  In the present paper, we construct an algorithm for the evaluation of real
Riemann zeta function $\zeta(s)$ for all real $s$, $s>1$, in polynomial time
and linear space on Turing machines in Ko-Friedman model. The algorithms is
based on a series expansion of real Riemann zeta function $\zeta(s)$ (the
series globally convergents) and uses algorithms for the evaluation of real
function $(1+x)^h$ and hypergeometric series in polynomial time and linear
space.
  The algorithm from the present paper modified in an obvious way to work with
the complex numbers can be used to evaluate complex Riemann zeta function
$\zeta(s)$ for $s=\sigma+\mathbf{i}t$, $\sigma\ne 1$ (so, also for the case of
$\sigma<1$), in polynomial time and linear space in $n$ wherein $2^{-n}$ is a
precision of the computation; the modified algorithm will be also polynomial
time and linear space in $\lceil \log_2(t)\rceil$ and exponential time and
exponential space in $\lceil \log_2(\sigma)\rceil$.
