{
  "id": "1405.6982",
  "version": "v1",
  "published": "2014-05-27T17:30:16.000Z",
  "updated": "2014-05-27T17:30:16.000Z",
  "title": "Non-vanishing of Dirichlet series with periodic coefficients",
  "authors": [
    "Tapas Chatterjee",
    "M. Ram Murty"
  ],
  "comment": "22 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "For any periodic function $f:{\\mathbb N} \\to {\\mathbb C}$ with period $q$, we study the Dirichlet series $L(s,f):=\\sum_{n\\geq 1} f(n)/n^s.$ It is well-known that this admits an analytic continuation to the entire complex plane except at $s=1$, where it has a simple pole with residue $$\\rho:= q^{-1}\\sum_{1\\leq a\\leq q} f(a).$$ Thus, the function is analytic at $s=1$ when $\\rho=0$ and in this case, we study its non-vanishing using the theory of linear forms in logarithms and Dirichlet $L$-series. In this way, we give new proofs of an old criterion of Okada for the non-vanishing of $L(1,f)$ as well as a classical theorem of Baker, Birch and Wirsing. We also give some new necessary and sufficient conditions for the non-vanishing of $L(1,f)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-27T17:30:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "11M20"
    ],
    "keywords": [
      "dirichlet series",
      "periodic coefficients",
      "non-vanishing",
      "entire complex plane",
      "sufficient conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.6982C"
    }
  }
}