Chebyshev approximations for the Riemann zeta function

Authors:
W. J. Cody, K. E. Hillstrom and Henry C. Thacher

Journal:
Math. Comp. **25** (1971), 537-547

MSC:
Primary 65D20

DOI:
https://doi.org/10.1090/S0025-5718-1971-0295535-9

MathSciNet review:
0295535

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper presents well-conditioned rational Chebyshev approximations, involving at most one exponentiation, for computation of either or , 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.

**[1]**N. W. Clark & W. J. Cody,*Self-Contained Exponentiation, Proc. FJCC*, AFIPS Press, Montvale, N. J., 1969, pp. 701-706.**[2]**W. J. Cody, ``A survey of practical rational and polynomial approximation of functions,''*SIAM Rev.*, v. 12, 1970, pp. 400-423. MR**0267725 (42:2627)****[3]**R. F. King & D. L. Phillips, ``The logarithmic error and Newton's method for the square root,''*Comm. ACM*, v. 12, 1969, pp. 87-88. MR**0285109 (44:2333)****[4]**D. H. Lehmer, ``On the maxima and minima of Bernoulli polynomials,''*Amer. Math. Monthly*, v. 47, 1940, pp. 533-538. MR**2**, 43. MR**0002378 (2:43a)****[5]**B. Markman, ``The Riemann zeta function,''*BIT*, v. 5, 1965, pp. 138-141.**[6]**P. H. Sterbenz & C. T. Fike, ``Optimal starting approximations for Newton's method,''*Math. Comp.*, v. 23, 1969, pp. 313-318. MR**39**#6511. MR**0245199 (39:6511)****[7]**H. C. Thacher, Jr., ``On expansions of the Riemann zeta function,''*Math. Comp.*(To appear.)

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1971-0295535-9

Keywords:
Rational Chebyshev approximations,
Riemann zeta function,
Hurwitz zeta function,
logarithmic error,
near degeneracy

Article copyright:
© Copyright 1971
American Mathematical Society