{
  "id": "2107.02158",
  "version": "v1",
  "published": "2021-07-05T17:41:32.000Z",
  "updated": "2021-07-05T17:41:32.000Z",
  "title": "Quantitative bounds for Gowers uniformity of the Möbius and von Mangoldt functions",
  "authors": [
    "Terence Tao",
    "Joni Teräväinen"
  ],
  "comment": "53 pages",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "We establish quantitative bounds on the $U^k[N]$ Gowers norms of the M\\\"obius function $\\mu$ and the von Mangoldt function $\\Lambda$ for all $k$, with error terms of shape $O((\\log\\log N)^{-c})$. As a consequence, we obtain quantitative bounds for the number of solutions to any linear system of equations of finite complexity in the primes, with the same shape of error terms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-05T17:41:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B30",
      "11N37",
      "37A44"
    ],
    "keywords": [
      "von mangoldt function",
      "gowers uniformity",
      "error terms",
      "gowers norms",
      "linear system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}