The author presents a new class of remarkable algorithms for the computation of the Riemann zeta-function \(\zeta(s)\) to arbitrary precision in arbitrary domains. These algorithm do not compete with the Riemann-Siegel formula but they do improve the standard algorithms based on the Euler-Maclaurin summation formula. The author proves: let \(p_n(x)=\sum_{k=0}^na_kx^k\) be an arbitrary polynomial of degree \(n\) with \(p_n(-1)\neq 0\), then \[ \begin{split} \zeta(s)={-1\over (1-2^{1-s})p_n(-1)}\sum_{j=0}^{n-1}{(-1)^j\left( \sum_{k=0}^j(-1)^ka_k-p_n(-1)\right)\over (1+j)^s}+\\ {1\over p_n(-1)(1-2^{1-s})\Gamma(s)}\int_0^1{p_n(x)|\log x|^{s-1}\over 1+x} dx.\end{split} \] The idea now is to choose \(p_n\) so that the error, i.e. the second term on the right hand side, is as small as possible.
For the entire collection see [Zbl 0954.00025].