FP//LINSPACE computability of Riemann zeta function in Ko-Friedman model

Abstract

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 $\lceil \log_2(\sigma)\rceil$.