{
  "id": "2204.06261",
  "version": "v1",
  "published": "2022-04-13T09:13:36.000Z",
  "updated": "2022-04-13T09:13:36.000Z",
  "title": "On Signs of Fourier Coefficients of Hecke-Maass Cusp Forms on $\\mathrm{GL}_3$",
  "authors": [
    "Jesse Jääsaari"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We consider sign changes of Fourier coefficients of Hecke-Maass cusp forms for the group $\\mathrm{SL}_3(\\mathbb Z)$. When the underlying form is self-dual, we show that there are $\\gg_\\varepsilon X^{5/6-\\varepsilon}$ sign changes among the coefficients $\\{A(m,1)\\}_{m\\leq X}$ and that there is a positive proportion of sign changes for many self-dual forms. Similar result concerning the positive proportion of sign changes also hold for the real-valued coefficients $A(m,m)$ for generic $\\mathrm{GL}_3$ cusp forms, a result which is based on a new effective Sato-Tate type theorem for a family of $\\mathrm{GL}_3$ cusp forms we establish. In addition, non-vanishing of the Fourier coefficients is studied under the Ramanujan-Petersson conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-13T09:13:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hecke-maass cusp forms",
      "fourier coefficients",
      "sign changes",
      "positive proportion",
      "effective sato-tate type theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}