Asymptotic Properties of Stieltjes Constants
Polish Academy of Sciences
Institute for the History of Sciences
Nowy Świat 72, 00-330 Warsaw, PolandE-mail: krzysiek2357@gmail.com
Received:
Received: 12 September 2022; revised: 9 November 2022; accepted: 9 November 2022; published online: 10 November 2022
DOI: 10.12921/cmst.2022.0000021
Abstract:
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.
Key words:
Nørlund-Rice integral, saddle point method, Stieltjes constants
References:
[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.
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.
Key words:
Nørlund-Rice integral, saddle point method, Stieltjes constants
References:
[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.
