Asymptotic Properties of Stieltjes Constants

• Autorzy: Maślanka Krzysztof

Identyfikatory

DOI

10.12921/cmst.2022.0000021

• Języki publikacji: EN

Abstrakty

EN

We present a new asymptotic formula for the Stieltjes constants which is both simpler and more accurate than several others published in the literature (see e.g. [1–3]). More importantly, it is also a good starting point for a detailed analysis of some surprising regularities in these important constants.

Słowa kluczowe

EN

Stieltjes constants; saddle point method; Nørlund-Rice integral

• 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. 4
• Strony: 123--131
• Opis fizyczny: Bibliogr. 17 poz., rys.

Twórcy

autor

Maślanka Krzysztof

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

Bibliografia

• [1] C. Knessl, M.W. Coffey, An Effective Asymptotic Formula for the Stieltjes Constants, Mathematics of Computation 80(273), 379–386 (2011).
• [2] L. Fekih-Ahmed, A New Effective Asymptotic Formula for the Stieltjes Constants, arXiv:1407.5567v3 (2014).
• [3] R.B. Paris, An Asymptotic Expansion for the Stieltjes Constants, Mathematica Aeterna 5, 707–716 (2015).
• [4] K. Maślanka, M. Wolf, Are the Stieltjes constants irrational? Some computer experiments, Computational Methods in Science and Technology 26(3), 77–87 (2020).
• [5] K. Maślanka, A. Koleżyński, The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm, Computational Methods in Science and Technology 28(2), 47–59 (2022).
• [6] Wikipedia, Stieltjes constants, https://en.wikipedia.org/wiki/Stieltjes_constants.
• [7] Wolfram MathWorld, Lambert W-Function, https://math world.wolfram.com/LambertW-Function.html.
• [8] Wolfram MathWorld, Stirling Number of the First Kind, https://mathworld.wolfram.com/StirlingNumberoftheFirstKi nd.html.
• [9] P. Flajolet, R. Sedgewick, Mellin transforms and asymptotics: Finite differences and Rice’s integrals, Theoretical Computer Science 144, 101–124 (1995).
• [10] D.E. Knuth, The Art of Computer Programming 3: Sorting and Searching, Addision-Wesley, Reading, MA (1998).
• [11] N.E. Nørlund, Vorlesungen über Differenzenrechnung, Springer, Berlin (1924). Reprinted: Chelsea Publishing Company, New York (1954).
• [12] H.M. Edwards, Riemann’s Zeta Function, Dover Publications (2001).
• [13] P. Flajolet, L. Vepstas, On Differences of Zeta Values, arXiv:math/0611332v2 (2007).
• [14] A. Erdélyi, Asymptotic Expansions, Dover Publications (1956).
• [15] Wolfram Research, Inc., Mathematica, Version 13.1, Champaign, Illinois (2022).
• [16] A. LeClair, An electrostatic depiction of the validity of the Riemann Hypothesis and a formula for the N-th zero at large N, arxiv:1305.2613v5 (2013).
• [17] A. Jasiński, https://oeis.org/A114523; https://oeis.org/A114 524, [In:] N. Sloan, The On-Line Encyclopedia of Integer Sequences.

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).