{
  "id": "2208.10459",
  "version": "v1",
  "published": "2022-08-22T17:26:24.000Z",
  "updated": "2022-08-22T17:26:24.000Z",
  "title": "Modular forms and an explicit Chebotarev variant of the Brun-Titchmarsh theorem",
  "authors": [
    "Daniel Hu",
    "Hari R. Iyer",
    "Alexander Shashkov"
  ],
  "comment": "39 pages, comments welcome!",
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove an explicit Chebotarev variant of the Brun--Titchmarsh theorem. This leads to explicit versions of the best-known unconditional upper bounds toward conjectures of Lang and Trotter for the coefficients of holomorphic cuspidal newforms. In particular, we prove that $$\\lim_{x \\to \\infty} \\frac{\\#\\{1 \\leq n \\leq x \\mid \\tau(n) \\neq 0\\}}{x} > 1-1.15 \\times 10^{-12},$$ where $\\tau(n)$ is Ramanujan's tau-function. This is the first known positive unconditional lower bound for the proportion of positive integers $n$ such that $\\tau(n) \\neq 0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-22T17:26:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R44",
      "11N36",
      "11F30"
    ],
    "keywords": [
      "explicit chebotarev variant",
      "brun-titchmarsh theorem",
      "modular forms",
      "best-known unconditional upper bounds",
      "holomorphic cuspidal newforms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}