{
  "id": "2502.03437",
  "version": "v1",
  "published": "2025-02-05T18:32:15.000Z",
  "updated": "2025-02-05T18:32:15.000Z",
  "title": "The Second Moment of Sums of Hecke Eigenvalues I",
  "authors": [
    "Ned Carmichael"
  ],
  "comment": "28 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 $(\\lambda_f(n))_{n\\geq1}$ 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 $k$, in the regime where the length of the sums $x$ is smaller than $k^2$. We observe interesting transitions in the size of the sums when $x\\approx k$ and $x\\approx k^2$. In subsequent work (part II), it will be shown that once $x$ is larger than $k^2$ (where the latter transition occurs), the average size of the sums $S(x,f)$ becomes dramatically smaller.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-02-05T18:32:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F30",
      "11N37",
      "11F11"
    ],
    "keywords": [
      "hecke eigenvalues",
      "second moment",
      "hecke cusp form",
      "transition occurs",
      "subsequent work"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}