{"id": "card_question_riemann", "kind": "reference", "title": "The Riemann hypothesis", "body": "The Riemann hypothesis. Every non-trivial zero of the Riemann zeta function has real part one half. Its chart is the tick stick stick_riemann_hypothesis: what is sealed, what is cited and what stays open are read live at /stick?id=stick_riemann_hypothesis. Open.", "source": {"label": "Clay Mathematics Institute, the Millennium Prize Problems (2000)", "url": "https://www.claymath.org/millennium-problems/", "ref": "stick_riemann_hypothesis", "domain": "mathematics", "authority_tier": "reference"}, "shelf": "millennium", "box": "question", "bands": ["millennium", "open question", "clay", "riemann", "zeta", "primes", "number theory"], "subject": "The Riemann hypothesis", "connections": [{"to_card_id": "card_spine_millennium_sources", "relationship": "member_of", "evidence": "one of the seven questions the Millennium shelf is about"}, {"to_card_id": "card_floor_millennium", "relationship": "open_end_of", "evidence": "an open question hanging off the floor of what is proven"}, {"to_card_id": "card_joint_gue_statistics", "relationship": "connects_at", "evidence": "Zeta zeros and the lattice Dirac operator share GUE statistics. montgomery 1973; odlyzko 1987; verbaarschot 1994; berry keating 1999. Sealed: the Riemann stick's spacing witness (seal deefbb6d)"}, {"to_card_id": "card_joint_l_functions", "relationship": "connects_at", "evidence": "One machinery: Euler product, functional equation, critical line. wiles 1995. Sealed: xi_functional_equation_at_s_0_3 (Riemann stick) and the 11a1 instance (BSD stick)"}, {"to_card_id": "card_joint_weil_conjectures", "relationship": "connects_at", "evidence": "The Weil conjectures (Deligne 1974): the Riemann hypothesis over finite fields, a theorem. deligne 1974; mulmuley 2011. Cited, not sealed: no arithmetic to recompute"}, {"to_card_id": "card_joint_diophantine_rh", "relationship": "connects_at", "evidence": "The Riemann hypothesis is one Diophantine equation with no solutions. davis matiyasevich robinson 1976. Cited, not sealed: no arithmetic to recompute"}, {"to_card_id": "card_floor_logarithm", "relationship": "open_end_of", "evidence": "the open question this chain reaches"}, {"to_card_id": "card_chain_riemann_1859", "relationship": "connects_at", "evidence": "where the logarithm enters: the explicit formula runs through log zeta - the product over the primes turned into a sum (sealed on the stick: Li's coefficients from log xi)"}, {"to_card_id": "card_chain_von_mangoldt_1905", "relationship": "connects_at", "evidence": "where the logarithm enters: the zero count is (T/2pi) log(T/2pi) - sealed on the stick at the 100,000th zero"}, {"to_card_id": "card_chain_conrey_1989", "relationship": "builds_on", "evidence": "a later work standing on an earlier one"}, {"to_card_id": "card_src_mill_platt_trudgian_2021", "relationship": "builds_on", "evidence": "a later work standing on an earlier one"}, {"to_card_id": "card_floor_riemann", "relationship": "open_end_of", "evidence": "the open question this chain reaches"}, {"to_card_id": "card_chart_de_bruijn_newman", "relationship": "charted_by", "evidence": "the stick seals the window 0 <= Lambda <= 0.22; RH is Lambda = 0 exactly - an equivalence the Riemann question already carries"}, {"to_card_id": "card_chart_zeta", "relationship": "charted_by", "evidence": "the stick seals zeta(2) = pi^2/6 (Basel) and the functional equation - the object the Riemann question is about"}, {"to_card_id": "card_chart_prime_waveforms", "relationship": "charted_by", "evidence": "the stick seals the explicit formula psi(x) = x - sum over zeros x^rho/rho - the zeros as the frequencies of the prime staircase, the heart of the Riemann question"}, {"to_card_id": "card_instr_number_theory", "relationship": "connects_at", "evidence": "instruments the map: number_theory checks the zero count, Robin, Schoenfeld, Lagarias, Nicolas and Li criteria - the Riemann stick's witnesses are its verdicts, sealed"}, {"to_card_id": "card_instr_riemann_accel", "relationship": "connects_at", "evidence": "instruments the map: the Riemann-Siegel acceleration is how the zeros are swept fast enough to count to height 10^7"}], "author": "engine", "created_at": 0.0, "updated_at": 0.0, "visibility": "public", "lifecycle_stage": "public", "volatility": "permanent", "surface": "secular", "generated": false, "call": "millennium.question", "facets": {"subject": ["clay", "millennium", "number theory", "open question", "primes", "riemann", "zeta"]}, "presentation": {"glyph": "?", "kind_label": "A question", "by": "Clay Mathematics Institute, the Millennium Prize Problems (2000)", "authority": "reference", "posted": "", "standing": "", "link": {"url": "https://www.claymath.org/millennium-problems/", "host": "claymath.org", "provider": "", "titled": "The Riemann hypothesis", "waybill_line": "", "reach": "", "embed": ""}}, "neighbors": [{"id": "card_spine_millennium_sources", "title": "The Millennium sources — the papers behind the seven sticks", "relationship": "on the shelf of", "why": "one of the seven questions the Millennium shelf is about", "href": "/card/card_spine_millennium_sources", "resolved": true}, {"id": "card_floor_millennium", "title": "The Millennium floor - seven open questions, and where they connect", "relationship": "open end of", "why": "an open question hanging off the floor of what is proven", "href": "/card/card_floor_millennium", "resolved": true}, {"id": "card_joint_gue_statistics", "title": "Zeta zeros and the lattice Dirac operator share GUE statistics", "relationship": "connects at", "why": "Zeta zeros and the lattice Dirac operator share GUE statistics. montgomery 1973; odlyzko 1987; verbaarschot 1994; berry keating 1999. 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