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

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.