{
  "id": "2003.00901",
  "version": "v1",
  "published": "2020-02-28T09:31:58.000Z",
  "updated": "2020-02-28T09:31:58.000Z",
  "title": "Pseudodifferential Operators on $\\mathbf{Q}_p$ and $L$-Series",
  "authors": [
    "Parikshit Dutta",
    "Debashis Ghoshal"
  ],
  "comment": "1+13 pages, 1 figure",
  "categories": [
    "math.NT",
    "hep-th"
  ],
  "abstract": "We define a family of pseudodifferential operators on the Hilbert space $L^2(\\mathbf{Q}_p)$ of complex valued square-integrable functions on the $p$-adic number field $\\mathbf{Q}_p$. The Riemann zeta-function and the related Dirichlet $L$-functions can be expressed as a trace of these operators on a subspace of $L^2(\\mathbf{Q}_p)$. We also extend this to the $L$-functions associated with modular (cusp) forms. Wavelets on $L^2(\\mathbf{Q}_p)$ are common sets of eigenfunctions of these operators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-28T09:31:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "47G30",
      "65T60"
    ],
    "keywords": [
      "pseudodifferential operators",
      "adic number field",
      "hilbert space",
      "common sets",
      "complex valued square-integrable functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}