{
  "id": "1911.05295",
  "version": "v1",
  "published": "2019-11-13T05:02:38.000Z",
  "updated": "2019-11-13T05:02:38.000Z",
  "title": "An improved asymptotic formula for the distribution of irreducible polynomials in arithmetic progressions over Fq",
  "authors": [
    "Zhang Zihan",
    "Han Dongchun"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\mathbb{F}_{q}$ be a finite field with $q$ elements and $\\mathbb{F}_{q}[x]$ the ring of polynomials over $\\mathbb{F}_{q}$. Let $l(x), k(x)$ be coprime polynomials in $\\mathbb{F}_{q}[x]$ and $\\Phi(k)$ the Euler function in $\\mathbb{F}_{q}[x]$. Let $\\pi(l, k; n)$ be the number of monic irreducible polynomials of degree $n$ in $\\mathbb{F}_{q}[x]$ which are congruent to $l(x)$ module $k(x)$. For any positive integer $n$, we denote by $\\Omega(n)$ the least prime divisor of $n$. In this paper, we show that $$\\pi(l, k; n)=\\frac{1}{\\Phi(k)}\\frac{q^{n}}{n}+O\\left(n^{\\alpha}\\right)+O\\left(\\frac{q^{\\frac{n}{\\Omega{(n)}}}}{n}\\right),$$ where $\\alpha$ only depends on the choice of $k(x)\\in\\Fq$. Note that the above error term improves the one implied by Weil's conjecture. Our approach is completely elementary.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-13T05:02:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "asymptotic formula",
      "arithmetic progressions",
      "distribution",
      "finite field",
      "coprime polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}