{ "id": "marked-type-ii-estimates-for-prime-predecessors", "title": "Marked Type II estimates for prime predecessors", "description": "OpenAI's one-sided marked Type II estimate for prime predecessors: bilinear sums of the prime indicator minus its rough-number average against arbitrary rough coefficients, at the predecessors mn - 1 and weighted by marks from fixed bands of primes, are smaller than every power of log x. A coefficient criterion; a zero-free strip at large height and long prime polynomials; Cauchy's inequality and a determinant equation; physical states, root residues and independent lines; lifespans, a memory of pending primes and contraction on clean and dirty edges; birth distinctness; the major arcs by characters, Mellin separation and sparse frequencies.", "units": [ { "id": "the-marked-type-ii-estimate", "title": "The marked Type II estimate", "reader": "the-marked-type-ii-estimate.html", "source": "src/the-marked-type-ii-estimate.md", "source_sha256": "69cd516ad638407122b29f2a495441c4c4cda9bfd1bb8818b1186c5f2cd1095f", "status": "author_self_checked" }, { "id": "primes-divisors-mean-squares-and-rough-numbers", "title": "Primes, divisors, mean squares and rough numbers", "reader": "primes-divisors-mean-squares-and-rough-numbers.html", "source": "src/primes-divisors-mean-squares-and-rough-numbers.md", "source_sha256": 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"Counting primes by elementary means: Chebyshev and Mertens (course NT-ZETA, lesson NT-ZETA-02, Theorems 3.1 and 4.2)", "The prime number theorem (NT-ZETA-09, Theorem 3.1)", "The Siegel–Walfisz theorem (NT-DIRL-11, Theorem 2.1)", "Mean values of Dirichlet polynomials (NT-ZETA-16, Theorem 2.1)", "Exponential sums: Kusmin–Landau, the second-derivative test and van der Corput's inequality (NT-ZETA-07, Theorems 2.1, 2.2 and 3.1)", "Jensen's formula (NT-ZETA-05, Theorem 1.1 and inequality (4)); Landau's lemma (NT-ZETA-13, Theorem 1.1); the zeta function near s = 1 and the positivity identity (NT-ZETA-08, (1.1) and Lemma 3.1)", "Dirichlet series and Euler products (NT-ZETA-01, Lemma A.2: Cauchy's theorem), used in lesson 6", "Schwartz functions and Fourier inversion (course AN-01, prerequisite page prerequisites/U011-free-foundations/schwartz-fourier-foundations-U017, Section F4), used in lessons 6 and 17", "Lebesgue integration and smoothing, selected programme proofs (course AN-01, prerequisite page prerequisites/U011-free-foundations/banach-foundation-bridges, Sections 15.1 and 15.2: Tonelli's theorem, Cauchy's and Hölder's inequalities for integrals), used in lesson 17", "Measure and Hilbert space tools for Haar integration (course harmonic-analysis-on-locally-compact-groups, Theorem 3.1: Hölder's inequality), used in lessons 4 and 9" ], "status": "author_self_checked", "proof_providers": "providers.json", "provenance": "provenance.json", "licence": "CC0-1.0" }