An amortized-complexity method to compute the Riemann zeta function

TL;DR

The paper introduces a practical, elementary method for computing the Riemann zeta function at many points within a specific interval, achieving near-optimal complexity with advantages in simplicity and implementation.

Contribution

It presents a new amortized-complexity algorithm for computing the Riemann zeta function efficiently over short intervals, avoiding complex tools like FFT.

Findings

Method achieves $T^{1/4+o(1)}$ complexity per point.

Implementation results match theoretical complexity predictions.

Method is simple, does not require large storage or FFT.

Abstract

A practical method to compute the Riemann zeta function is presented. The method can compute $\zeta(1/2+it)$ at any $\lfloor T^{1/4} \rfloor$ points in $[T,T+T^{1/4}]$ using an average time of $T^{1/4+o(1)}$ per point. This is the same complexity as the Odlyzko-Sch\"onhage algorithm over that interval. Although the method far from competes with the Odlyzko-Sch\"onhage algorithm over intervals much longer than $T^{1/4}$, it still has the advantages of being elementary, simple to implement, it does not use the fast Fourier transform or require large amounts of storage space, and its error terms are easy to control. The method has been implemented, and results of timing experiments agree with its theoretical amortized complexity of $T^{1/4+o(1)}$.