{
  "id": "1511.08116",
  "version": "v1",
  "published": "2015-11-25T16:55:08.000Z",
  "updated": "2015-11-25T16:55:08.000Z",
  "title": "The Lerch zeta function IV. Hecke operators",
  "authors": [
    "Jeffrey C. Lagarias",
    "Wen-Ching Winnie Li"
  ],
  "comment": "40 pages, preliminary version",
  "categories": [
    "math.NT"
  ],
  "abstract": "This paper studies algebraic and analytic structures associated with the Lerch zeta function. It defines a family of two-variable Hecke operators $\\{ T_m: \\, m \\ge 1\\}$ given by $T_m(f)(a, c) = \\frac{1}{m} \\sum_{k=0}^{m-1} f(\\frac{a+k}{m}, mc)$ acting on certain spaces of real-analytic functions, including Lerch zeta functions for various parameter values. It determines the action of various related operators on these function spaces. It characterizes Lerch zeta functions (for fixed $s$ in the following way. It shows that there is for each $s \\in {\\bf C}$ a two-dimensional vector space spanned by linear combinations of Lerch zeta functions is characterized as a maximal space of simultaneous eigenfunctions for this family of Hecke operators. This result is an analogue of a result of Milnor for the Hurwitz zeta function. We also relate these functions to a linear partial differential operator in the $(a, c)$-variables having the Lerch zeta function as an eigenfunction.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-25T16:55:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M35"
    ],
    "keywords": [
      "linear partial differential operator",
      "characterizes lerch zeta functions",
      "hurwitz zeta function",
      "paper studies algebraic",
      "two-dimensional vector space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151108116L"
    }
  }
}