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1: 25.10 Zeros
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25.10.1
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25.10.2
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►More than 41% of all the zeros in the critical strip lie on the critical line (Bui et al. (2011)).
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2: Bibliography B
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Rogers-Ramanujan identities in the hard hexagon model.
J. Statist. Phys. 26 (3), pp. 427–452.
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Rogers-Ramanujan Identities: A Century of Progress from Mathematics to Physics.
In Proceedings of the International Congress of Mathematicians,
Vol. III (Berlin, 1998),
pp. 163–172.
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Phase-space projection identities for diffraction catastrophes.
J. Phys. A 13 (1), pp. 149–160.
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A cubic counterpart of Jacobi’s identity and the AGM.
Trans. Amer. Math. Soc. 323 (2), pp. 691–701.
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More than 41% of the zeros of the zeta function are on the critical line.
Acta Arith. 150 (1), pp. 35–64.
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3: 24.10 Arithmetic Properties
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►where .
…valid when and , where is a fixed integer.
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24.10.8
►valid for fixed integers , and for all such that
and .
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24.10.9
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4: 26.21 Tables
§26.21 Tables
►Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients for up to 50 and up to 25; extends Table 26.4.1 to ; tabulates Stirling numbers of the first and second kinds, and , for up to 25 and up to ; tabulates partitions and partitions into distinct parts for up to 500. ►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts , partitions into parts , and unrestricted plane partitions up to 100. It also contains a table of Gaussian polynomials up to . ►Goldberg et al. (1976) contains tables of binomial coefficients to and Stirling numbers to .5: 27.16 Cryptography
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►Thus, and .
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►By the Euler–Fermat theorem (27.2.8), ; hence .
But , so is the same as modulo .
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6: 36.9 Integral Identities
7: 22.9 Cyclic Identities
§22.9 Cyclic Identities
… ►§22.9(ii) Typical Identities of Rank 2
… ► ►§22.9(iii) Typical Identities of Rank 3
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§24.5(ii) Other Identities
… ►§24.5(iii) Inversion Formulas
►In each of (24.5.9) and (24.5.10) the first identity implies the second one and vice-versa. …9: 15.17 Mathematical Applications
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§15.17(iv) Combinatorics
►In combinatorics, hypergeometric identities classify single sums of products of binomial coefficients. …10: 27.15 Chinese Remainder Theorem
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►The Chinese remainder theorem states that a system of congruences , always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod ), where is the product of the moduli.
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