{
  "id": "1407.2656",
  "version": "v2",
  "published": "2014-07-09T23:42:09.000Z",
  "updated": "2015-09-15T02:29:27.000Z",
  "title": "The Error Term in the Sato-Tate Conjecture",
  "authors": [
    "Jesse Thorner"
  ],
  "comment": "9 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $f(z)=\\sum_{n=1}^\\infty a(n)e^{2\\pi i nz}\\in S_k^{new}(\\Gamma_0(N))$ be a newform of even weight $k\\geq2$ that does not have complex multiplication. Then $a(n)\\in\\mathbb{R}$ for all $n$, so for any prime $p$, there exists $\\theta_p\\in[0,\\pi]$ such that $a(p)=2p^{(k-1)/2}\\cos(\\theta_p)$. Let $\\pi(x)=\\#\\{p\\leq x\\}$. For a given subinterval $I\\subset[0,\\pi]$, the now-proven Sato-Tate Conjecture tells us that as $x\\to\\infty$, \\[ \\#\\{p\\leq x:\\theta_p\\in I\\}\\sim \\mu_{ST}(I)\\pi(x),\\quad \\mu_{ST}(I)=\\int_{I} \\frac{2}{\\pi}\\sin^2(\\theta)~d\\theta. \\] Let $\\epsilon>0$. Assuming that the symmetric power $L$-functions of $f$ are automorphic, we prove that as $x\\to\\infty$, \\[ \\#\\{p\\leq x:\\theta_p\\in I\\}=\\mu_{ST}(I)\\pi(x)+O\\left(\\frac{x}{(\\log x)^{9/8-\\epsilon}}\\right), \\] where the implied constant is effectively computable and depends only on $k,N,$ and $\\epsilon$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-09T23:42:09.000Z",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-09-15T02:29:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F30",
      "11M41"
    ],
    "keywords": [
      "error term",
      "now-proven sato-tate conjecture tells",
      "complex multiplication",
      "symmetric power",
      "subinterval"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.2656T"
    }
  }
}