{
  "id": "1506.06161",
  "version": "v1",
  "published": "2015-06-19T21:24:21.000Z",
  "updated": "2015-06-19T21:24:21.000Z",
  "title": "The Lerch zeta function III. Polylogarithms and special values",
  "authors": [
    "Jeffrey C. Lagarias",
    "W. -C. Winnie Li"
  ],
  "comment": "55 pages, 6 figures",
  "categories": [
    "math.NT"
  ],
  "abstract": "This paper studies algebraic and analytic structures associated with the Lerch zeta function, extending the complex variables viewpoint taken in part II. The Lerch transcendent $\\Phi(s, z, c)$ is obtained from the Lerch zeta function $\\zeta(s, a, c)$ by the change of variable $z=e^{2 \\pi i a}$. We show that it analytically continues to a maximal domain of holomorphy in three complex variables $(s, z, c)$ as a multivalued function defined over the base manifold ${\\bf C} \\times P^1({\\bf C} \\smallsetminus \\{0, 1, \\infty\\}) \\times ({\\bf C}\\smallsetminus {\\bf Z})$. and compute the monodromy functions defining the multivaluedness. For positive integer values s=m and c=1 this function is closely related to the classical m-th order polylogarithm $Li_m(z)$ We study its behavior as a function of two variables $(z, c)$ for special values where s=m is an integer. For $m \\ge 1$ it gives a one-parameter deformation of the polylogarithm, and satisfies a linear ODE with coefficients depending on c, of order m+1 of Fuchsian type. We determine its (m+1)-dimensional monodromy representation, which is a non-algebraic deformation of the monodromy of $Li_m(z).$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-19T21:24:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M35",
      "33B30"
    ],
    "keywords": [
      "lerch zeta function",
      "special values",
      "complex variables viewpoint taken",
      "classical m-th order polylogarithm",
      "paper studies algebraic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 55,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150606161L"
    }
  }
}