{
  "id": "the-first-moment-of-cubic-gauss-sums",
  "title": "The first moment of cubic Gauss sums",
  "description": "Patterson's first-moment conjecture for cubic Gauss sums, proved unconditionally after OpenAI's 2026 manuscript: the normalized cubic Gauss sums over the primary primes of Z[omega] of norm at most X add up to (6/5) c_* X^(5/6)/log X, with c_* = (2 pi)^(2/3)/(3 Gamma(2/3)). Cubic Gauss sums and their cubes; large sieves for cubic characters; the level Voronoi formula; exceptional character moments; corrected dispersion; an exact decomposition of the prime sum; the sharp cutoff; fixed angular modes and the powers G^k with k not divisible by 3.",
  "units": [
    {
      "id": "cubic-gauss-sums-and-pattersons-conjecture",
      "title": "Cubic Gauss sums and Patterson's conjecture",
      "reader": "cubic-gauss-sums-and-pattersons-conjecture.html",
      "source": "src/cubic-gauss-sums-and-pattersons-conjecture.md",
      "source_sha256": "0f5216653391433f875ba504fa0adde5e90579a320bd01f9efb803669e2325ec",
      "status": "author_self_checked"
    },
    {
      "id": "large-sieves-for-cubic-characters",
      "title": "Large sieves for cubic characters",
      "reader": "large-sieves-for-cubic-characters.html",
      "source": "src/large-sieves-for-cubic-characters.md",
      "source_sha256": "723782c84eac5887addeb3da830a49376cba07fadac352530e74667159d183c6",
      "status": "author_self_checked"
    },
    {
      "id": "the-level-formula-and-type-i-sums",
      "title": "The level formula and Type I sums",
      "reader": "the-level-formula-and-type-i-sums.html",
      "source": "src/the-level-formula-and-type-i-sums.md",
      "source_sha256": "1d3d6453a4ffb188b0005f2d7eddb20be734dc48f446b263bf84fd4e8259bed6",
      "status": "author_self_checked"
    },
    {
      "id": "character-moments-at-two-exceptional-configurations",
      "title": "Character moments at two exceptional configurations",
      "reader": "character-moments-at-two-exceptional-configurations.html",
      "source": "src/character-moments-at-two-exceptional-configurations.md",
      "source_sha256": "c18e1a19d1951cbe41dd547cca31cf7ecb2673748a47e6f5e21efd26835b4fb5",
      "status": "author_self_checked"
    },
    {
      "id": "corrected-dispersion",
      "title": "Corrected dispersion",
      "reader": "corrected-dispersion.html",
      "source": "src/corrected-dispersion.md",
      "source_sha256": "37f83af591fd33b2ff780e8e4055d72735376c30e03b5e44db8275ade9609972",
      "status": "author_self_checked"
    },
    {
      "id": "an-exact-decomposition-of-the-prime-sum",
      "title": "An exact decomposition of the prime sum",
      "reader": "an-exact-decomposition-of-the-prime-sum.html",
      "source": "src/an-exact-decomposition-of-the-prime-sum.md",
      "source_sha256": "d2e50906222a174aedcf89eb96d8ac128145a3f0f089b7cda0bc053258bc892d",
      "status": "author_self_checked"
    },
    {
      "id": "bilinear-sums-and-the-sharp-cutoff",
      "title": "Bilinear sums and the sharp cutoff",
      "reader": "bilinear-sums-and-the-sharp-cutoff.html",
      "source": "src/bilinear-sums-and-the-sharp-cutoff.md",
      "source_sha256": "609c720191ff53de4f4b8bf41642a459393ef2f4557dcc4f4edcaa6c9b3d3cb7",
      "status": "author_self_checked"
    },
    {
      "id": "angular-modes-and-powers-of-gauss-sums",
      "title": "Angular modes and powers of Gauss sums",
      "reader": "angular-modes-and-powers-of-gauss-sums.html",
      "source": "src/angular-modes-and-powers-of-gauss-sums.md",
      "source_sha256": "9cc077d5c5b247292047500ae9a74f3112fb99a0d3d89e55fc0e54c9689e29a3",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "power residue symbols, the cubic reciprocity law and ideal reciprocity with conductors (course Class field theory, lessons Power residue symbols and reciprocity laws; Ray class fields, conductors and ideal reciprocity)",
    "Gauss sums of Dirichlet characters and the prime number theorem for arithmetic progressions (course Dirichlet L-functions and primes in progressions, NT-DIRL-03 and NT-DIRL-09)",
    "Mertens' estimates, the Gamma function and Stirling's formula, mean values of Dirichlet polynomials (course The Riemann zeta function, NT-ZETA-02, NT-ZETA-03, NT-ZETA-16)",
    "Hecke characters and the functional equation of Hecke L-functions (course Adeles, ideles and Tate's thesis, NT-ADL-06 and NT-ADL-10)",
    "Fourier inversion and Poisson summation on lattices (course Harmonic analysis on locally compact abelian groups)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}