{
  "id": "1512.01299",
  "version": "v1",
  "published": "2015-12-04T01:45:07.000Z",
  "updated": "2015-12-04T01:45:07.000Z",
  "title": "The Second Moment of Sums of Coefficients of Cusp Forms",
  "authors": [
    "Thomas A. Hulse",
    "Chan Ieong Kuan",
    "David Lowry-Duda",
    "Alexander Walker"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $f$ and $g$ be weight $k$ holomorphic cusp forms and let $S_f(n)$ and $S_g(n)$ denote the sums of their first $n$ Fourier coefficients. Hafner and Ivi\\'c~\\cite{HafnerIvic89}, building on~\\cite{chandrasekharan1962functional} and~\\cite{chandrasekharan1964mean}, proved asymptotics for $\\sum_{n \\leq X} \\lvert S_f(n) \\rvert^2$ and proved that the Classical Conjecture, that $S_f(X) \\ll X^{\\frac{k-1}{2} + \\frac{1}{4} + \\epsilon}$, holds on average over long intervals. In this paper, we introduce and obtain meromorphic continuations for the Dirichlet series $D(s, S_f \\times S_g) = \\sum S_f(n)\\overline{S_g(n)} n^{-(s+k-1)}$ and $D(s, S_f \\times \\overline{S_g}) = \\sum_n S_f(n)S_g(n) n^{-(s + k - 1)}$. Using these meromorphic continuations, we prove asymptotics for the smoothed second moment sums $\\sum S_f(n)\\overline{S_g(n)} e^{-n/X}$, proving a smoothed generalization of~\\cite{HafnerIvic89}. We also attain asymptotics for analogous smoothed second moment sums of normalized Fourier coefficients, proving smoothed generalizations of what would be attainable from~\\cite{chandrasekharan1962functional}. Our methodology extends to a wide variety of weights and levels, and comparison with~\\cite{chandrasekharan1962functional} indicates very general cancellation between the Rankin-Selberg $L$-function $L(s, f\\times g)$ and shifted convolution sums of the coefficients of $f$ and $g$. In forthcoming works, the authors apply the results of this paper to prove the Classical Conjecture on $\\lvert S_f(n) \\rvert^2$ is true on short intervals, and to prove sign change results on $\\{S_f(n)\\}_{n \\in \\mathbb{N}}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-04T01:45:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F30"
    ],
    "keywords": [
      "meromorphic continuations",
      "fourier coefficients",
      "classical conjecture",
      "holomorphic cusp forms",
      "analogous smoothed second moment sums"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151201299H"
    }
  }
}