeta functions.",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-12-08T08:38:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M35",
      "11M06",
      "11M36",
      "11J72"
    ],
    "keywords": [
      "natural first sums",
      "riemann zeta functions",
      "hurwitz zeta functions",
      "study sums",
      "exploit relationship"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.6270S"
    }
  }
}