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

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Maślanka K. , Koleżyński A.

Computational Methods in Science and Technology

2022

tom 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.

Are the Stieltjes constants irrational ? : Some computer experiments

88%

Maślanka K. D. , Wolf M.

Computational Methods in Science and Technology

2020

tom Vol. 26, No. 3

77--87

Khnichin’s theorem is a surprising and still relatively little known result. It can be used as a specific criterion for determining whether or not any given number is irrational. In this paper we apply this theorem as well as the Gauss-Kuzmin theorem to several thousand high precision (up to more than 53 000 significant digits) initial Stieltjes constants γn, n = = 0, 1, 2, . . . , 5000 in order to confirm that, as is commonly believed, they are irrational numbers (and even transcendental). We also study the normality of these important constants.