{
  "id": "2408.14815",
  "version": "v1",
  "published": "2024-08-27T06:54:55.000Z",
  "updated": "2024-08-27T06:54:55.000Z",
  "title": "The fourth moment of truncated Eisenstein series",
  "authors": [
    "Goran Djanković",
    "Rizwanur Khan"
  ],
  "comment": "27 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We asymptotically evaluate the fourth moment of truncated Eisenstein series with large Laplacian eigenvalue. With suitable normalization our main term equals 3, which coincides with the fourth moment of a standard normal random variable, as predicted by the Random Wave Conjecture. This is the first time that the fourth moment in this conjecture has been verified for Eiseinstein series. Our paper builds upon the authors' previous work in which the fourth moment of Eisenstein series was established with a regularized integral. That however was an alternative set-up, yielding an asymptotic main term different from 3, and as such did not address the Random Wave Conjecture as originally framed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-27T06:54:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F12",
      "11M99",
      "81Q50",
      "58J51"
    ],
    "keywords": [
      "fourth moment",
      "truncated eisenstein series",
      "random wave conjecture",
      "main term equals",
      "asymptotic main term"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}