{
  "id": "1407.1577",
  "version": "v2",
  "published": "2014-07-07T04:21:25.000Z",
  "updated": "2014-11-11T05:20:02.000Z",
  "title": "Benford's Law for Coefficients of Newforms",
  "authors": [
    "Marie Jameson",
    "Jesse Thorner",
    "Lynnelle Ye"
  ],
  "comment": "10 pages. Referee comments implemented. To appear in International Journal of Number Theory",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $f(z)=\\sum_{n=1}^\\infty \\lambda_f(n)e^{2\\pi i n z}\\in S_{k}^{new}(\\Gamma_0(N))$ be a normalized Hecke eigenform of even weight $k\\geq2$ on $\\Gamma_0(N)$ without complex multiplication. Let $\\mathbb{P}$ denote the set of all primes. We prove that the sequence $\\{\\lambda_f(p)\\}_{p\\in\\mathbb{P}}$ does not satisfy Benford's Law in any base $b\\geq2$. However, given a base $b\\geq2$ and a string of digits $S$ in base $b$, the set \\[ A_{\\lambda_f}(b,S):=\\{\\text{$p$ prime : the first digits of $\\lambda_f(p)$ in base $b$ are given by $S$}\\} \\] has logarithmic density equal to $\\log_b(1+S^{-1})$. Thus $\\{\\lambda_f(p)\\}_{p\\in\\mathbb{P}}$ follows Benford's Law with respect to logarithmic density. Both results rely on the now-proven Sato-Tate Conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-07T04:21:25.000Z",
      "comment": "10 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-11-11T05:20:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F30",
      "11K06",
      "11B83"
    ],
    "keywords": [
      "coefficients",
      "satisfy benfords law",
      "now-proven sato-tate conjecture",
      "logarithmic density equal",
      "first digits"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.1577J"
    }
  }
}