{
  "id": "1205.4610",
  "version": "v1",
  "published": "2012-05-21T14:20:46.000Z",
  "updated": "2012-05-21T14:20:46.000Z",
  "title": "Almost-prime $k$-tuples",
  "authors": [
    "James Maynard"
  ],
  "comment": "26 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $k\\ge 2$ and $\\Pi(n)=\\prod_{i=1}^k(a_in+b_i)$ for some integers $a_i, b_i$ ($1\\le i\\le k$). Suppose that $\\Pi(n)$ has no fixed prime divisors. Weighted sieves have shown for infinitely many integers $n$ that $\\Omega(\\Pi(n))\\le r_k$ holds for some integer $r_k$ which is asymptotic to $k\\log{k}$. We use a new kind of weighted sieve to improve the possible values of $r_k$ when $k\\ge 4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-05-21T14:20:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N36",
      "11N05",
      "11N35"
    ],
    "keywords": [
      "almost-prime",
      "weighted sieve",
      "fixed prime divisors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1205.4610M"
    }
  }
}