Simple and accurate approximations to the Riemann zeta function

TL;DR

This paper develops simple, high-precision approximations to the Riemann zeta function $\zeta(s)$ in vertical strips, including the critical strip, by augmenting the Riemann-Siegel main sum with a tractable remainder term. The approximations are organized as $\zeta_p(s)=F(s;N_t,p)$ with $N_t=\left\lfloor \sqrt{t/(2\pi)}\right\rfloor$, and rely on precomputed coefficients $\{\omega_{p,j}\}$ and $\{\lambda_{p,j}\}$ to form $\mathcal{I}_{M,p}(s)$; these coefficients are obtained via a Gaussian-quadrature discretization of a Mordell-type integral, yielding an explicit algorithm for computing $\omega_{p,j}$ and $\lambda_{p,j}$. The authors demonstrate through extensive numerics that the approximation error diminishes rapidly as $p$ grows, providing extremely accurate results for large $t$ and across $0\le\sigma\le 1$, including accurate approximations to $\zeta'(s)$. They also discuss double vs quadruple precision behavior, provide implementation details (including MATLAB code and downloadable coefficient tables up to $p=150$), and show how the method scales to high-precision computations valuable for zeros and related analytic-number-theory tasks.

Abstract

We develop approximations for the Riemann zeta function that enable high-precision computation within the critical strip and other vertical strips. These approximations combine the main sum of the Riemann-Siegel formula with a simple approximation of the remainder term, which involves only elementary functions and certain precomputed coefficients obtained via Gaussian quadrature. Additionally, we provide approximations for the derivative of the Riemann zeta function and present extensive numerical evidence demonstrating the accuracy of these approximations.

Figures (9)

• Figure 1: The numbers $\lambda_{p,j}$ and $|\omega_{p,j}|$ for $p\in \{20,40\}$.
• Figure 2: The values of $\Delta_p(t)$ for $p \in \{10,20,30,40,50\}$. The black dots on plot (b) correspond to values of $\Delta_p(t_n)$.
• Figure 3: The values of $\log_{10} (\Delta_p(t))$ for $p \in \{90,120,150\}$.
• Figure 4: The values of $|\zeta(1/2+{\textnormal{i}} t) - \zeta_p(1/2+{\textnormal{i}} t)|$ for $p \in \{3,5\}$ and $t$ close to $10^{10}$. The gray vertical lines show the locations of $t_n=2\pi n^2$ for $39894\le n \le 39897$.
• Figure 5: The values of $\Delta^{(1)}_p(t)$ for $p \in \{10,20,30,40,50\}$.

Theorems & Definitions (2)

• Proposition 1
• proof