{
  "id": "0809.0076",
  "version": "v1",
  "published": "2008-08-30T17:09:12.000Z",
  "updated": "2008-08-30T17:09:12.000Z",
  "title": "Matrices related to Dirichlet series",
  "authors": [
    "David A. Cardon"
  ],
  "comment": "17 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We attach a certain $n \\times n$ matrix $A_n$ to the Dirichlet series $L(s)=\\sum_{k=1}^{\\infty}a_k k^{-s}$. We study the determinant, characteristic polynomial, eigenvalues, and eigenvectors of these matrices. The determinant of $A_n$ can be understood as a weighted sum of the first $n$ coefficients of the Dirichlet series $L(s)^{-1}$. We give an interpretation of the partial sum of a Dirichlet series as a product of eigenvalues. In a special case, the determinant of $A_n$ is the sum of the M\\\"obius function. We disprove a conjecture of Barrett and Jarvis regarding the eigenvalues of $A_n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-08-30T17:09:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dirichlet series",
      "determinant",
      "eigenvalues",
      "special case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0809.0076C"
    }
  }
}