{
  "id": "2103.13542",
  "version": "v1",
  "published": "2021-03-25T01:08:48.000Z",
  "updated": "2021-03-25T01:08:48.000Z",
  "title": "Moments of the Hurwitz zeta Function on the Critical Line",
  "authors": [
    "Anurag Sahay"
  ],
  "comment": "comments/suggestions welcome!",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the moments $M_k(T;\\alpha) = \\int_T^{2T} |\\zeta(s,\\alpha)|^{2k}\\,dt$ of the Hurwitz zeta function $\\zeta(s,\\alpha)$ on the critical line, $s = 1/2 + it$. We conjecture, in analogy with the Riemann zeta function, that $M_k(T;\\alpha) \\sim c_k(\\alpha) T (\\log T)^{k^2}$. In the case of $\\alpha\\in\\mathbb{Q}$, we use heuristics from analytic number theory and random matrix theory to compute $c_k(\\alpha)$. In the process, we investigate moments of products of Dirichlet $L$-functions on the critical line. We provide several pieces of evidence for our conjectures, in particular by proving some of them for the cases $k = 1,2$ and $\\alpha \\in \\mathbb{Q}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-25T01:08:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M35"
    ],
    "keywords": [
      "hurwitz zeta function",
      "critical line",
      "riemann zeta function",
      "analytic number theory",
      "random matrix theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}