{
  "id": "2407.16937",
  "version": "v1",
  "published": "2024-07-24T02:16:21.000Z",
  "updated": "2024-07-24T02:16:21.000Z",
  "title": "A converse to a theorem of Gauss on Gauss sums",
  "authors": [
    "Jonathan W. Bober",
    "Leo Goldmakher"
  ],
  "comment": "4 pages. Comments very welcome!",
  "categories": [
    "math.NT",
    "math.RT"
  ],
  "abstract": "In this note we prove (under mild hypotheses) that $f$ is a nontrivial character of $\\mathbb{F}_p$ if and only if the Fourier transform of $f$ has magnitude 1 somewhere in $\\mathbb{F}_p^\\times$. This implies a converse to a theorem of Gauss on the magnitude of the Gauss sum, in addition to other consequences.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-24T02:16:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11L05",
      "11T24",
      "20C15"
    ],
    "keywords": [
      "gauss sum",
      "mild hypotheses",
      "nontrivial character",
      "fourier transform"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}