{
  "id": "2301.05044",
  "version": "v1",
  "published": "2023-01-12T14:17:42.000Z",
  "updated": "2023-01-12T14:17:42.000Z",
  "title": "On almost-prime $k$-tuples",
  "authors": [
    "Bin Chen"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\tau$ denote the divisor function and $\\mathcal{H}=\\{h_{1},...,h_{k}\\}$ be an admissible set. We prove that there are infinitely many $n$ for which the product $\\prod_{i=1}^{k}(n+h_{i})$ is square-free and $\\sum_{i=1}^{k}\\tau(n+h_{i})\\leq \\lfloor \\rho_{k}\\rfloor$, where $\\rho_{k}$ is asymptotic to $\\frac{2126}{2853} k^{2}$. It improves a previous result of M. Ram Murty and A. Vatwani, replacing $2126/2853$ by $3/4$. The main ingredients in our proof are the higher rank Selberg sieve and Irving-Wu-Xi estimate for the divisor function in arithmetic progressions to smooth moduli.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-12T14:17:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N05",
      "11N35",
      "11N36"
    ],
    "keywords": [
      "almost-prime",
      "divisor function",
      "higher rank selberg sieve",
      "main ingredients",
      "ram murty"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}