{
  "id": "2011.10140",
  "version": "v1",
  "published": "2020-11-19T22:56:39.000Z",
  "updated": "2020-11-19T22:56:39.000Z",
  "title": "Determining optimal test functions for $2$-level densities",
  "authors": [
    "Elżbieta Bołdyriew",
    "Fangu Chen",
    "Charles Devlin VI",
    "Steven J. Miller",
    "Jason Zhao"
  ],
  "comment": "15 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Katz and Sarnak conjectured a correspondence between the $n$-level density statistics of zeros from families of $L$-functions with eigenvalues from random matrix ensembles, and in many cases the sums of smooth test functions, whose Fourier transforms are finitely supported over scaled zeros in a family, converge to an integral of the test function against a density $W_{n, G}$ depending on the symmetry $G$ of the family (unitary, symplectic or orthogonal). This integral bounds the average order of vanishing at the central point of the corresponding family of $L$-functions. We can obtain better estimates on this vanishing in two ways. The first is to do more number theory, and prove results for larger $n$ and greater support; the second is to do functional analysis and obtain better test functions to minimize the resulting integrals. We pursue the latter here when $n=2$, minimizing \\[ \\frac{1}{\\Phi(0, 0)} \\int_{{\\mathbb R}^2} W_{2,G} (x, y) \\Phi(x, y) dx dy \\] over test functions $\\Phi : {\\mathbb R}^2 \\to [0, \\infty)$ with compactly supported Fourier transform. We study a restricted version of this optimization problem, imposing that our test functions take the form $\\phi(x) \\psi(y)$ for some fixed admissible $\\psi(y)$ and $\\operatorname{supp}{\\widehat \\phi} \\subseteq [-1, 1]$. Extending results from the 1-level case, namely the functional analytic arguments of Iwaniec, Luo and Sarnak and the differential equations method introduced by Freeman and Miller, we explicitly solve for the optimal $\\phi$ for appropriately chosen fixed test function $\\psi$. We conclude by discussing further improvements on estimates by the method of iteration.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-19T22:56:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "determining optimal test functions",
      "fourier transform",
      "level density statistics",
      "better test functions",
      "functional analytic arguments"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}