{
  "id": "2112.11412",
  "version": "v2",
  "published": "2021-12-21T18:18:59.000Z",
  "updated": "2022-02-05T18:26:01.000Z",
  "title": "Siegel zeros, twin primes, Goldbach's conjecture, and primes in short intervals",
  "authors": [
    "Kaisa Matomäki",
    "Jori Merikoski"
  ],
  "comment": "The submitted version. Several typos and inaccuracies corrected, including some in the statements of our results",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the distribution of prime numbers under the unlikely assumption that Siegel zeros exist. In particular we prove for \\[ \\sum_{n \\leq X} \\Lambda(n) \\Lambda(\\pm n+h) \\] an asymptotic formula which holds uniformly for $h = O(X)$. Such an asymptotic formula has been previously obtained only for fixed $h$ in which case our result quantitatively improves those of Heath-Brown (1983) and Tao and Ter\\\"av\\\"ainen (2021). Since our main theorems work also for large $h$ we can derive new results concerning connections between Siegel zeros and the Goldbach conjecture and between Siegel zeros and primes in almost all very short intervals.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-02-05T18:26:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "siegel zeros",
      "short intervals",
      "twin primes",
      "goldbachs conjecture",
      "asymptotic formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}