{
  "as_of": "2026-09-05",
  "references": [
    {
      "title": "Lecture Notes 3 for Cambridge Part III: Elementary Methods in Analytic Number Theory",
      "author": "Adam J. Harper",
      "date": "2015-04-02",
      "url": "https://warwick.ac.uk/fac/sci/maths/people/staff/harper/elementarymethodsnotes3.pdf",
      "used_for": [
        "Selberg fundamental lemma in remainder form",
        "naive beta reproduction",
        "signed beta -> beta - c beta^2 proof sketch"
      ]
    },
    {
      "title": "Distribution of prime numbers",
      "publisher": "Encyclopedia of Mathematics",
      "url": "https://encyclopediaofmath.org/wiki/Distribution_of_prime_numbers",
      "used_for": [
        "historical Bombieri-Wirsing log-power elementary PNT error terms",
        "historical elementary-method context"
      ]
    },
    {
      "title": "Wirsing's Elementary Proofs of the Prime Number Theorem with Remainder Terms",
      "author": "Harold G. Diamond",
      "year": 2023,
      "doi": "10.1007/978-3-031-31617-3_8",
      "used_for": "historical remainder-term calibration"
    },
    {
      "title": "An elementary proof of the prime number theorem with a remainder term",
      "authors": [
        "H. G. Diamond",
        "J. Steinig"
      ],
      "venue": "Inventiones Mathematicae 11 (1970), 199-258",
      "used_for": "historical stretched-exponential-in-log elementary PNT error"
    }
  ]
}