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arXiv · 2210.04609

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

Krzysztof Maślanka·Andrzej Koleżyński
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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. $ζ(1+\varepsilon),ζ(1+2\varepsilon),ζ(1+3\varepsilon),...$ 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.

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Krzysztof Maślanka, Andrzej Koleżyński. 2022-09-27. The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm. https://doi.org/10.12921/cmst.2022.0000014

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