{
  "id": "2502.03436",
  "version": "v1",
  "published": "2025-02-05T18:31:38.000Z",
  "updated": "2025-02-05T18:31:38.000Z",
  "title": "The Second Moment of Sums of Hecke Eigenvalues II",
  "authors": [
    "Ned Carmichael"
  ],
  "comment": "34 pages. Comments welcome",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $f$ be a Hecke cusp form of weight $k$ for $\\mathrm{SL}_2(\\mathbb{Z})$, and let $(\\lambda_f(n))_{n\\geq 1}$ denote its (suitably normalised) sequence of Hecke eigenvalues. We compute the first and second moments of the sums $S(x,f)=\\sum_{x\\leq n\\leq 2x} \\lambda_f(n)$, on average over forms $f$ of large weight $k$. It is proved that when the length of the sums $x$ is larger than $k^2$, the second moment is roughly of size $x^{1/2}$. This is in sharp contrast to the regime where $x$ is slightly smaller than $k^2$, where it was shown in preceding work (part I) that the second moment is of size $x$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-02-05T18:31:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F30",
      "11N37",
      "11F11"
    ],
    "keywords": [
      "second moment",
      "hecke eigenvalues",
      "hecke cusp form",
      "large weight",
      "sharp contrast"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}