{
  "id": "1912.01716",
  "version": "v1",
  "published": "2019-12-03T22:14:28.000Z",
  "updated": "2019-12-03T22:14:28.000Z",
  "title": "On eigenfunctions of the kernel $\\frac{1}{2} + \\lfloor \\frac{1}{xy} \\rfloor - \\frac{1}{xy}$",
  "authors": [
    "Nigel Watt"
  ],
  "comment": "58 pages, LaTeX. Submitted to Functiones et Approximatio Commentarii Mathematici",
  "categories": [
    "math.NT",
    "math.CA"
  ],
  "abstract": "The integral kernel $K(x,y) := \\frac{1}{2} + \\lfloor \\frac{1}{xy} \\rfloor - \\frac{1}{xy}$ ($0<x,y\\leq 1$) has connections with the Riemann zeta-function and a (recently observed) connection with the Mertens function. In this paper we begin a general study of the eigenfunctions of $K$. Our proofs utilise some classical real analysis (including Lebesgue's theory of integration) and elements of the established theory of square integrable symmetric integral kernels.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-03T22:14:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33E30",
      "45M05",
      "47B35",
      "11M06"
    ],
    "keywords": [
      "eigenfunctions",
      "square integrable symmetric integral kernels",
      "connection",
      "lebesgues theory",
      "mertens function"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 58,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}