{
  "id": "1406.7326",
  "version": "v1",
  "published": "2014-06-27T22:08:06.000Z",
  "updated": "2014-06-27T22:08:06.000Z",
  "title": "Bounding sums of the Möbius function over arithmetic progressions",
  "authors": [
    "Lynnelle Ye"
  ],
  "comment": "29 pages; undergraduate thesis version",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $M(x)=\\sum_{1\\le n\\le x}\\mu(n)$ where $\\mu$ is the M\\\"obius function. It is well-known that the Riemann Hypothesis is equivalent to the assertion that $M(x)=O(x^{1/2+\\epsilon})$ for all $\\epsilon>0$. There has been much interest and progress in further bounding $M(x)$ under the assumption of the Riemann Hypothesis. In 2009, Soundararajan established the current best bound of \\[ M(x)\\ll\\sqrt{x}\\exp\\left((\\log x)^{1/2}(\\log\\log x)^c\\right) \\] (setting $c$ to $14$, though this can be reduced). Halupczok and Suger recently applied Soundararajan's method to bound more general sums of the M\\\"obius function over arithmetic progressions, of the form \\[ M(x;q,a)=\\sum_{\\substack{n\\le x \\\\ n\\equiv a\\pmod{q}}}\\mu(n). \\] They were able to show that assuming the Generalized Riemann Hypothesis, $M(x;q,a)$ satisfies \\[ M(x;q,a)\\ll_{\\epsilon}\\sqrt{x}\\exp\\left((\\log x)^{3/5}(\\log\\log x)^{16/5+\\epsilon}\\right) \\] for all $q\\le\\exp\\left(\\frac{\\log 2}2\\lfloor(\\log x)^{3/5}(\\log\\log x)^{11/5}\\rfloor\\right)$, with $a$ such that $(a,q)=1$, and $\\epsilon>0$. In this paper, we improve Halupczok and Suger's work to obtain the same bound for $M(x;q,a)$ as Soundararajan's bound for $M(x)$ (with a $1/2$ in the exponent of $\\log x$), with no size or divisibility restriction on the modulus $q$ and residue $a$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-27T22:08:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N56"
    ],
    "keywords": [
      "arithmetic progressions",
      "möbius function",
      "bounding sums",
      "current best bound",
      "generalized riemann hypothesis"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.7326Y"
    }
  }
}