{
  "id": "0705.0732",
  "version": "v2",
  "published": "2007-05-05T07:22:23.000Z",
  "updated": "2007-09-24T07:18:08.000Z",
  "title": "Integrals Over Polytopes, Multiple Zeta Values and Polylogarithms, and Euler's Constant",
  "authors": [
    "Jonathan Sondow",
    "Sergey Zlobin"
  ],
  "comment": "19 pages, to appear in Mat Zametki. Ver 2.: Added Remark 3 on a Chen (Drinfeld-Kontsevich) iterated integral; simplified Proposition 2; gave reference for (19); corrected [16]; fixed typo",
  "journal": "Math. Notes (in English) 84 (2008) pp. 568-583; Mat. Zametki (in Russian) 84:4 (2008) pp. 609-626",
  "doi": "10.1134/S0001434608090290",
  "categories": [
    "math.NT",
    "math.CV"
  ],
  "abstract": "Let $T$ be the triangle with vertices (1,0), (0,1), (1,1). We study certain integrals over $T$, one of which was computed by Euler. We give expressions for them both as a linear combination of multiple zeta values, and as a polynomial in single zeta values. We obtain asymptotic expansions of the integrals, and of sums of certain multiple zeta values with constant weight. We also give related expressions for Euler's constant. In the final section, we evaluate more general integrals -- one is a Chen (Drinfeld-Kontsevich) iterated integral -- over some polytopes that are higher-dimensional analogs of $T$. This leads to a relation between certain multiple polylogarithm values and multiple zeta values.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-09-24T07:18:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11Y60",
      "30B10",
      "30D30",
      "33B15",
      "33B30"
    ],
    "keywords": [
      "multiple zeta values",
      "eulers constant",
      "multiple polylogarithm values",
      "single zeta values",
      "higher-dimensional analogs"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0705.0732S"
    }
  }
}