{
  "id": "1412.7812",
  "version": "v1",
  "published": "2014-12-25T11:37:25.000Z",
  "updated": "2014-12-25T11:37:25.000Z",
  "title": "Cauchy Means of Dirichlet polynomials",
  "authors": [
    "Michel Weber"
  ],
  "comment": "15 pages",
  "categories": [
    "math.NT",
    "math.CA"
  ],
  "abstract": "We study Cauchy means of Dirichlet polynomials $$\\int_\\R \\Big|\\sum_{n=1}^N \\frac{1}{ n^{\\s+ ist}} \\Big|^{2q} \\frac{\\dd t}{\\pi( t^2+1)}.$$ These integrals were investigated when $q=1,\\s= 1, s=1/2 $ by Wilf, using integral operator theory and Widom's eigenvalue estimates. We show the optimality of some upper bounds obtained by Wilf. We also obtain new estimates for the case $q\\ge 1$, $\\s\\ge 0$ and $s>0$. We complete Wilf's approach by relating it with other approaches (having notably connection with Brownian motion), allowing simple proofs, and also prove new results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-25T11:37:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B35",
      "47A70",
      "15B05",
      "30B50",
      "60J65"
    ],
    "keywords": [
      "dirichlet polynomials",
      "integral operator theory",
      "complete wilfs approach",
      "widoms eigenvalue estimates",
      "study cauchy means"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.7812W"
    }
  }
}