{
  "id": "2109.10136",
  "version": "v1",
  "published": "2021-09-21T12:43:03.000Z",
  "updated": "2021-09-21T12:43:03.000Z",
  "title": "Linear independence of odd zeta values using Siegel's lemma",
  "authors": [
    "Stéphane Fischler"
  ],
  "comment": "37 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove that among 1 and the odd zeta values $\\zeta(3)$, $\\zeta(5)$, ..., $\\zeta(s)$, at least $ 0.21 \\sqrt{s / \\log s}$ are linearly independent over the rationals, for any sufficiently large odd integer $s$. This improves on the lower bound $(1-\\varepsilon) \\log s / (1+\\log 2)$ obtained by Ball-Rivoal in 2001. Up to the numerical constant $0.21$, it gives as a corollary a new proof of the lower bound on the number of irrationals in this family proved in 2020 by Lai-Yu. The proof is based on Siegel's lemma to construct non-explicit linear forms in odd zeta values, instead of using explicit well-poised hypergeometric series. Siegel's linear independence criterion (instead of Nesterenko's) is applied, with a multiplicity estimate (namely a generalization of Shidlovsky's lemma). The result is also adapted to deal with values of the first $s$ polylogarithms at a fixed algebraic point in the unit disk, improving bounds of Rivoal and Marcovecchio.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-21T12:43:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J72",
      "11M06"
    ],
    "keywords": [
      "odd zeta values",
      "siegels lemma",
      "lower bound",
      "siegels linear independence criterion",
      "construct non-explicit linear forms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}