{
  "id": "2411.01845",
  "version": "v1",
  "published": "2024-11-04T06:41:07.000Z",
  "updated": "2024-11-04T06:41:07.000Z",
  "title": "A note on zero-density approaches for the difference between consecutive primes",
  "authors": [
    "Valeriia Starichkova"
  ],
  "comment": "15 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this note, we generalise two results on prime numbers in short intervals. The first result is Ingham's theorem which connects the zero-density estimates with short intervals where the prime number theorem holds, and the second result is due to Heath-Brown and Iwaniec, which derives the weighted zero-density estimates used for obtaining the lower bound for the number of primes in short intervals. The generalised versions of these results make the connections between the zero-free regions, zero-density estimates, and the primes in short intervals more transparent. As an example, the generalisation of Ingham's theorem implies that, under the Density Hypothesis, the prime number theorem holds in $[x - \\sqrt{x}\\exp(\\log^{2/3+\\varepsilon}x), x]$, which refines upon the classic interval $[x - x^{1/2+ \\varepsilon}, x]$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-04T06:41:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N05",
      "11M26"
    ],
    "keywords": [
      "prime number theorem holds",
      "zero-density approaches",
      "short intervals",
      "consecutive primes",
      "difference"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}