{"id": "card_chain_euler_1737", "kind": "reference", "title": "Euler 1737 — Variae observationes circa series infinitas (E72)", "body": "L. Euler (1737). Variae observationes circa series infinitas (E72). Comm. Acad. Sci. Petrop. 9 (1737, publ. 1744) 160–188. Canonical: https://scholarlycommons.pacific.edu/euler-works/72/. Free copy: https://scholarlycommons.pacific.edu/euler-works/72/. License as found: public domain (the author died more than a century ago, or the work was published before 1930). What it gave the chain: the Euler product over the primes; its logarithm turns the product into a sum, and the sum of 1/p diverges like log log x.", "source": {"label": "L. Euler (1737), Comm. Acad. Sci. Petrop. 9 (1737, publ. 1744) 160–188", "url": "https://scholarlycommons.pacific.edu/euler-works/72/", "domain": "mathematics", "authority_tier": "reference"}, "shelf": "codex", "box": "chain", "bands": ["chain", "record", "1737", "euler", "mathematics"], "subject": "Variae observationes circa series infinitas (E72)", "connections": [{"to_card_id": "card_chain_mercator_1668", "relationship": "builds_on", "evidence": "the Euler product over the primes; its logarithm turns the product into a sum, and the sum of 1/p diverges like log log x"}, {"to_card_id": "card_chain_riemann_1859", "relationship": "enables", "evidence": "zeta continued to the plane, the functional equation, the explicit formula through log zeta, and the hypothesis"}, {"to_card_id": "card_floor_riemann", "relationship": "part_of", "evidence": "a root of this chain - one of the two trees it began from"}, {"to_card_id": "card_chain_dirichlet_1837", "relationship": "enables", "evidence": "L-functions: Euler's product carried into arithmetic progressions"}], "author": "engine", "created_at": 0.0, "updated_at": 0.0, "visibility": "public", "lifecycle_stage": "public", "volatility": "permanent", "surface": "secular", "generated": false, "call": "codex.chain", "facets": {"subject": ["1737", "euler", "mathematics", "record"]}, "presentation": {"glyph": "•", "kind_label": "chain", "by": "L. Euler (1737), Comm. Acad. Sci. Petrop. 9 (1737, publ. 1744) 160–188", "authority": "reference", "posted": "", "standing": "", "link": {"url": "https://scholarlycommons.pacific.edu/euler-works/72/", "host": "scholarlycommons.pacific.edu", "provider": "", "titled": "Euler 1737 — Variae observationes circa series infinitas (E72)", "waybill_line": "", "reach": "", "embed": ""}}, "neighbors": [{"id": "card_chain_mercator_1668", "title": "Mercator 1668 — Logarithmotechnia", "relationship": "builds on", "why": "the Euler product over the primes; its logarithm turns the product into a sum, and the sum of 1/p diverges like log log x", "href": "/card/card_chain_mercator_1668", "resolved": true}, {"id": "card_chain_riemann_1859", "title": "Riemann 1859 — Über die Anzahl der Primzahlen unter einer gegebenen Grösse", "relationship": "enables", "why": "zeta continued to the plane, the functional equation, the explicit formula through log zeta, and the hypothesis", "href": "/card/card_chain_riemann_1859", "resolved": true}, {"id": "card_floor_riemann", "title": "The Riemann chain - from Euler's product and Legendre's count to the critical line", "relationship": "part of", "why": "a root of this chain - one of the two trees it began from", "href": "/card/card_floor_riemann", "resolved": true}, {"id": "card_chain_dirichlet_1837", "title": "Dirichlet 1837 — Beweis des Satzes, dass jede unbegrenzte arithmetische Progression ... unendlich viele Primzahlen enthält", "relationship": "enables", "why": "L-functions: Euler's product carried into arithmetic progressions", "href": "/card/card_chain_dirichlet_1837", "resolved": true}], "house": {"door": "FIND", "kind": "card", "trail": "connections", "seal": "/card/card_chain_euler_1737", "next_step": {"do": "read the source itself", "door": "FIND", "web": "/reader.html?card=card_chain_euler_1737"}, "ends": "a verdict or a card · the trail · a seal · one next step"}}