{
  "id": "2405.10888",
  "version": "v1",
  "published": "2024-05-17T16:25:44.000Z",
  "updated": "2024-05-17T16:25:44.000Z",
  "title": "The fourth moment of the Hurwitz zeta function",
  "authors": [
    "Winston Heap",
    "Anurag Sahay"
  ],
  "comment": "34 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove a sharp upper bound for the fourth moment of the Hurwitz zeta function $\\zeta(s,\\alpha)$ on the critical line when the shift parameter $\\alpha$ is irrational and of irrationality exponent strictly less than 3. As a consequence, we determine the order of magnitude of the $2k$th moment for all $0 \\leqslant k \\leqslant 2$ in this case. In contrast to the Riemann zeta function and other $L$-functions from arithmetic, these grow like $T (\\log T)^k$. This suggests, and we conjecture, that the value distribution of $\\zeta(s,\\alpha)$ on the critical line is Gaussian.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-17T16:25:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hurwitz zeta function",
      "fourth moment",
      "critical line",
      "sharp upper bound",
      "riemann zeta function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}