The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm

TL;DR

This paper introduces a simple and efficient algorithm for high-precision calculation of Stieltjes constants, achieving up to 80,000 significant digits by leveraging hypergeometric-like expansions of the Riemann zeta function and high-precision zeta evaluations.

Contribution

The authors develop a novel, fast method for computing Stieltjes constants at very high precision using a sequence of high-precision zeta function values and hypergeometric-like expansions.

Findings

Achieved calculation of Stieltjes constants with up to 80,000 significant digits.

Demonstrated the efficiency of the method using PARI/GP for high-precision zeta evaluations.

Provided a practical approach for high-precision numerical analysis of special constants.

Abstract

We present a simple but efficient method of calculating Stieltjes constants at a very high level of precision, up to about 80000 significant digits. This method is based on the hypergeometric-like expansion for the Riemann zeta function presented by one of the authors in 1997 \cite{Maslanka 1}. The crucial ingredient in this method is a sequence of high-precision numerical values of the Riemann zeta function computed in equally spaced real arguments, i.e. $\zeta(1+\varepsilon),\zeta(1+2\varepsilon),\zeta(1+3\varepsilon),...$ where $\varepsilon$ is some real parameter. (Practical choice of $\varepsilon$ is described in the main text.) Such values of zeta may be readily obtained using the PARI/GP program, which is especially suitable for this.