{
  "id": "1808.09944",
  "version": "v1",
  "published": "2018-08-29T17:50:49.000Z",
  "updated": "2018-08-29T17:50:49.000Z",
  "title": "On a conjecture of Livingston",
  "authors": [
    "Siddhi Pathak"
  ],
  "journal": "Canadian Math. Bull., Vol. 60 (1), (2017), 184-195",
  "doi": "10.4153/CMB-2016-065-1",
  "categories": [
    "math.NT"
  ],
  "abstract": "In an attempt to resolve a folklore conjecture of Erd\\H{o}s regarding the non-vanishing at $s=1$ of the $L$-series attached to a periodic arithmetical function with period $q$ and values in $\\{-1, 1 \\}$, Livingston conjectured the $\\overline{\\mathbb{Q}}$ - linear independence of logarithms of certain algebraic numbers. In this paper, we disprove Livingston's conjecture for composite $q \\geq 4$, highlighting that a new approach is required to settles Erd\\H{o}s's conjecture. We also prove that the conjecture is true for prime $q \\geq 3$, and indicate that more ingredients are needed to settle Erd\\H{o}s's conjecture for prime $q$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-29T17:50:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J86",
      "11J72"
    ],
    "keywords": [
      "disprove livingstons conjecture",
      "folklore conjecture",
      "algebraic numbers",
      "periodic arithmetical function",
      "linear independence"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}