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On an Evaluation Method for Zeta Constants Based on a Number Theoretic Approach

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Abstract

New formulas for zeta constants are obtained based on a number theoretic approach that is used in proving irrationality of some classical constants. Using these formulas, one can approximate zeta constants and their combinations by rational fractions and construct a new efficient evaluation method for them.

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References

  1. 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. Transm. (Engl. Transl.), 1995, vol. 31, no. 4, pp. 353–362]. http://mi.mathnet.ru/eng/ppi294

    MathSciNet  Google Scholar 

  2. Borwein, J.M., Bradley, D.M., and Crandall, R.E., Computational Strategies for the Riemann Zeta Function, J. Comput. Appl. Math., 2000, vol. 121, no. 1–2, pp. 247–296. https://doi.org/10.1016/S0377-0427(00)00336-8

    Article  MathSciNet  Google Scholar 

  3. Skorokhodov, S.L., Padé Approximants and Numerical Analysis of the Riemann Zeta Function, Zh. Vychisl. Mat. Mat. Fiz., 2003, vol. 43, no. 9, pp. 1330–1352 [Comput. Math. Math. Phys. (Engl. Transl.), 2003, vol. 43, no. 9, pp. 1277–1298]. http://mi.mathnet.ru/eng/zvmmf961

    MathSciNet  MATH  Google Scholar 

  4. Zudilin, V.V., Binomial Sums Related to 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]. https://doi.org/10.1023/B:MATN.0000023341.93824.fb

    Article  Google Scholar 

  5. Karatsuba, E.A., On One Method for Fast Approximation of Zeta Constants by Rational Fractions, Probl. Peredachi Inf., 2014, vol. 50, no. 2, pp. 77–95 [Probl. Inf. Transm. (Engl. Transl.), 2014, vol. 50, no. 2, pp. 186–202]. https://doi.org/10.1134/S0032946014020057

    MathSciNet  MATH  Google Scholar 

  6. Matiyasevich, Yu.V., Riemann’s Zeta Function and Finite Dirichlet Series, Alg. Analiz, 2015, vol. 27, no. 6, pp. 174–198 [St. Petersburg Math. J. (Engl. Transl.), 2016, vol. 27, no. 6, pp. 985–1002]. https://doi.org/10.1090/spmj/1431

    MathSciNet  Google Scholar 

  7. Karatsuba, E.A., On One Method for Constructing a Family of Approximations of Zeta Constants by Rational Fractions, Probl. Peredachi Inf., 2015, vol. 51, no. 4, pp. 78–91 [Probl. Inf. Transm. (Engl. Transl.), 2015, vol. 51, no. 4, pp. 378–390]. https://doi.org/10.1134/S0032946015040079

    MathSciNet  Google Scholar 

  8. Hermite, C., Sur la fonction exponentielle, C. R. Acad. Sci. Paris, 1873, vol. 77, pp. 18–24.

    MATH  Google Scholar 

  9. Beukers, F., A Note on the Irrationality of ζ(2) and ζ(3), Bull. London Math. Soc., 1979, vol. 11, no. 3, pp. 268–272. https://doi.org/10.1112/blms/11.3.268

    Article  MathSciNet  Google Scholar 

  10. Beukers, F., Legendre Polynomials in Irrationality Proofs, Bull. Austral. Math. Soc., 1980, vol. 22, no. 3, pp. 431–438. https://doi.org/10.1017/S0004972700006742

    Article  MathSciNet  Google Scholar 

  11. Dvornicich, R. and Viola, C., Some Remarks on Beukers’ Integrals, Number Theory, Vol. II (Budapest, 1987), Győry, K. and Halász, G., Eds., Colloq. Math. Soc. János Bolyai, vol. 51, Amsterdam: North-Holland, 1990, pp. 637–657.

  12. Hata, M., A Note on Beukers’ Integral, J. Austral. Math. Soc. Ser. A, 1995, vol. 58, no. 2, pp. 143–153. https://doi.org/10.1017/S1446788700038192

    Article  MathSciNet  Google Scholar 

  13. Hessami Pilehrood, Kh., Hessami Pilehrood, T., and Tauraso, R., Congruences Concerning Jacobi Polynomials and Apéry-like Formulae, Int. J. Number Theory, 2012, vol. 8, no. 7, pp. 1789–1811. https://doi.org/10.1142/S1793042112501035

    Article  MathSciNet  Google Scholar 

  14. Zudilin, V.V., On the Irrationality of the Values of the Riemann Zeta Function, Izv. Ross. Akad. Nauk Ser. Mat., 2002, vol. 66, no. 3, pp. 49–102 [Izv. Math. (Engl. Transl.), 2002, vol. 66, no. 3, pp. 489–542]. https://doi.org/10.1070/IM2002v066n03ABEH000387

    Article  MathSciNet  Google Scholar 

  15. Chu, W. and De Donno, L., Hypergeometric Series and Harmonic Number Identities, Adv. in Appl. Math., 2005, vol. 34, no. 1, pp. 123–137. https://doi.org/10.1016/j.aam.2004.05.003

    Article  MathSciNet  Google Scholar 

  16. Spieß, J., Some Identities Involving Harmonic Numbers, Math. Comp., 1990, vol. 55, no. 192, pp. 839–863. https://doi.org/10.2307/2008451

    Article  MathSciNet  Google Scholar 

  17. Wang, W. and Jia, C., Harmonic Number Identities via the Newton–Andrews Method, Ramanujan J., 2014, vol. 35, no. 2, pp. 263–285. https://doi.org/10.1007/s11139-013-9511-1

    Article  MathSciNet  Google Scholar 

  18. Bateman, H. and Erdélyi, A., Higher Transcendental Functions, New York: McGraw-Hill, 1953, vol. 1. Translated under the title Vysshie transtsendentnye funktsii, Moscow: Nauka, 1965, vol. 1.

    MATH  Google Scholar 

  19. Whittaker, E.T. and Watson, G.N., A Course of Modern Analysis, Part 2: The Transcendental Functions, Cambridge: Cambridge Univ. Press, 1927, 4th ed. Translated under the title Kurs sovremennogo analiza, Part 2: Transtsendentnye funktsii, Moscow: Gostekhizdat, 1934.

    Google Scholar 

  20. Karatsuba, E.A., On an Identity with Binomial Coefficients, Mat. Zametki, 2019, vol. 105, no. 1, pp. 149–152 [Math. Notes (Engl. Transl.), 2019, vol. 105, no. 1–2, pp. 145–147]. https://doi.org/10.1134/S0001434619010176

    Article  MathSciNet  Google Scholar 

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Funding

The research was supported by the Russian Foundation for Basic Research, project no. 19-07-00750.

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Authors and Affiliations

  1. Dorodnitsyn Computing Center of the Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Moscow, Russia

    E. A. Karatsuba

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  1. E. A. Karatsuba
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Translated from Problemy Peredachi Informatsii, 2021, Vol. 57, No. 3, pp. 73–89 https://doi.org/10.31857/S0555292321030050.

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Karatsuba, E. On an Evaluation Method for Zeta Constants Based on a Number Theoretic Approach. Probl Inf Transm 57, 265–280 (2021). https://doi.org/10.1134/S0032946021030054

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