{"id": "card_floor_riemann", "kind": "note", "title": "The Riemann chain - from Euler's product and Legendre's count to the critical line", "body": "Two trees. The analytic: Euler's product over the primes (1737), Dirichlet's L-functions (1837). The arithmetic: Legendre's guess at the prime count (1798), Gauss's logarithmic integral (counted 1792, written 1849), Chebyshev's bounds (1852). They become one in Riemann's eight pages of 1859: zeta on the whole plane, the explicit formula, the hypothesis. Then the prime number theorem (Hadamard, de la Vallée Poussin, 1896), the zero count (von Mangoldt 1905), zeros on the line - infinitely many (Hardy 1914), a positive proportion (Selberg 1942), a third (Levinson 1974), two fifths (Conrey 1989) - and the computations: Turing's method (1953), Odlyzko's tables (1987), Platt and Trudgian to 3·10^12 (2021). The open end stands on the last links.", "source": {"label": "Narrow Highway - a chain on the one map (operator seed)", "url": "", "ref": "docs/MILLENNIUM_PREPAREDNESS.md", "authority_tier": "engine_derived"}, "shelf": "codex", "box": "floor", "bands": ["floor", "chain", "two trees", "riemann", "primes", "zeta", "prime number theorem"], "subject": "The Riemann chain - from Euler's product and Legendre's count to the critical line", "connections": [{"to_card_id": "card_k_floor_of_discovery", "relationship": "part_of", "evidence": "the riemann chain rests on the one Floor of Discovery"}, {"to_card_id": "card_chain_euler_1737", "relationship": "has_part", "evidence": "a root of this chain - one of the two trees it began from"}, {"to_card_id": "card_chain_legendre_1798", "relationship": "has_part", "evidence": "a root of this chain - one of the two trees it began from"}, {"to_card_id": "card_question_riemann", "relationship": "has_open_end", "evidence": "the open question this chain reaches"}], "author": "engine", "created_at": 0.0, "updated_at": 0.0, "visibility": "public", "lifecycle_stage": "public", "volatility": "permanent", "surface": "secular", "generated": false, "call": "codex.floor", "facets": {"subject": ["chain", "prime number theorem", "primes", "riemann", "two trees", "zeta"]}, "presentation": {"glyph": "•", "kind_label": "floor", "by": "Narrow Highway - a chain on the one map (operator seed)", "authority": "engine_derived", "posted": "", "standing": ""}, "neighbors": [{"id": "card_k_floor_of_discovery", "title": "The Floor of Discovery — one floor, and by its design the fear of God", "relationship": "part of", "why": "the riemann chain rests on the one Floor of Discovery", "href": "/card/card_k_floor_of_discovery", "resolved": true}, {"id": "card_chain_euler_1737", "title": "Euler 1737 — Variae observationes circa series infinitas (E72)", "relationship": "has part", "why": "a root of this chain - one of the two trees it began from", "href": "/card/card_chain_euler_1737", "resolved": true}, {"id": "card_chain_legendre_1798", "title": "Legendre 1798 — Essai sur la théorie des nombres", "relationship": "has part", "why": "a root of this chain - one of the two trees it began from", "href": "/card/card_chain_legendre_1798", "resolved": true}, {"id": "card_question_riemann", "title": "The Riemann hypothesis", "relationship": "has open end", "why": "the open question this chain reaches", "href": "/card/card_question_riemann", "resolved": true}], "house": {"door": "FIND", "kind": "card", "trail": "connections", "seal": "/card/card_floor_riemann", "next_step": {"do": "follow a connection: what this card rests on, and what rests on it", "door": "FIND", "tool": "card_connections", "params": {"id": "card_floor_riemann"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}