{
  "id": "large-gaps-between-primes",
  "title": "Large gaps between primes",
  "description": "Large gaps between consecutive primes. The average gap, and the theorem that for every C > 0 a positive proportion of the gaps p_(n+1) - p_n exceed C log p_n, which answers a question of Erdős and Prachar on the increases of p_n/n. Smooth divisor sums and their averages, with a prime factor through the Bombieri-Vinogradov theorem; the singular series and Gallagher's theorem on its average; correlations of smooth divisor sums by Euler products near the pole of the zeta function; a square of signed divisor sums whose prime mass in an adjacent interval is controlled by neighbouring dimensions; and an alternating choice of test functions that makes this mass arbitrarily small.",
  "units": [
    {
      "id": "prime-gaps-and-adjacent-intervals",
      "title": "Prime gaps and adjacent intervals",
      "reader": "prime-gaps-and-adjacent-intervals.html",
      "source": "src/prime-gaps-and-adjacent-intervals.md",
      "source_sha256": "9930f843f5630a83654b398affcb085af16ca22a4dcde9ef2b3f512dfac93dfb",
      "status": "author_self_checked"
    },
    {
      "id": "divisor-sums-and-primes-in-progressions",
      "title": "Divisor sums and primes in progressions",
      "reader": "divisor-sums-and-primes-in-progressions.html",
      "source": "src/divisor-sums-and-primes-in-progressions.md",
      "source_sha256": "00e01018c10881c9cc794c994cd8947e463a40b186c796b70cd14713fc0ee1da",
      "status": "author_self_checked"
    },
    {
      "id": "the-singular-series-and-its-average",
      "title": "The singular series and its average",
      "reader": "the-singular-series-and-its-average.html",
      "source": "src/the-singular-series-and-its-average.md",
      "source_sha256": "39582893498dc62eaf55cda20cb4a27edf148874e9f5d4108ecfba8f6bd323e8",
      "status": "author_self_checked"
    },
    {
      "id": "correlations-of-smooth-divisor-sums",
      "title": "Correlations of smooth divisor sums",
      "reader": "correlations-of-smooth-divisor-sums.html",
      "source": "src/correlations-of-smooth-divisor-sums.md",
      "source_sha256": "e64e714775a33e6e0d48257cd48419c2d7289c1664e69fa547bba0109b8afa04",
      "status": "author_self_checked"
    },
    {
      "id": "a-square-with-little-prime-mass-in-the-next-interval",
      "title": "A square with little prime mass in the next interval",
      "reader": "a-square-with-little-prime-mass-in-the-next-interval.html",
      "source": "src/a-square-with-little-prime-mass-in-the-next-interval.md",
      "source_sha256": "db9f1075badd37a797a01a83d80add53564b49ebb25d013027287cb7c82718bf",
      "status": "author_self_checked"
    },
    {
      "id": "cancellation-between-adjacent-dimensions",
      "title": "Cancellation between adjacent dimensions",
      "reader": "cancellation-between-adjacent-dimensions.html",
      "source": "src/cancellation-between-adjacent-dimensions.md",
      "source_sha256": "9490641fead0e44444dbd39ef7efd3e1e7f0bc8508a31dcacce12e3197649580",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "NT-ZETA lessons on Dirichlet series (Euler products, zeta near s = 1), on Chebyshev's and Mertens's estimates, and on the prime number theorem",
    "NT-DIRL lesson on the Bombieri-Vinogradov theorem (Corollary 4.1)",
    "Fourier inversion on the Schwartz space (elliptic-operators-and-boundary-problems, prerequisite bridges, Theorem 1.1)",
    "Core course D10 (Fubini's theorem)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}