{
  "id": "2005.10714",
  "version": "v1",
  "published": "2020-05-21T15:11:16.000Z",
  "updated": "2020-05-21T15:11:16.000Z",
  "title": "Small values of $| L^\\prime/L(1,χ) |$",
  "authors": [
    "Youness Lamzouri",
    "Alessandro Languasco"
  ],
  "comment": "20 pages, 3 figures, 1 table",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper, we investigate the quantity $m_q:=\\min_{\\chi\\ne \\chi_0} | L^\\prime/L(1,\\chi)|$, as $q\\to \\infty$ over the primes, where $L(s,\\chi)$ is the Dirichlet $L$-function attached to a non trivial Dirichlet character modulo $q$. Our main result shows that $m_q \\ll \\log\\log q/\\sqrt{\\log q}$. We also compute $m_q$ for every odd prime $q$ up to $10^7$. As a consequence we numerically verified that for every odd prime $q$, $3 \\le q \\le 10^7$, we have $c_1/q< m_q<5/\\sqrt{q}$, with $c_1=21/200$. In particular, this shows that $L^\\prime(1,\\chi) \\ne 0$ for every non trivial Dirichlet character $\\chi$ mod $q$ where $3\\leq q\\leq 10^7$ is prime, answering a question of Gun, Murty and Rath in this range. We also provide some statistics and scatter plots regarding the $m_q$-values, see Section 6. The programs used and the computational results described here are available at the following web address: \\url{http://www.math.unipd.it/~languasc/smallvalues.html}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-21T15:11:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E45",
      "11M41"
    ],
    "keywords": [
      "small values",
      "non trivial dirichlet character modulo",
      "odd prime",
      "web address",
      "computational results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}