{
  "id": "1810.09763",
  "version": "v1",
  "published": "2018-10-23T10:39:42.000Z",
  "updated": "2018-10-23T10:39:42.000Z",
  "title": "Linear Independence of Harmonic Numbers over the field of Algebraic Numbers",
  "authors": [
    "Tapas Chatterjee",
    "Sonika Dhillon"
  ],
  "comment": "To appear in the Ramanujan Journal",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $H_n =\\sum\\limits_{k=1}^n \\frac{1}{k}$ be the $n$-th harmonic number. Euler extended it to complex arguments and defined $H_r$ for any complex number $r$ except for the negative integers. In this paper, we give a new proof of the transcendental nature of $H_r$ for rational $r$. For some special values of $q>1,$ we give an upper bound for the number of linearly independent harmonic numbers $H_{a/q}$ with $ 1 \\leq a \\leq q$ over the field of algebraic numbers. Also, for any finite set of odd primes $J$ with $|J|=n,$ define $$W_J=\\overline{\\mathbb{Q}}-\\text {span of } \\{ H_1, \\ H_{a_{j_i}/q_i} | \\ 1 \\leq a_{j_i} \\leq q_i -1, \\ 1 \\leq j_i \\leq q_i-1, \\ \\ \\forall q_i \\in J\\}.$$ Finally, we show that $$\\text{ dim }_{\\overline{\\mathbb{Q}}} ~W_J=\\sum\\limits_{\\substack{i=1 \\\\ q_i \\in J}}^n \\frac{\\phi (q_i )}{2} + 2.$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-23T10:39:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J81",
      "11J86",
      "11J91"
    ],
    "keywords": [
      "algebraic numbers",
      "linear independence",
      "linearly independent harmonic numbers",
      "th harmonic number",
      "finite set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}