{
  "id": "1610.08551",
  "version": "v1",
  "published": "2016-10-26T21:14:02.000Z",
  "updated": "2016-10-26T21:14:02.000Z",
  "title": "Computations of the Mertens Function and Improved Bounds on the Mertens Conjecture",
  "authors": [
    "Greg Hurst"
  ],
  "comment": "15 pages, 7 figures, 4 tables",
  "categories": [
    "math.NT"
  ],
  "abstract": "The Mertens function is defined as $M(x) = \\sum_{n \\leq x} \\mu(n)$, where $\\mu(n)$ is the M\\\"obius function. The Mertens conjecture states $|M(x)/\\sqrt{x}| < 1$ for $x > 1$, which was proven false in 1985 by showing $\\liminf M(x)/\\sqrt{x} < -1.009$ and $\\limsup M(x)/\\sqrt{x} > 1.06$. The same techniques used were revisited here with present day hardware and algorithms, giving improved lower and upper bounds of $-1.837625$ and $1.826054$. In addition, $M(x)$ was computed for all $x \\leq 10^{16}$, recording all extrema, all zeros, and $10^8$ values sampled at a regular interval. Lastly, an algorithm to compute $M(x)$ in $O(x^{2/3+\\varepsilon})$ time was used on all powers of two up to $2^{73}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-26T21:14:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "mertens function",
      "computations",
      "mertens conjecture states",
      "regular interval",
      "proven false"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}