{
  "id": "1909.08301",
  "version": "v1",
  "published": "2019-09-18T09:14:05.000Z",
  "updated": "2019-09-18T09:14:05.000Z",
  "title": "Zeros of $L(s)+L(2s)+\\cdots+L(Ns)$ in the region of absolute convergence",
  "authors": [
    "Łukasz Pańkowski",
    "Mattia Righetti"
  ],
  "comment": "7 pages, 1 figure",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper we show that for every Dirichlet $L$-function $L(s,\\chi)$ and every $N\\geq 2$ the Dirichlet series $L(s,\\chi)+L(2s,\\chi)+\\cdots+L(Ns,\\chi)$ have infinitely many zeros for $\\sigma>1$. Moreover we show that for many general $L$-functions with an Euler product the same holds if $N$ is sufficiently large, or if $N=2$. On the other hand we show with an example the the method doesn't work in general for $N=3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-18T09:14:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M41",
      "11M26"
    ],
    "keywords": [
      "absolute convergence",
      "euler product",
      "dirichlet series",
      "sufficiently large"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}