{
  "id": "2311.13788",
  "version": "v1",
  "published": "2023-11-23T03:14:34.000Z",
  "updated": "2023-11-23T03:14:34.000Z",
  "title": "Non-Linear Additive Twists of $\\mathrm{GL}_{3}$ Hecke Eigenvalues",
  "authors": [
    "Ikuya Kaneko",
    "Wing Hong Leung"
  ],
  "comment": "26 pages. LaTeX2e",
  "categories": [
    "math.NT"
  ],
  "abstract": "We bound non-linear additive twists of $\\mathrm{GL}_{3}$ Hecke eigenvalues, improving upon the work of Kumar-Mallesham-Singh (2022). The proof employs the DFI circle method with standard manipulations (Voronoi, Cauchy-Schwarz, lengthening, and additive reciprocity). The main novelty includes the conductor lowering mechanism, albeit sacrificing some savings to remove an analytic oscillation, followed by the iteration ad infinitum of Cauchy-Schwarz and Poisson. The resulting character sums are estimated via the work of Adolphson-Sperber (1993). As an application, we prove nontrivial bounds for the first moment of $\\mathrm{GL}_{3}$ Hardy's function, which corresponds to the cubic moment of Hardy's function studied by Ivi\\'{c} (2012).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-23T03:14:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F66",
      "11M41",
      "11F55"
    ],
    "keywords": [
      "hecke eigenvalues",
      "hardys function",
      "bound non-linear additive twists",
      "dfi circle method",
      "iteration ad infinitum"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}