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A zero-free strip Re s > 1 - 10^-6 for the Hecke L-functions of every cyclotomic field containing the twelfth roots of unity, from cubic Gauss sums: S-integers, sextic reciprocity, Kummer characters, the cubic theta function of Kazhdan and Patterson, a Weyl reflection, a quadratic large sieve, a Poisson probe and its Euler product, zero detection; uniform complete splitting in Kummer fields; primes p = crQ + 1 from a block sieve, Bombieri-Vinogradov, marked bilinear estimates and Buchstab's densities.", "units": [ { "id": "artins-conjecture-and-the-index-of-a-prime", "title": "Artin's conjecture and the index of a prime", "reader": "artins-conjecture-and-the-index-of-a-prime.html", "source": "src/artins-conjecture-and-the-index-of-a-prime.md", "source_sha256": "5e882ae1e379499cacbf4afd644c2813f17cdfaf9e078569a2d7f96725a7a95f", "status": "author_self_checked" }, { "id": "s-integers-and-sixth-power-classes", "title": "S-integers and sixth-power classes", "reader": 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zero-free strip", "reader": "the-zero-free-strip.html", "source": "src/the-zero-free-strip.md", "source_sha256": "18c5618bc2b4b5aba2ad435ab16910e62dab232574cf1c5807ecac7a2cd1fd55", "status": "author_self_checked" }, { "id": "uniform-complete-splitting-in-kummer-fields", "title": "Uniform complete splitting in Kummer fields", "reader": "uniform-complete-splitting-in-kummer-fields.html", "source": "src/uniform-complete-splitting-in-kummer-fields.md", "source_sha256": "b5340da0719b596a5f286155693db30dc6e2da299efd512d9abe1177bb55de5b", "status": "author_self_checked" }, { "id": "a-block-sieve-and-rough-number-densities", "title": "A block sieve and rough-number densities", "reader": "a-block-sieve-and-rough-number-densities.html", "source": "src/a-block-sieve-and-rough-number-densities.md", "source_sha256": "339751e5de8bab405a15358c0bff5f4f2f210358924edb09cb263e353349bf94", "status": "author_self_checked" }, { "id": "marked-type-ii-estimates", "title": "Marked Type II estimates and rough factors", "reader": "marked-type-ii-estimates.html", "source": "src/marked-type-ii-estimates.md", "source_sha256": "b073bd5fb97bd0c5efa97b7de97449df6d9b649adce45813d8dfa6d1abdf90c4", "status": "author_self_checked" }, { "id": "weights-mass-and-distribution-in-a-progression", "title": "Weights, mass and distribution in a progression", "reader": "weights-mass-and-distribution-in-a-progression.html", "source": "src/weights-mass-and-distribution-in-a-progression.md", "source_sha256": "522b567b7733b5feef047e9e4077c69aefe488e86d970873ee0f281b1db85c27", "status": "author_self_checked" }, { "id": "primes-with-controlled-predecessors", "title": "Primes with controlled predecessors", "reader": "primes-with-controlled-predecessors.html", "source": "src/primes-with-controlled-predecessors.md", "source_sha256": "7f71f8bc79d0874e85dddf0cfc5887f7c752fb5f2211d37b7edc3b7b7668102c", "status": "author_self_checked" } ], "prerequisites": [ "Mertens' estimates, the Gamma function, entire functions of order one, convexity in the critical strip, the prime number theorem, Hadamard's three circles theorem (course The Riemann zeta function, NT-ZETA-02, NT-ZETA-03, NT-ZETA-05, NT-ZETA-06, NT-ZETA-09, NT-ZETA-18)", "the Siegel-Walfisz and Bombieri-Vinogradov theorems (course Dirichlet L-functions and primes in progressions, NT-DIRL-11, NT-DIRL-15)", "explicit formulas for L-functions (course Explicit formulas and positivity, NT-EXPF-02)", "adeles, ideles, Poisson summation on the adeles, Hecke characters, Tate's thesis and Hecke L-functions (course Adeles, ideles and Tate's thesis, NT-ADL-02 to NT-ADL-10)", "units, discriminants and the different, ramification, cyclotomic fields, counting ideals of bounded norm (course Number fields)", "Kummer theory, Hilbert symbols, power residue symbols and reciprocity laws, the global reciprocity law, ring class fields, Artin L-functions (course Class field theory)", "Hensel's lemma, places in extensions and the product formula, the multiplicative group of a local field (course Local fields, NT-LOC-03, NT-LOC-05, NT-LOC-11)", "Dirichlet series and Euler products (NT-ZETA-01, Lemmas A.1-A.3: dominated convergence, Cauchy's theorem and estimates, isolated zeros and the residue formula), used in lessons 5, 7, 13 and 14", "Schwartz functions and Fourier inversion (course AN-01, prerequisite page prerequisites/U011-free-foundations/schwartz-fourier-foundations-U017, Sections F4 and F5), used in lessons 5, 7, 10, 11, 13 and 18", "Lebesgue integration and smoothing, selected programme proofs (course AN-01, prerequisite page prerequisites/U011-free-foundations/banach-foundation-bridges, Section 15.1: Tonelli's and Fubini's theorems), used in lesson 7", "Additive characters, self-dual measures and Poisson summation on the adeles (NT-ADL-05, Proposition 5.2: Fourier inversion at the finite places), used in lesson 7", "Bernoulli series and Poisson summation (course AN-01, reader page AN01-U056, Lemma 0.1: Fourier series of smooth periodic functions), used in lesson 11" ], "status": "author_self_checked", "proof_providers": "providers.json", "provenance": "provenance.json", "licence": "CC0-1.0" }