{
  "id": "1311.7016",
  "version": "v1",
  "published": "2013-11-27T15:50:04.000Z",
  "updated": "2013-11-27T15:50:04.000Z",
  "title": "Quadratic Non-residues in Short Intervals",
  "authors": [
    "Sergei V. Konyagin",
    "Igor E. Shparlinski"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We use the Burgess bound and combinatorial sieve to obtain an upper bound on the number of primes $p$ in a dyadic interval $[Q,2Q]$ for which a given interval $[u+1,u+\\psi(Q)]$ does not contain a quadratic non-residue modulo $p$. The bound is nontrivial for any function $\\psi(Q)\\to\\infty$ as $Q\\to\\infty$. This is an analogue of the well known estimates on the smallest quadratic non-residue modulo $p$ on average over primes $p$, which corresponds to the choice $u=0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-27T15:50:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "short intervals",
      "smallest quadratic non-residue modulo",
      "dyadic interval",
      "upper bound",
      "combinatorial sieve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.7016K"
    }
  }
}