{
  "id": "2008.05778",
  "version": "v1",
  "published": "2020-08-13T09:50:46.000Z",
  "updated": "2020-08-13T09:50:46.000Z",
  "title": "Uniform estimates for almost primes over finite fields",
  "authors": [
    "Dor Elboim",
    "Ofir Gorodetsky"
  ],
  "comment": "13 pages, 1 figure. Comments are welcome",
  "categories": [
    "math.NT",
    "math.CO",
    "math.PR"
  ],
  "abstract": "We establish a new asymptotic formula for the number of polynomials of degree $n$ with $k$ prime factors over a finite field $\\mathbb{F}_q$. The error term tends to $0$ uniformly in $n$ and in $q$, and $k$ can grow beyond $\\log n$. Previously, asymptotic formulas were known either for fixed $q$, through the works of Warlimont and Hwang, or for small $k$, through the work of Arratia, Barbour and Tavar\\'e. As an application, we estimate the total variation distance between the number of cycles in a random permutation on $n$ elements and the number of prime factors of a random polynomial of degree $n$ over $\\mathbb{F}_q$. The distance tends to $0$ at rate $1/(q\\sqrt{\\log n})$. Previously this was only understood when either $q$ is fixed and $n$ tends to $\\infty$, or $n$ is fixed and $q$ tends to $\\infty$, by results of Arratia, Barbour and Tavar\\'{e}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-13T09:50:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite field",
      "uniform estimates",
      "prime factors",
      "asymptotic formula",
      "error term tends"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}