The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm
Maślanka Krzysztof 1, Koleżyński Andrzej 2
1 Polish Academy of Sciences
Institute for the History of Sciences
Nowy Świat 72, 00-330 Warsaw, Poland
E-mail: krzysiek2357@gmail.com2 University of Science and Technology
Faculty of Materials Science and Ceramics
Mickiewicza 30, 30-059 Cracow, Poland
E-mail: kolezyn@agh.edu.pl
Received:
Received: 6 June 2022; revised: 13 June 2022; accepted: 15 June 2022; published online: 25 June 2022
DOI: 10.12921/cmst.2022.0000014
Abstract:
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.
Key words:
experimental mathematics, PARI/GP computer algebra system, Riemann zeta function, Stieltjes constants
References:
[1] PARI/GP version 2.14.0, 64-bit (2022), available from https://pari.math.u-bordeaux.fr/download.html.
[2] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing, New York, Dover (1972).
[3] O.R. Ainsworth, L.W. Howell, The Generalized Euler-Mascheroni Constants, NASA Technical Paper 2264 (1984).
[4] O.R. Ainsworth, L.W. Howell, An Integral Representation of the Generalized Euler-Mascheroni Constants, NASA Technical Paper 2456 (1985).
[5] L. Báez-Duarte, On Maslanka’s Representation for the Riemann Zeta Function, International Journal of Mathematics and Mathematical Sciences 2010, 714147 (2010).
[6] L. Báez-Duarte, A New Necessary and Sufficient Condition for the Riemann Hypothesis (2003).
[7] I.V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations, arXiv:1401.3724v3 (2015).
[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).
[9] I.V. Blagouchine, F. Johansson, Computing Stieltjes Constants Using Complex Integration, arXiv:1804.01679v3 (2018).
[10] M. Coffey, The Stieltjes constants, their relation to the ηj coefficients, and representation of the Hurwitz zeta function, arXiv:0706.0343v2 (2009).
[11] L. Euler. Variae observationes circa series infinitas, Commentarii academiae scientiarum Petropolitanae 9, 160–188 (1744).
[12] P. Flajolet, L. Vepstas, On Differences of Zeta Values, arXiv:math.CA/0611332v1 (2006).
[13] J. Franel, Short untitled announcement communicated by Ernesto Cesàro, L’Intermédiaire des mathematiciens 1, 153–154 (1895).
[14] C. Hermite, T.J. Stieltjes, Correspondance d’Hermite et de Stieltjes 1, (8 novembre 1882–22 juillet 1889) (1905).
[15] J.L.W.V. Jensen, Sur la fonction ζ(s) de Riemann, Comptes rendus hebdomadaires des séances de l’Académie des sciences, p. 1156.
[16] F. Johansson, Rigorous high-precision computation of the Hurwitz zeta function and its derivatives, Numerical Algorithms 69, 253–270 (2014).
[17] J.B. Keiper, Power series expansions of Riemann’s ξ function, Mathematics of Computation 58, 765–765 (1992).
[18] R. Kreminski, Newton-Cotes integration for approximating Stieltjes (generalized Euler) constants, Mathematics of Computation 72 (2002).
[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.
[22] S. Wolfram, Jerry Keiper (1953–1995), The Mathematica Journal 5 (1995). Obituary available at https://www.stephenw olfram.com/publications/jerry-keiper/.
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.
Key words:
experimental mathematics, PARI/GP computer algebra system, Riemann zeta function, Stieltjes constants
References:
[1] PARI/GP version 2.14.0, 64-bit (2022), available from https://pari.math.u-bordeaux.fr/download.html.
[2] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing, New York, Dover (1972).
[3] O.R. Ainsworth, L.W. Howell, The Generalized Euler-Mascheroni Constants, NASA Technical Paper 2264 (1984).
[4] O.R. Ainsworth, L.W. Howell, An Integral Representation of the Generalized Euler-Mascheroni Constants, NASA Technical Paper 2456 (1985).
[5] L. Báez-Duarte, On Maslanka’s Representation for the Riemann Zeta Function, International Journal of Mathematics and Mathematical Sciences 2010, 714147 (2010).
[6] L. Báez-Duarte, A New Necessary and Sufficient Condition for the Riemann Hypothesis (2003).
[7] I.V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations, arXiv:1401.3724v3 (2015).
[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).
[9] I.V. Blagouchine, F. Johansson, Computing Stieltjes Constants Using Complex Integration, arXiv:1804.01679v3 (2018).
[10] M. Coffey, The Stieltjes constants, their relation to the ηj coefficients, and representation of the Hurwitz zeta function, arXiv:0706.0343v2 (2009).
[11] L. Euler. Variae observationes circa series infinitas, Commentarii academiae scientiarum Petropolitanae 9, 160–188 (1744).
[12] P. Flajolet, L. Vepstas, On Differences of Zeta Values, arXiv:math.CA/0611332v1 (2006).
[13] J. Franel, Short untitled announcement communicated by Ernesto Cesàro, L’Intermédiaire des mathematiciens 1, 153–154 (1895).
[14] C. Hermite, T.J. Stieltjes, Correspondance d’Hermite et de Stieltjes 1, (8 novembre 1882–22 juillet 1889) (1905).
[15] J.L.W.V. Jensen, Sur la fonction ζ(s) de Riemann, Comptes rendus hebdomadaires des séances de l’Académie des sciences, p. 1156.
[16] F. Johansson, Rigorous high-precision computation of the Hurwitz zeta function and its derivatives, Numerical Algorithms 69, 253–270 (2014).
[17] J.B. Keiper, Power series expansions of Riemann’s ξ function, Mathematics of Computation 58, 765–765 (1992).
[18] R. Kreminski, Newton-Cotes integration for approximating Stieltjes (generalized Euler) constants, Mathematics of Computation 72 (2002).
[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.
[22] S. Wolfram, Jerry Keiper (1953–1995), The Mathematica Journal 5 (1995). Obituary available at https://www.stephenw olfram.com/publications/jerry-keiper/.
