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

• Autorzy: Maślanka Krzysztof , Koleżyński Andrzej

Identyfikatory

DOI

10.12921/cmst.2022.0000014

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Riemann zeta function; Stieltjes constants; experimental mathematics; PARI/GP computer algebra system

• Wydawca: Institute of Bioorganic Chemistry Scientific Publishers OWN, Polish Academy of Sciences
• Czasopismo: Computational Methods in Science and Technology
• Rocznik: 2022
• Tom: Vol. 28, No. 2
• Strony: 47--57
• Opis fizyczny: Bibliogr. 22 poz., rys.

Twórcy

autor

Maślanka Krzysztof

• Polish Academy of Sciences Institute for the History of Sciences Nowy Swiat 72, 00-330 Warsaw, Poland

autor

Koleżyński Andrzej

• University of Science and Technology Faculty of Materials Science and Ceramics Mickiewicza 30, 30-059 Cracow, Poland

Bibliografia

• [1] PARI/GP version 2.14.0, 64-bit (2022), available from https://pari.math.u-bordeaux.fr/download.html.
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• [8] I.V. Blagouchine, Expansions of generalized Euler’s constants into the series of polynomials in π−2 and into the formal enveloping series with rational coefficients only, arXiv:1501.00740v4 (2016).
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• [19] K. Maślanka, The Beauty of Nothingness: Essay on the Zeta Function of Riemann, Acta Cosmologica XXIII, 13–18 (1998); A hypergeometric-like Representation of Zeta function of Riemann, arXiv:math-ph/0105007v1 (2001); see also http://functions.wolfram.com, citation index: 10.01.06.0012.01 and 10.01.17.0003.01.
• [20] K. Maślanka, Báez Duarte’s Criterion for the Riemann Hypothesis and Rice’s Integrals, arXiv:math/0603713v2 (2006).
• [21] B. Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsberichte der Berliner Akademie, 671–680 (1859). English translation available at http://www.maths.tcd.ie/pub/HistMath/People/Riemann.
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Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).