{
  "id": "uniform-exclusion-of-landau-siegel-zeros",
  "title": "Uniform exclusion of Landau–Siegel zeros",
  "description": "OpenAI's theorem (2026) that (1 - beta) log q >= c for an absolute constant c > 0 and every real zero beta of every primitive real Dirichlet L-function of conductor q >= 3, the logarithmic form of the Landau–Siegel problem. The prime bias forced by a real zero near 1, an interpolation theorem for polynomials with separately bounded degrees on a projected integer box, a weighted determinant of conjugates in Q(sqrt d, sqrt 2), and its two bounds: Hadamard's inequality, and divisibility by the Frobenius congruence at the primes with chi(p) = -1. A zero-free region with no exceptional zero follows.",
  "units": [
    {
      "id": "real-zeros-and-the-prime-bias",
      "title": "Real zeros and the prime bias",
      "reader": "real-zeros-and-the-prime-bias.html",
      "source": "src/real-zeros-and-the-prime-bias.md",
      "source_sha256": "0abd8c01b8c69f078d304e3882d31ce71433e8361a78fecfeae8d3b01710e922",
      "status": "author_self_checked"
    },
    {
      "id": "interpolation-on-a-projected-integer-box",
      "title": "Interpolation on a projected integer box",
      "reader": "interpolation-on-a-projected-integer-box.html",
      "source": "src/interpolation-on-a-projected-integer-box.md",
      "source_sha256": "09e8ae6fad182877cf331f07bd9d9748a0b1a37723da4f8170234cea672a1c93",
      "status": "author_self_checked"
    },
    {
      "id": "a-weighted-determinant-in-a-biquadratic-field",
      "title": "A weighted determinant in a biquadratic field",
      "reader": "a-weighted-determinant-in-a-biquadratic-field.html",
      "source": "src/a-weighted-determinant-in-a-biquadratic-field.md",
      "source_sha256": "1599d81b5d3f066f80159e5c86df519e76b25e61ff60e97c00aa3e71b9aafda6",
      "status": "author_self_checked"
    },
    {
      "id": "two-bounds-for-the-determinant",
      "title": "Two bounds for the determinant",
      "reader": "two-bounds-for-the-determinant.html",
      "source": "src/two-bounds-for-the-determinant.md",
      "source_sha256": "2c1d96f54beef84be323e408b030c998696f2131955786b558c934b0a78d3b88",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "complex analysis and Dirichlet L-functions as in the programme courses The Riemann zeta function and Dirichlet L-functions and primes in progressions, cited at each use",
    "linear algebra over a field: determinants, rank, Hadamard's inequality"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}