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

TL;DR

This paper presents a polynomial-time, linear-space algorithm for evaluating the real Riemann zeta function for s>1 within the Ko-Friedman model, extending to complex arguments with similar efficiency.

Contribution

It introduces a novel algorithm based on series expansions that computes the Riemann zeta function efficiently in polynomial time and linear space, applicable to both real and complex inputs.

Findings

Algorithm computes real zeta function in polynomial time and linear space.

Extension to complex zeta function maintains polynomial time and space for certain parameters.

Algorithm's efficiency is based on series expansions and hypergeometric series evaluation.

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