{
  "id": "1711.05358",
  "version": "v1",
  "published": "2017-11-14T23:43:05.000Z",
  "updated": "2017-11-14T23:43:05.000Z",
  "title": "Linear and quadratic uniformity of the Möbius function over $\\mathbb{F}_q[t]$",
  "authors": [
    "Pierre-Yves Bienvenu",
    "Thái Hoàng Lê"
  ],
  "comment": "23 pages, 1 figure",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "We examine correlations of the M\\\"obius function over $\\mathbb{F}_q[t]$ with linear or quadratic phases, that is, averages of the form \\begin{equation} \\label{eq:average} \\frac{1}{q^n}\\sum_{\\text{deg }f<n} \\mu(f)\\chi(Q(f)) \\end{equation} for an additive character $\\chi$ over $\\mathbb{F}_q$ and a polynomial $Q\\in\\mathbb{F}_q[x_0,\\ldots,x_{n-1}]$ of degree at most 2 in the coefficients $x_0,\\ldots, x_{n-1}$ of $f=\\sum_{i< n}x_i t^i$. Like in the integers, it is reasonable to expect that, due to the random-like behaviour of $\\mu$, such sums should exhibit considerable cancellation. In this paper we show that such a correlation is bounded by $O_\\epsilon \\left( q^{(-1/6+\\epsilon)n} \\right)$ for any $\\epsilon >0$ if $Q$ is linear and $O_A \\left( n^{-A} \\right)$ for any $A>0$ if $Q$ is quadratic. The latter result is unconditional if $Q(f)$ is a linear form in the coefficients of $f^2$, that is, a Hankel quadratic form, whereas for general quadratic forms, it is conditional on a new conjecture of an additive-combinatorial nature.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-14T23:43:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B30",
      "11T55"
    ],
    "keywords": [
      "quadratic uniformity",
      "möbius function",
      "hankel quadratic form",
      "general quadratic forms",
      "additive-combinatorial nature"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}