{
  "id": "2110.06855",
  "version": "v3",
  "published": "2021-10-13T16:47:28.000Z",
  "updated": "2022-05-18T20:44:46.000Z",
  "title": "Applications of zero-free regions on averages and shifted convolution sums of Hecke eigenvalues",
  "authors": [
    "Jiseong Kim"
  ],
  "comment": "A few minor typos, grammatical mistakes are fixed. Corollary 1.4, Corollary 1.6 are added",
  "categories": [
    "math.NT"
  ],
  "abstract": "By assuming the Vinogradov-Korobov type zero-free regions and the generalized Ramanujan-Petersson conjecture, we give some nontrivial upper bounds of almost all short sums of Fourier coefficients of Hecke-Maass cusp forms for $SL(n,\\mathbb{Z})$. As an application, we obtain the nontrivial upper bounds for the averages of shifted sums for $SL(n,\\mathbb{Z}) \\times SL(n,\\mathbb{Z}).$ We also give some results on sign changes by assuming some zero-free regions and the generalized Ramanujan-Petersson conjecture.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2022-05-18T20:44:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F30",
      "11N36"
    ],
    "keywords": [
      "shifted convolution sums",
      "hecke eigenvalues",
      "nontrivial upper bounds",
      "application",
      "generalized ramanujan-petersson conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}