{
  "id": "2408.10399",
  "version": "v1",
  "published": "2024-08-19T20:31:10.000Z",
  "updated": "2024-08-19T20:31:10.000Z",
  "title": "On the sign changes of $ψ(x)-x$",
  "authors": [
    "Maciej Grześkowiak",
    "Jerzy Kaczorowski",
    "Łukasz Pańkowski",
    "Maciej Radziejewski"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We improve the lower bound for $V(T)$, the number of sign changes of the error term $\\psi(x)-x$ in the Prime Number Theorem in the interval $[1,T]$ for large $T$. We show that \\[ \\liminf_{T\\to\\infty}\\frac{V(T)}{\\log T}\\geq\\frac{\\gamma_{0}}{\\pi}+\\frac{1}{60} \\] where $\\gamma_{0}=14.13\\ldots$ is the imaginary part of the lowest-lying non-trivial zero of the Riemann zeta-function. The result is based on a new density estimate for zeros of the associated $k$-function, over $4\\cdot10^{21}$ times better than previously known estimates of this type.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-19T20:31:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M26",
      "11N05",
      "42A75"
    ],
    "keywords": [
      "sign changes",
      "prime number theorem",
      "lower bound",
      "density estimate",
      "error term"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}