Chebyshev approximations for the Riemann zeta function

Abstract

This paper presents well-conditioned rational Chebyshev approximations, involving at most one exponentiation, for computation of either ζ ( s ) \zeta (s) or ζ ( s ) − 1 , .5 ≦ s ≦ 55 \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.