{
  "id": "2407.01045",
  "version": "v1",
  "published": "2024-07-01T07:54:54.000Z",
  "updated": "2024-07-01T07:54:54.000Z",
  "title": "Disproving a weaker form of Hooley's conjecture",
  "authors": [
    "Mounir Hayani"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Hooley conjectured that $G(x;q) \\ll x\\log q$, as soon as $q\\to +\\infty$, where $G(x;q)$ represents the variance of primes $p \\leq x$ in arithmetic progressions modulo $q$, weighted by $\\log p$. In this paper, we study $G_\\eta(x;q)$, a function similar to $G(x;q)$, but including the weighting factor $\\eta\\left(\\frac{p}{x}\\right)$, which has a dampening effect on the values of $G_\\eta$. Our study is motivated by the disproof of Hooley's conjecture by Fiorilli and Martin in the range $q \\asymp \\log \\log x$. Even though this weighting factor dampens the values, we still prove that an estimation of the form $G_\\eta(x;q) \\ll x\\log q$ is false in the same range.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-01T07:54:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hooleys conjecture",
      "weaker form",
      "arithmetic progressions modulo",
      "disproving",
      "function similar"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}