{
  "id": "2409.12819",
  "version": "v1",
  "published": "2024-09-19T14:46:42.000Z",
  "updated": "2024-09-19T14:46:42.000Z",
  "title": "Residue Class Patterns of Consecutive Primes",
  "authors": [
    "Cheuk Fung Lau"
  ],
  "comment": "23 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "For $m,q \\in \\mathbb{N}$, we call an $m$-tuple $(a_1,\\ldots,a_m) \\in \\prod_{i=1}^m (\\mathbb{Z}/q\\mathbb{Z})^\\times$ good if there are infinitely many consecutive primes $p_1,\\ldots,p_m$ satisfying $p_i \\equiv a_i \\pmod{q}$ for all $i$. We show that given any $m$ sufficiently large, $q$ squarefree, and $A \\subseteq (\\mathbb{Z}/q\\mathbb{Z})^\\times$ with $|A|=\\lfloor 71(\\log m)^3 \\rfloor$, we can form at least one non-constant good $m$-tuple $(a_1,\\ldots,a_m) \\in \\prod_{i=1}^m A$. Using this, we can provide a lower bound for the number of residue class patterns attainable by consecutive primes, and for $m$ large and $\\varphi(q) \\gg (\\log m)^{10}$ this improves on the lower bound obtained from direct applications of Shiu (2000) and Dirichlet (1837). The main method is modifying the Maynard-Tao sieve found in Banks, Freiberg, and Maynard (2015), where instead of considering the 2nd moment we considered the $r$-th moment, where $r$ is an integer depending on $m$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-19T14:46:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N05",
      "11N13",
      "11N36"
    ],
    "keywords": [
      "consecutive primes",
      "lower bound",
      "th moment",
      "residue class patterns attainable",
      "direct applications"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}