{
  "id": "1703.10582",
  "version": "v1",
  "published": "2017-03-30T17:30:23.000Z",
  "updated": "2017-03-30T17:30:23.000Z",
  "title": "Large sums of Hecke eigenvalues of holomorphic cusp forms",
  "authors": [
    "Youness Lamzouri"
  ],
  "comment": "19 pages, submitted",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $f$ be a Hecke cusp form of weight $k$ for the full modular group, and let $\\{\\lambda_f(n)\\}_{n\\geq 1}$ be the sequence of its normalized Fourier coefficients. Motivated by the problem of the first sign change of $\\lambda_f(n)$, we investigate the range of $x$ (in terms of $k$) for which there are cancellations in the sum $S_f(x)=\\sum_{n\\leq x} \\lambda_f(n)$. We first show that $S_f(x)=o(x\\log x)$ implies that $\\lambda_f(n)<0$ for some $n\\leq x$. We also prove that $S_f(x)=o(x\\log x)$ in the range $\\log x/\\log\\log k\\to \\infty$ assuming the Riemann hypothesis for $L(s, f)$, and furthermore that this range is best possible unconditionally. More precisely, we establish the existence of many Hecke cusp forms $f$ of large weight $k$, for which $S_f(x)\\gg_A x\\log x$, when $x=(\\log k)^A.$ Our results are $GL_2$ analogues of work of Granville and Soundararajan for character sums, and could also be generalized to other families of automorphic forms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-30T17:30:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "holomorphic cusp forms",
      "large sums",
      "hecke eigenvalues",
      "hecke cusp form",
      "full modular group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}