{
  "id": "2012.00610",
  "version": "v1",
  "published": "2020-11-30T10:25:08.000Z",
  "updated": "2020-11-30T10:25:08.000Z",
  "title": "Estimates of the bounds of $π(x)$ and $π((x+1)^{2}-x^{2})$",
  "authors": [
    "Connor Paul Wilson"
  ],
  "comment": "8 pages, 0 figures",
  "categories": [
    "math.NT"
  ],
  "abstract": "We show the following bounds on the prime counting function $\\pi(x)$ using principles from analytic number theory, giving an estimate: $$2 \\log 2 \\geq \\limsup_{x \\rightarrow \\infty} \\frac{\\pi(x)}{x / \\log x} \\geq \\liminf_{x \\rightarrow \\infty} \\frac{\\pi(x)}{x / \\log x} \\geq \\log 2$$ for all $x$ sufficiently large. We also conjecture about the bounding of $\\pi((x+1)^{2}-x^{2})$, as is relevant to Legendre's conjecture about the number of primes in the aforementioned interval such that: $$ \\left \\lfloor\\frac{1}{2}\\left(\\frac{\\left(x+1\\right)^{2}}{\\log\\left(x+1\\right)}-\\frac{x^{2}}{\\log x}\\right)-\\frac{\\left(\\log x\\right)^{2}}{\\log\\left(\\log x\\right)}\\right \\rfloor \\leq \\pi((x+1)^{2}-x^{2}) \\leq $$ $$ \\left \\lfloor\\frac{1}{2}\\left(\\frac{\\left(x+1\\right)^{2}}{\\log\\left(x+1\\right)}-\\frac{x^{2}}{\\log x}\\right) + \\log^{2}x\\log\\log x \\right \\rfloor$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-30T10:25:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "analytic number theory",
      "legendres conjecture",
      "prime counting function",
      "principles",
      "sufficiently large"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}