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21: 25.15 Dirichlet -functions
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►This result plays an important role in the proof of Dirichlet’s theorem on primes in arithmetic progressions (§27.11).
Related results are:
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22: 15.19 Methods of Computation
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►The accuracy is controlled and validated by a running error analysis coupled with interval arithmetic.
23: 33.23 Methods of Computation
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►Use of extended-precision arithmetic increases the radial range that yields accurate results, but eventually other methods must be employed, for example, the asymptotic expansions of §§33.11 and 33.21.
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24: Bibliography O
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Error Analysis of Complex Arithmetic.
In Computational Aspects of Complex Analysis (Braunlage, 1982), H. Werner, L. Wuytack, E. Ng, and H. J. Bünger (Eds.),
NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., Vol. 102, pp. 279–292.
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Numerical Computing with IEEE Floating Point Arithmetic.
Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
25: Bibliography S
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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean.
SIAM J. Math. Anal. 20 (6), pp. 1514–1528.
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A Course in Arithmetic.
Graduate Texts in Mathematics, Vol. 7, Springer-Verlag, New York.
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Staudt and arithmetical properties of Bernoulli numbers.
Historia Sci. (2) 5 (1), pp. 69–74.
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Algorithm 786: Multiple-precision complex arithmetic and functions.
ACM Trans. Math. Software 24 (4), pp. 359–367.
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Algorithm 693: A FORTRAN package for floating-point multiple-precision arithmetic.
ACM Trans. Math. Software 17 (2), pp. 273–283.
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26: Bibliography
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Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, , and the Ladies Diary.
Amer. Math. Monthly 95 (7), pp. 585–608.
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The MasPar MP-1 as a computer arithmetic laboratory.
J. Res. Nat. Inst. Stand. Technol. 101 (2), pp. 165–174.
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Arb: A C Library for Arbitrary Precision Ball Arithmetic.
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27: 27.4 Euler Products and Dirichlet Series
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►The fundamental theorem of arithmetic is linked to analysis through the concept of the Euler product.
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28: 3.10 Continued Fractions
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►The quotient-difference algorithm is frequently unstable and may require high-precision arithmetic or exact arithmetic.
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29: Bibliography L
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Algorithm 567: Extended-range arithmetic and normalized Legendre polynomials [A1], [C1].
ACM Trans. Math. Software 7 (1), pp. 141–146.
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An underflow-induced graphics failure solved by SLI arithmetic.
In IEEE Symposium on Computer Arithmetic, E. E. Swartzlander, M. J. Irwin, and G. A. Jullien (Eds.),
Washington, D.C., pp. 10–17.
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