Abstract
We present a new method for deriving both known and new fast approximations of zeta constants ζ(n), n ≥ 2, n is an integer, by rational fractions.
Similar content being viewed by others
References
Catalan, E., Mémoire sur la transformation des séries et sur quelques intégrales définies, Mémoires couronnés et mémoires des savants étrangers, publies par l’Académie Royale des Sciences, des Lettres et des Beaux-arts de Belgique, T. XXXIII (1865-1867), Bruxelles: Hayez, 1867, pp. 1–50.
Markoff, A.A., Mémoire sur la transformation des séries peu convergentes en séries très convergentes, Mémoires de l’Académie Impériale des Sciences de St.-Pétersbourg, 1890, vol. XXXVII, no. 9, pp. 1–22.
Ogigova, H., Les lettres de Gh. Hermite à A. Markoff, 1885–1899, Rev. Histoire Sci. Appl., 1967, vol. 20, no. 1, pp. 2–32.
Apéry, R., Irrationalité de ζ(2) et ζ(3), in Journées arithmétiques de Luminy, Marseille, France, June 20–24, 1978, Astérisque, 1979, vol. 61, pp. 11–13.
van der Poorten, A.J., A Proof that Euler Missed... Apéry’s Proof of the Irrationality of ζ(3). An Informal Report, Math. Intelligencer, 1978/79, vol. 1, no. 4, pp. 195–203.
Koecher, M., Letter to the Editor, Math. Intelligencer, 1979/80, vol. 2, no. 2, pp. 62–64.
Leshchiner, D., Some New Identities for ζ(k), J. Number Theory, 1981, vol. 13, no. 3, pp. 355–362.
Berndt, B.C., Ramanujan’s Notebooks, New York: Springer, 1985, vol. 1.
Butzer, P.L., Markett, C., and Schmidt, M., Stirling Numbers, Central Factorial Numbers, and Representations of the Riemann Zeta Function, Results Math., 1991, vol. 19, no. 3–4, pp. 257–274.
Nan-Yue, Z. and Williams, K.S., Values of the Riemann Zeta Function and Integrals Involving log(2 sinh(θ/2)) and log(2 sin(θ/2)), Pacific J. Math., 1995, vol. 168, no. 2, pp. 271–289.
Borwein, J.M. and Bradley, D.M., Empirically Determined Apéry-like Formulae for ζ(4n + 3), Experiment. Math., 1997, vol. 6, no. 3, pp. 181–194
Almkvist, G. and Granville, A., Borwein and Bradley’s Apéry-like Formulae for ζ(4n +3), Experiment. Math., 1999, vol. 8, no. 2, pp. 197–203
Zudilin, V.V., On Third-Order Apéry-type Recursion for ζ(5), Mat. Zametki, 2002, vol. 72, no. 5, pp. 796–800 [Math. Notes (Engl. Transl.), 2002, vol. 72, no. 5–6, pp. 733–737]
Rivoal, T., Simultaneous Generation of Koecher and Almkvist-Granville’s Apéry-like Formulae, Experiment. Math., 2004, vol. 13, no. 4, pp. 503–508.
Zudilin, V.V., Binomial Sums Associated with Rational Approximations to ζ(4), Mat. Zametki, 2004, vol. 75, no. 4, pp. 637–640 [Math. Notes (Engl. Transl.), 2004, vol. 75, no. 3–4, pp. 594–597].
Bailey, D.H., Borwein, J.M., and Bradley, D.M., Experimental Determination of Apéry-like Identities for ζ(2n + 2), Experiment. Math., 2006, vol. 15, no. 3, pp. 281–289
Hessami Pilehrood, Kh. and Hessami Pilehrood, T., Simultaneous Generation for Zeta Values by the Markov-WZ Method, Discrete Math. Theor. Comput. Sci., 2008, vol. 10, no. 3, pp. 115–123.
Chan, H.H. and Zudilin, W., New Representations for Apéry-like Sequences, Mathematika, 2010, vol. 56, no. 1, pp. 107–117.
Karatsuba, E.A., Fast Computation of Transcendental Functions, Dokl. Akad. Nauk SSSR, 1991, vol. 318, no. 2, pp. 278–279 [Soviet Math. Dokl. (Engl. Transl.), 1991, vol. 43, no. 3, pp. 693–694].
Karatsuba, E.A., Fast Evaluation of Transcendental Functions, Probl. Peredachi Inf., 1991, vol. 27, no. 4, pp. 76–99 [Probl. Inf. Trans. (Engl. Transl.), 1991, vol. 27, no. 4, pp. 339–360].
Karatsuba, E.A., A New Method for the Fast Calculation of Transcendental Functions, Uspekhi Mat. Nauk, 1991, vol. 46, no. 2 (278), pp. 219–220 [Russian Math. Surveys (Engl. Transl.), 1991, vol. 46, no. 2, pp. 246–247].
Karatsuba, E.A., Fast Calculation of ζ(3), Probl. Peredachi Inf., 1993, vol. 29, no. 1, pp. 68–73 [Probl. Inf. Trans. (Engl. Transl.), 1993, vol. 29, no. 1, pp. 58–62].
Karatsuba, C.A., Fast Evaluation of Bessel Functions, Integral Transform. Spec. Funct., 1993, vol. 1, no. 4, pp. 269–276.
Karatsuba, E.A., Fast Computation of the Riemann Zeta Function ζ(s) for Integer Values of s, Probl. Peredachi Inf., 1953, vol. 31, no. 4, pp. 69–80 [Probl. Inf. Trans. (Engl. Transl.), 1995, vol. 31, no. 4, pp. 353–362].
Karatsuba, E.A., On the Fast Calculation of the Riemann Zeta Function for Integer Argument, Dokl. Ross. Akad. Nauk, 1996, vol. 349, no. 4, p. 463.
Karatsuba, E.A., Fast Computation of the Values of the Hurwitz Zeta Function and Dirichlet L-Series, Probl. Peredachi Inf., 1998, vol. 34, no. 4, pp. 62–75 [Probl. Inf. Trans. (Engl. Transl.), 1998, vol. 34, no. 4, pp. 342–353].
Karatsuba, E.A., Fast Evaluation of Hypergeometric Function by FEE, Proc. 3rd CMFT Conf. on Computational Methods and Function Theory, Nicosia, Cyprus, Oct. 13–17, 1997, Papamichael, N., Ruscheweyh, S., and Saff, E.B., Eds., Singapore: World Sci., 1999, pp. 303–314.
Karatsuba, E.A., On the Computation of the Euler Constant γ, J. Numer. Algorithms, 2000, vol. 24, no. 1–2, pp. 83–97.
Karatsuba, E.A., Fast Computation of Some Special Integrals of Mathematical Physics, Scientific Computing, Validated Numerics, Interval Methods, Krämer, W. and von Gudenberg, J.W., Eds., Boston: Kluwer, 2001, pp. 29–41.
Karatsuba, E.A., Fast Computation of ζ(3) and of Some Special Integrals Using the Ramanujan Formula and Polylogarithms, BIT Numer. Math., 2001, vol. 41, no. 4, pp. 722–730.
Temme, N.M., Special Functions. An Introduction to the Classical Functions of Mathematical Physics, New York: Wiley, 1996.
Benderskii, Yu.V., Fast Computations, Dokl. Akad. Nauk SSSR, 1975, vol. 223, no. 5, pp. 1041–1043.
Kummer, E.E., Eine neue Methode, die numerischen Summen langsam convergirender Reihen zu berechnen, J. Reine Angew. Math., 1837, vol. 16, pp. 206–214.
Fikhtengol’ts, G.M., Kurs differentsial’nogo i integral’nogo ischisleniya (A Course in Differential and Integral Calculus), Moscow: Fizmatgiz, 1959, vol. 2.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © E.A. Karatsuba, 2014, published in Problemy Peredachi Informatsii, 2014, Vol. 50, No. 2, pp. 77–95.
Rights and permissions
About this article
Cite this article
Karatsuba, E.A. On one method for fast approximation of zeta constants by rational fractions. Probl Inf Transm 50, 186–202 (2014). https://doi.org/10.1134/S0032946014020057
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0032946014020057