{
  "id": "2107.03556",
  "version": "v1",
  "published": "2021-07-08T01:28:14.000Z",
  "updated": "2021-07-08T01:28:14.000Z",
  "title": "Some Remarks on Small Values of $τ(n)$",
  "authors": [
    "Kaya Lakein",
    "Anne Larsen"
  ],
  "comment": "8 pages, 2 tables",
  "categories": [
    "math.NT"
  ],
  "abstract": "A natural variant of Lehmer's conjecture that the Ramanujan $\\tau$-function never vanishes asks whether, for any given integer $\\alpha$, there exist any $n \\in \\mathbb{Z}^+$ such that $\\tau(n) = \\alpha$. A series of recent papers excludes many integers as possible values of the $\\tau$-function using the theory of primitive divisors of Lucas numbers, computations of integer points on curves, and congruences for $\\tau(n)$. We synthesize these results and methods to prove that if $0 < |\\alpha| < 100$ and $\\alpha \\notin T := \\{2^k, -24,-48, -70,-90, 92, -96\\}$, then $\\tau(n) \\neq \\alpha$ for all $n > 1$. Moreover, if $\\alpha \\in T$ and $\\tau(n) = \\alpha$, then $n$ is square-free with prescribed prime factorization. Finally, we show that a strong form of the Atkin-Serre conjecture implies that $|\\tau(n)| > 100$ for all $n > 2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-08T01:28:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F30"
    ],
    "keywords": [
      "small values",
      "atkin-serre conjecture implies",
      "vanishes asks",
      "natural variant",
      "lehmers conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}