{"query": "The zeta function", "count": 20, "results": [{"id": "card_chart_zeta", "title": "The zeta function", "shelf": "codex", "surface": "secular", "snippet": "The zeta function. A chart on the one map: this is the tick stick stick_the_zeta_function, placed at the node it charts. Basis: the stick seals zeta(2) = pi^2/6 (Basel) and the functional equation - t", "authority_tier": "engine_derived", "source": "Narrow Highway - a chart: a tick stick placed on the one map", "readable": false, "generated": false}, {"id": "card_n_ebe4c25c22ae", "title": "The Riemann hypothesis — sealed all around, refused at the center", "shelf": "science", "surface": "secular", "snippet": "The deepest open question about the primes, and the cleanest demonstration of the engine's\nhonesty. The FACTS seal: zeta(2)=pi²/6 (Basel), zeta(4)=pi⁴/90, zeta(6)=pi⁶/945, zeta(-1)=-1/12,\nthe trivial ", "authority_tier": "engine_derived", "source": "Concordance assay — 2026-07-09", "readable": false, "generated": false}, {"id": "card_alm_connection_riemann_hypothesis_open_zeta_facts", "title": "Almanac: The Riemann Hypothesis -- an open question about where the zeta function vanishes, with the established facts sealed and the conjecture marked honestly", "shelf": "almanac", "surface": "secular", "snippet": "SITUATION\n  The Riemann zeta function, zeta(s) = the sum of 1/k^s, is the deepest known bridge between the smooth world of analysis and the prime numbers. Several of its truths are firmly ESTABLISHED ", "authority_tier": "reference", "source": "The Almanac — verified-only practical wisdom (sealed)", "readable": false, "generated": false}, {"id": "card_src_mill_odlyzko_1987", "title": "A. M. Odlyzko 1987 — On the distribution of spacings between zeros of the zeta function", "shelf": "millennium", "surface": "secular", "snippet": "A. M. Odlyzko (1987). On the distribution of spacings between zeros of the zeta function. Math. Comp. 48 (1987) 273–308. DOI 10.1090/S0025-5718-1987-0866115-0. Canonical: https://doi.org/10.1090/S0025", "authority_tier": "reference", "source": "A. M. Odlyzko (1987), Math. Comp. 48 (1987) 273–308", "readable": false, "generated": false}, {"id": "card_chain_turing_1953", "title": "Turing 1953 — Some calculations of the Riemann zeta-function", "shelf": "codex", "surface": "secular", "snippet": "A. M. Turing (1953). Some calculations of the Riemann zeta-function. Proc. London Math. Soc. (3) 3 (1953) 99–117. DOI 10.1112/plms/s3-3.1.99. Canonical: https://doi.org/10.1112/plms/s3-3.1.99. Cited b", "authority_tier": "reference", "source": "A. M. Turing (1953), Proc. London Math. Soc. (3) 3 (1953) 99–117", "readable": false, "generated": false}, {"id": "card_chain_conrey_1989", "title": "Conrey 1989 — More than two fifths of the zeros of the Riemann zeta function are on the critical line", "shelf": "codex", "surface": "secular", "snippet": "J. B. Conrey (1989). More than two fifths of the zeros of the Riemann zeta function are on the critical line. J. reine angew. Math. 399 (1989) 1–26. DOI 10.1515/crll.1989.399.1. Canonical: https://doi", "authority_tier": "reference", "source": "J. B. Conrey (1989), J. reine angew. Math. 399 (1989) 1–26", "readable": false, "generated": false}, {"id": "card_alm_connection_zeta_even_powers_are_pi", "title": "Almanac: Inverse even powers fold into pi; inverse odd powers escape into mystery -- the two faces of the zeta function", "shelf": "almanac", "surface": "secular", "snippet": "SITUATION\n  Sum the inverse powers of the whole numbers and a startling split appears. For every EVEN power, the sum is a rational number times a power of pi: zeta(2) = 1 + 1/4 + 1/9 + ... = pi^2/6 (t", "authority_tier": "reference", "source": "The Almanac — verified-only practical wisdom (sealed)", "readable": false, "generated": false}, {"id": "card_src_mill_odlyzko_zeros_tables", "title": "A. M. Odlyzko 1988 — Tables of zeros of the Riemann zeta function", "shelf": "millennium", "surface": "secular", "snippet": "A. M. Odlyzko (1988-2001). Tables of zeros of the Riemann zeta function. University of Minnesota (zeros1: the first 100,000 zeros, accurate to 3e-9; zeros2: the first 100 to over 1000 places; zeros3/4", "authority_tier": "reference", "source": "A. M. Odlyzko (1988-2001), University of Minnesota (zeros1: the first 100,000 zeros, accurate to 3e-9; zeros2: the first 100 to over 1000 places; z", "readable": false, "generated": false}, {"id": "card_src_mill_montgomery_1973", "title": "H. L. Montgomery 1973 — The pair correlation of zeros of the zeta function", "shelf": "millennium", "surface": "secular", "snippet": "H. L. Montgomery (1973). The pair correlation of zeros of the zeta function. Proc. Symp. Pure Math. 24 (AMS, 1973) 181–193. DOI 10.1090/pspum/024/9944. Canonical: https://doi.org/10.1090/pspum/024/994", "authority_tier": "reference", "source": "H. L. Montgomery (1973), Proc. Symp. Pure Math. 24 (AMS, 1973) 181–193", "readable": false, "generated": false}, {"id": "card_chain_selberg_1942", "title": "Selberg 1942 — On the zeros of Riemann's zeta-function", "shelf": "codex", "surface": "secular", "snippet": "A. Selberg (1942). On the zeros of Riemann's zeta-function. Skr. Norske Vid.-Akad. Oslo I (1942) no. 10, 1–59. Cited by its record. License as found: cited by its record. What it gave the chain: a pos", "authority_tier": "reference", "source": "A. Selberg (1942), Skr. Norske Vid.-Akad. Oslo I (1942) no. 10, 1–59", "readable": false, "generated": false}, {"id": "card_chain_levinson_1974", "title": "Levinson 1974 — More than one third of zeros of Riemann's zeta-function are on sigma = 1/2", "shelf": "codex", "surface": "secular", "snippet": "N. Levinson (1974). More than one third of zeros of Riemann's zeta-function are on sigma = 1/2. Adv. Math. 13 (1974) 383–436. DOI 10.1016/0001-8708(74)90074-7. Canonical: https://doi.org/10.1016/0001-", "authority_tier": "reference", "source": "N. Levinson (1974), Adv. Math. 13 (1974) 383–436", "readable": false, "generated": false}, {"id": "card_src_mill_odlyzko_schonhage_1988", "title": "A. M. Odlyzko 1988 — Fast algorithms for multiple evaluations of the Riemann zeta function", "shelf": "millennium", "surface": "secular", "snippet": "A. M. Odlyzko, A. Schönhage (1988). Fast algorithms for multiple evaluations of the Riemann zeta function. Trans. Amer. Math. Soc. 309 (1988) 797–809. DOI 10.1090/S0002-9947-1988-0961614-2. Canonical:", "authority_tier": "reference", "source": "A. M. Odlyzko, A. Schönhage (1988), Trans. Amer. Math. Soc. 309 (1988) 797–809", "readable": false, "generated": false}, {"id": "card_floor_riemann", "title": "The Riemann chain - from Euler's product and Legendre's count to the critical line", "shelf": "codex", "surface": "secular", "snippet": "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,", "authority_tier": "engine_derived", "source": "Narrow Highway - a chain on the one map (operator seed)", "readable": false, "generated": false}, {"id": "card_joint_weil_conjectures", "title": "The Weil conjectures (Deligne 1974): the Riemann hypothesis over finite fields, a theorem", "shelf": "codex", "surface": "secular", "snippet": "The Weil conjectures are the analogue of the Riemann hypothesis for the zeta functions of varieties over FINITE FIELDS, and they are a theorem (Deligne 1974): every eigenvalue of Frobenius on the i-th", "authority_tier": "engine_derived", "source": "Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)", "readable": false, "generated": false}, {"id": "card_floor_millennium", "title": "The Millennium floor - seven open questions, and where they connect", "shelf": "codex", "surface": "secular", "snippet": "The seven Millennium Prize Problems on the one map of reality. The FLOOR is what is proven or observed: the joints where two of the questions meet at one established thing - the GUE statistics shared ", "authority_tier": "engine_derived", "source": "Narrow Highway - the Millennium floor (operator seed)", "readable": false, "generated": false}, {"id": "card_n_ea9e3f380756", "title": "The primes and the nucleus — one statistical fingerprint", "shelf": "science", "surface": "secular", "snippet": "The night's deepest rhyme, and it is real, published, and unexplained. Montgomery (1973)\nand Dyson noticed that the SPACINGS between the Riemann zeta zeros follow the same distribution\nas the eigenval", "authority_tier": "engine_derived", "source": "Concordance assay — 2026-07-09", "readable": false, "generated": false}, {"id": "card_floor_logarithm", "title": "The logarithm - the instrument the joints share", "shelf": "codex", "surface": "secular", "snippet": "Two trees: Napier's table (1614) and Saint-Vincent's hyperbola (1647) - a number's logarithm as a tabulated count, and as an area. They become one function in Euler's Introductio (1748): e, and exp an", "authority_tier": "engine_derived", "source": "Narrow Highway - a chain on the one map (operator seed)", "readable": false, "generated": false}, {"id": "card_joint_gue_statistics", "title": "Zeta zeros and the lattice Dirac operator share GUE statistics", "shelf": "codex", "surface": "secular", "snippet": "The nearest-neighbour spacings of the zeros of zeta follow the Gaussian Unitary Ensemble of random matrix theory (Montgomery's pair correlation, 1973; Odlyzko's computation, 1987), and so does the low", "authority_tier": "engine_derived", "source": "Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)", "readable": false, "generated": false}, {"id": "card_joint_l_functions", "title": "One machinery: Euler product, functional equation, critical line", "shelf": "codex", "surface": "secular", "snippet": "Zeta is the simplest L-function; L(E,s) of an elliptic curve is another, given its analytic continuation by modularity (Wiles 1995). Both have an Euler product, a functional equation about a centre, a", "authority_tier": "engine_derived", "source": "Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)", "readable": false, "generated": false}, {"id": "card_chain_riemann_1859", "title": "Riemann 1859 — Über die Anzahl der Primzahlen unter einer gegebenen Grösse", "shelf": "codex", "surface": "secular", "snippet": "B. Riemann (1859). Über die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsber. Berliner Akad. (1859) 671–680. Canonical: https://www.claymath.org/collections/riemanns-1859-manuscript/. Fre", "authority_tier": "reference", "source": "B. Riemann (1859), Monatsber. Berliner Akad. (1859) 671–680", "readable": false, "generated": false}], "house": {"door": "FIND", "kind": "cards", "trail": "results", "seal": null, "next_step": {"do": "open the top card", "door": "FIND", "tool": "card_get", "params": {"id": "card_chart_zeta"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}