{
  "id": "1811.05246",
  "version": "v1",
  "published": "2018-11-13T12:19:06.000Z",
  "updated": "2018-11-13T12:19:06.000Z",
  "title": "On eigenvalues of the kernel ${1\\over 2}+\\lfloor {1\\over xy}\\rfloor - {1\\over xy}$ ($0<x,y\\leq 1$)",
  "authors": [
    "N. Watt"
  ],
  "comment": "8 Pages, Plain Tex, submitted to the Journal de Th\\'{e}orie des Nombres de Bordeaux",
  "categories": [
    "math.NT"
  ],
  "abstract": "We show that the kernel $K(x,y)={1\\over 2}+\\lfloor {1\\over xy}\\rfloor -{1\\over xy}$ ($0<x,y\\leq 1$) has infinitely many positive eigenvalues and infinitely many negative eigenvalues. Our interest in this kernel is motivated by the appearance of the quadratic form $\\sum_{m,n\\leq N} K\\bigl( {m\\over N} , {n\\over N}\\bigr) \\mu(m)\\mu(n)$ in an indentity involving the Mertens function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-13T12:19:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25",
      "45C05",
      "11A07"
    ],
    "keywords": [
      "quadratic form",
      "mertens function",
      "appearance"
    ],
    "note": {
      "typesetting": "Plain TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}