{
  "id": "2409.00813",
  "version": "v1",
  "published": "2024-09-01T19:04:38.000Z",
  "updated": "2024-09-01T19:04:38.000Z",
  "title": "Functional equation for LC-functions with even or odd modulator",
  "authors": [
    "Lahcen Lamgouni"
  ],
  "comment": "25 pages, 4 figures",
  "categories": [
    "math.NT",
    "math.CV"
  ],
  "abstract": "In a recent work, we introduced \\textit{LC-functions} $L(s,f)$, associated to a certain real-analytic function $f$ at $0$, extending the concept of the Hurwitz zeta function and its formula. In this paper, we establish the existence of a functional equation for a specific class of LC-functions. More precisely, we demonstrate that if the function $p_f(t):=f(t)(e^t-1)/t$, called the \\textit{modulator} of $L(s,f)$, exhibits even or odd symmetry, the \\textit{LC-function formula} -- a generalization of the Hurwitz formula -- naturally simplifies to a functional equation analogous to that of the Dirichlet L-function $L(s,\\chi)$, associated to a primitive character $\\chi$ of inherent parity. Furthermore, using this equation, we derive a general formula for the values of these LC-functions at even or odd positive integers, depending on whether the modulator $p_f$ is even or odd, respectively. Two illustrative examples of the functional equation are provided for distinct parity of modulators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-01T19:04:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M41"
    ],
    "keywords": [
      "functional equation",
      "odd modulator",
      "lc-functions",
      "hurwitz zeta function",
      "distinct parity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}