Chebyshev approximations for the Riemann zeta function

Cody, W. J.; Hillstrom, K. E.; Thacher, Henry C.

doi:10.1090/S0025-5718-1971-0295535-9

Chebyshev approximations for the Riemann zeta function
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by W. J. Cody, K. E. Hillstrom and Henry C. Thacher PDF

Math. Comp. 25 (1971), 537-547 Request permission

Abstract:

This paper presents well-conditioned rational Chebyshev approximations, involving at most one exponentiation, for computation of either $\zeta (s)$ or $\zeta (s) - 1,.5 \leqq s \leqq 55$, for up to 20 significant figures. The logarithmic error is required in one case. An algorithm for the Hurwitz zeta function, and an example of nearly double degeneracy are also given.

References

Self-Contained Exponentiation, Proc. FJCC

W. J. Cody, A survey of practical rational and polynomial approximation of functions, SIAM Rev. 12 (1970), 400–423. MR 267725, DOI 10.1137/1012082
Richard F. King and David L. Phillips, The logarithmic error and Newton’s method for the square root, Comm. ACM 12 (1969), 87–88. MR 0285109, DOI 10.1145/362848.362861
D. H. Lehmer, On the maxima and minima of Bernoulli polynomials, Amer. Math. Monthly 47 (1940), 533–538. MR 2378, DOI 10.2307/2303833

BIT

I. F. Ganžela and C. T. Fike, Sterbenz, P. H, Math. Comp. 23 (1969), 313–318. MR 245199, DOI 10.1090/S0025-5718-1969-0245199-6

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