{
  "id": "1308.6606",
  "version": "v2",
  "published": "2013-08-29T20:56:46.000Z",
  "updated": "2014-09-20T03:56:01.000Z",
  "title": "On the Typical Size and Cancelations Among the Coefficients of Some Modular Forms",
  "authors": [
    "Florian Luca",
    "Igor E. Shparlinski"
  ],
  "comment": "The second version contains some improvements and extensions of previous results, suggested by Maksym Radziwill, who is now a co-author",
  "categories": [
    "math.NT"
  ],
  "abstract": "We obtain a nontrivial upper bound for almost all elements of the sequences of real numbers which are multiplicative and at the prime indices are distributed according to the Sato--Tate density. Examples of such sequences come from coefficients of several $L$-functions of elliptic curves and modular forms. In particular, we show that $|\\tau(n)|\\le n^{11/2} (\\log n)^{-1/2+o(1)}$ for a set of $n$ of asymptotic density 1, where $\\tau(n)$ is the Ramanujan $\\tau$ function while the standard argument yields $\\log 2$ instead of $-1/2$ in the power of the logarithm. Another consequence of our result is that in the number of representations of $n$ by a binary quadratic form one has slightly more than square-root cancellations for almost all integers $n$. In addition we obtain a central limit theorem for such sequences, assuming a weak hypothesis on the rate of convergence to the Sato--Tate law. For Fourier coefficients of primitive holomorphic cusp forms such a hypothesis is known conditionally assuming the automorphy of all symmetric powers of the form and seems to be within reach unconditionally using the currently established potential automorphy.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-08-29T20:56:46.000Z",
      "abstract": "We obtain a nontrivial upper bound for almost all elements of the sequences of real numbers which are multiplicative and at the prime indices are distributed according to the Sato-Tate density. Examples of such sequences come from coefficients of several $L$-functions of elliptic curves and modular forms. In particular, we show that $|\\tau(n)|\\le n^{11/2} (\\log n)^{-1/2+o(1)})$ for a set of $n$ of asymptotic density 1, where $\\tau(n)$ is the Ramanujan $\\tau$ function. In comparison, the standard argument, based on the estimate for the number of prime divisors of a typical integer $n$, only leads to the bound $|\\tau(n)| \\le n^{11/2} (\\log n)^{\\log 2+ o(1)}$ for almost all $n$.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-09-20T03:56:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "modular forms",
      "coefficients",
      "typical size",
      "cancelations",
      "nontrivial upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1308.6606L"
    }
  }
}