Riemann–Siegel formula
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1: 25.10 Zeros
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§25.10(ii) Riemann–Siegel Formula
… ►Sign changes of are determined by multiplying (25.9.3) by to obtain the Riemann–Siegel formula: ►
25.10.3
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►Calculations based on the Riemann–Siegel formula reveal that the first ten billion zeros of in the critical strip are on the critical line (van de Lune et al. (1986)).
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2: 25.19 Tables
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Fletcher et al. (1962, §22.1) lists many sources for earlier tables of for both real and complex . §22.133 gives sources for numerical values of coefficients in the Riemann–Siegel formula, §22.15 describes tables of values of , and §22.17 lists tables for some Dirichlet -functions for real characters. For tables of dilogarithms, polylogarithms, and Clausen’s integral see §§22.84–22.858.
3: 25.18 Methods of Computation
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►The principal tools for computing are the expansion (25.2.9) for general values of , and the Riemann–Siegel formula (25.10.3) (extended to higher terms) for .
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4: Bibliography B
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A program for computing the Riemann zeta function for complex argument.
Comput. Phys. Comm. 20 (3), pp. 441–445.
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Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V.
Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.
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and the Riemann Zeros.
In Supersymmetry and Trace Formulae: Chaos and Disorder, I. V. Lerner, J. P. Keating, and D. E. Khmelnitskii (Eds.),
pp. 355–367.
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The Riemann zeros and eigenvalue asymptotics.
SIAM Rev. 41 (2), pp. 236–266.
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The Riemann-Siegel expansion for the zeta function: High orders and remainders.
Proc. Roy. Soc. London Ser. A 450, pp. 439–462.
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