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25 Zeta and Related FunctionsComputation

§25.20 Approximations

  • Cody et al. (1971) gives rational approximations for ζ(s) in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are 0.5s5, 5s11, 11s25, 25s55. Precision is varied, with a maximum of 20S.

  • Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of sζ(s+1) and ζ(s+k), k=2,3,4,5,8, for 0s1 (23D).

  • Luke (1969b, p. 306) gives coefficients in Chebyshev-series expansions that cover ζ(s) for 0s1 (15D), ζ(s+1) for 0s1 (20D), and lnξ(12+ix)25.4) for -1x1 (20D). For errata see Piessens and Branders (1972).

  • Morris (1979) gives rational approximations for Li2(x)25.12(i)) for 0.5x1. Precision is varied with a maximum of 24S.

  • Antia (1993) gives minimax rational approximations for Γ(s+1)Fs(x), where Fs(x) is the Fermi–Dirac integral (25.12.14), for the intervals -<x2 and 2x<, with s=-12,12,32,52. For each s there are three sets of approximations, with relative maximum errors 10-4,10-8,10-12.