{
  "id": "math/0311114",
  "version": "v4",
  "published": "2003-11-07T16:10:45.000Z",
  "updated": "2004-12-20T18:06:16.000Z",
  "title": "Hyperg{é}om{é}trie et fonction z{ê}ta de Riemann",
  "authors": [
    "C. Krattenthaler",
    "T. Rivoal"
  ],
  "comment": "AmS-LaTeX, 73 pages; completely rewritten: (1) The strategy for proving the theorems for the coefficient p_0 was changed. The effect is that our theorems now hold unconditionally. (2) A full proof of Zudilin's conjecture on the linear forms for zeta(4) coming from symmetric series is now contained. (3) These improvements made it necessary to completely restructure the article",
  "journal": "Mem. Amer. Math. Soc. 186, no. 875, Providence, R. I., 2007.",
  "categories": [
    "math.NT",
    "math.CA"
  ],
  "abstract": "We prove the second author's \"denominator conjecture\" [40] concerning the common denominators of coefficients of certain linear forms in zeta values. These forms were recently constructed to obtain lower bounds for the dimension of the vector space over $\\mathbb Q$ spanned by $1,\\zeta(m),\\zeta(m+2),...,\\zeta(m+2h)$, where $m$ and $h$ are integers such that $m\\ge2$ and $h\\ge0$. In particular, we immediately get the following results as corollaries: at least one of the eight numbers $\\zeta(5),\\zeta(7),...,\\zeta(19)$ is irrational, and there exists an odd integer $j$ between 5 and 165 such that 1, $\\zeta(3)$ and $\\zeta(j)$ are linearly independent over $\\mathbb{Q}$. This strengthens some recent results in [41] and [8], respectively. We also prove a related conjecture, due to Vasilyev [49], and as well a conjecture, due to Zudilin [55], on certain rational approximations of $\\zeta(4)$. The proofs are based on a hypergeometric identity between a single sum and a multiple sum due to Andrews [3]. We hope that it will be possible to apply our construction to the more general linear forms constructed by Zudilin [56], with the ultimate goal of strengthening his result that one of the numbers $\\zeta(5),\\zeta(7),\\zeta(9),\\zeta(11)$ is irrational.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2004-12-20T18:06:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J72",
      "11J82",
      "33C20"
    ],
    "keywords": [
      "denominator conjecture",
      "odd integer",
      "common denominators",
      "ultimate goal",
      "general linear forms"
    ],
    "tags": [
      "journal article",
      "monograph"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Mem. Amer. Math. Soc."
    },
    "note": {
      "typesetting": "LaTeX",
      "pages": 73,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math.....11114K"
    }
  }
}