{
  "id": "1512.05502",
  "version": "v1",
  "published": "2015-12-17T09:17:05.000Z",
  "updated": "2015-12-17T09:17:05.000Z",
  "title": "Short-Interval Averages of Sums of Fourier Coefficients of Cusp Forms",
  "authors": [
    "Thomas A. Hulse",
    "Chan Ieong Kuan",
    "David Lowry-Duda",
    "Alexander Walker"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $f$ be a weight $k$ holomorphic cusp form of level one, and let $S_f(n)$ denote the sum of the first $n$ Fourier coefficients of $f$. In analogy with Dirichlet's divisor problem, it is conjectured that $S_f(X) \\ll X^{\\frac{k-1}{2} + \\frac{1}{4} + \\epsilon}$. Understanding and bounding $S_f(X)$ has been a very active area of research. The current best bound for individual $S_f(X)$ is $S_f(X) \\ll X^{\\frac{k-1}{2} + \\frac{1}{3}} (\\log X)^{-0.1185}$ from Wu. Chandrasekharan and Narasimhan showed that the Classical Conjecture for $S_f(X)$ holds on average over intervals of length $X$. Jutila improved this result to show that the Classical Conjecture for $S_f(X)$ holds on average over short intervals of length $X^{\\frac{3}{4} + \\epsilon}$. Building on the results and analytic information about $\\sum \\lvert S_f(n) \\rvert^2 n^{-(s + k - 1)}$ from our recent paper, we further improve these results to show that the Classical Conjecture for $S_f(X)$ holds on average over short intervals of length approximately $X^{\\frac{2}{3} + \\epsilon}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-17T09:17:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F30"
    ],
    "keywords": [
      "fourier coefficients",
      "short-interval averages",
      "classical conjecture",
      "short intervals",
      "current best bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151205502H"
    }
  }
}