{
  "id": "2304.05178",
  "version": "v1",
  "published": "2023-04-11T12:35:05.000Z",
  "updated": "2023-04-11T12:35:05.000Z",
  "title": "On the derivatives of Hardy's function $Z(t)$",
  "authors": [
    "Hung M. Bui",
    "R. R. Hall"
  ],
  "comment": "17 pages, to appear in Bull. London Math. Soc",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $Z^{(k)}(t)$ be the $k$-th derivative of Hardy's $Z$-function. The numerics seem to suggest that if $k$ and $\\ell$ have the same parity, then the zeros of $Z^{(k)}(t)$ and $Z^{(\\ell)}(t)$ come in pairs which are very close to each other. That is to say that $Z^{(k)}(t)Z^{(\\ell)}(t)$ has constant sign for the majority, if not almost all, of values $t$. In this paper we show that this is true a positive proportion of times. We also study the sign of the product of four derivatives of Hardy's function, $Z^{(k)}(t)Z^{(\\ell)}(t)Z^{(m)}(t)Z^{(n)}(t)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-11T12:35:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11M26"
    ],
    "keywords": [
      "hardys function",
      "derivative",
      "constant sign",
      "positive proportion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}