{
  "id": "1004.2168",
  "version": "v1",
  "published": "2010-04-13T12:57:05.000Z",
  "updated": "2010-04-13T12:57:05.000Z",
  "title": "The Generating Function for the Dirichlet Series $L_m(s)$",
  "authors": [
    "William Y. C. Chen",
    "Neil J. Y. Fan",
    "Jeffrey Y. T. Jia"
  ],
  "comment": "18 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "The Dirichlet series $L_m(s)$ are of fundamental importance in number theory. Shanks defined the generalized Euler and class numbers in connection with these Dirichlet series, denoted by $\\{s_{m,n}\\}_{n\\geq 0}$. We obtain a formula for the exponential generating function $s_m(x)$ of $s_{m,n}$, where m is an arbitrary positive integer. In particular, for m>1, say, $m=bu^2$, where b is square-free and u>1, we prove that $s_m(x)$ can be expressed as a linear combination of the four functions $w(b,t)\\sec (btx)(\\pm \\cos ((b-p)tx)\\pm \\sin (ptx))$, where p is an integer satisfying $0\\leq p\\leq b$, $t|u^2$ and $w(b,t)=K_bt/u$ with $K_b$ being a constant depending on b. Moreover, the Dirichlet series $L_m(s)$ can be easily computed from the generating function formula for $s_m(x)$. Finally, we show that the main ingredient in the formula for $s_{m,n}$ has a combinatorial interpretation in terms of the m-signed permutations defined by Ehrenborg and Readdy. In principle, this answers a question posed by Shanks concerning a combinatorial interpretation for the numbers $s_{m,n}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-04-13T12:57:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B68",
      "05A05"
    ],
    "keywords": [
      "dirichlet series",
      "combinatorial interpretation",
      "class numbers",
      "fundamental importance",
      "arbitrary positive integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1004.2168C"
    }
  }
}