{
  "id": "2506.16766",
  "version": "v1",
  "published": "2025-06-20T05:59:36.000Z",
  "updated": "2025-06-20T05:59:36.000Z",
  "title": "On the error term of the fourth moment of the Riemann zeta-function",
  "authors": [
    "Neea Palojärvi",
    "Tim Trudgian"
  ],
  "comment": "12 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We examine the size of $E_{2}(T)$, the error term in the asymptotic formula for $\\int_{0}^{T} |\\zeta(1/2 + it)|^{4}\\, dt$ where $\\zeta(s)$ is the Riemann zeta-function. We make improvements in the powers of $\\log T$ in the known bounds for $E_{2}(T)$ and $\\int_{0}^{T} E_{2}(t)^{2}\\, dt$. As a consequence, we obtain small logarithmic improvements for $k$th moments where $8\\leq k\\leq 12$. In particular, we make a modest improvement on the 12th power moment for $\\zeta(s)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-06-20T05:59:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06"
    ],
    "keywords": [
      "riemann zeta-function",
      "error term",
      "fourth moment",
      "12th power moment",
      "small logarithmic improvements"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}