{
  "id": "1604.02067",
  "version": "v1",
  "published": "2016-04-07T16:35:06.000Z",
  "updated": "2016-04-07T16:35:06.000Z",
  "title": "Higher moments of arithmetic functions in short intervals: a geometric perspective",
  "authors": [
    "Daniel Hast",
    "Vlad Matei"
  ],
  "comment": "20 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We study the distribution in short intervals of certain arithmetic functions, including the von Mangoldt function and the M\\\"obius function, on polynomials over a finite field $\\mathbb{F}_q$. Using the Grothendieck-Lefschetz trace formula, we reinterpret each moment of these distributions as a point-counting problem on a highly singular complete intersection variety. We compute part of the $\\ell$-adic cohomology of these varieties, yielding an asymptotic bound on each moment for fixed degree $n$ in the limit as $q \\to \\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-07T16:35:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11T55",
      "11G25"
    ],
    "keywords": [
      "short intervals",
      "arithmetic functions",
      "higher moments",
      "geometric perspective",
      "highly singular complete intersection variety"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160402067H"
    }
  }
}