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

Maślanka Krzysztof, Koleżyński Andrzej

Computational Methods in Science and Technology

2022

Vol. 28, No. 2

47--57

We present a simple but efficient method of calculating Stieltjes constants at a very high level of precision, up to about 80 000 significant digits. This method is based on the hypergeometric-like expansion for the Riemann zeta function presented by one of the authors in 1997 [19]. 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 + ε), ζ(1 + 2ε), ζ(1 + 3ε), ... where ε is some real parameter. (Practical choice of ε 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.