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On the measure of approximation of the number π by algebraic numbers

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Abstract

By means of the method of the Laurent interpolation determinant, it is proved that, if ζ is an algebraic number, the real numbersd andL satisfy the inequalitiesd≥degζ,L≥L(ζ), andL≥3, and the numberd is sufficiently large, then the inequality

$$|\pi - \varsigma | \geqslant \exp ( - 21.4708d \cdot (\log L + d \cdot \log d) \cdot (1 + \log d))$$

holds. The constant 21.4708 in the above estimate for the measure of transcendence of the number π is the best among the known values.

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References

  1. N. I. Fel'dman, “The approximation of certain transcendental numbers. I,”Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.],15, No. 1, 53–74 (1951).

    MATH  Google Scholar 

  2. N. I. Fel'dman, “The measure of transcendence of the number π,”Izv. Akad., Nauk SSSR Ser. Mat. [Math. USSR. Izv.],24, No. 3, 357–368 (1960).

    MATH  Google Scholar 

  3. N. I. Fel'dman, “A constant in the estimation of the measure of transcendence of the number π,” in:Diophantine Approximations [in Russian], Pt. 1, Izd. Moskov. Univ., Moscow (1985).

    Google Scholar 

  4. M. Val'dshmidt [M. Waldschmidt] and Yu. M. Nesterenko, “On the approximation of the values of the exponential function and of the logarithm by algebraic numbers,” in:Diophantine Approximations. A Collection of Papers Dedicated to the Memory of Professor N. I. Fel'dman [in Russian], Vol. 2, Mathematical Transactions, Center for Applied Research at the Faculty of Mechanics and Mathematics of Moscow State University, Moscow (1996), pp. 23–42.

    Google Scholar 

  5. Kunrui Yu, “Linear forms in p-adic logarithms,” Pt. I,Acta Arith.,53, 105–113 (1989); Pt II,Compositio Math.,74, 15–113 (1990).

    Google Scholar 

  6. N. I. Fel'dman,Hilbert's Seventh Problem [in Russian], Izd. Moskov. Univ., Moscow (1982).

    MATH  Google Scholar 

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Authors and Affiliations

  1. M. V. Lomonosov Moscow State University, Moscow, USSR

    Y. M. aleksentsev

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  1. Y. M. aleksentsev
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Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 483–493, October, 1999.

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aleksentsev, Y.M. On the measure of approximation of the number π by algebraic numbers. Math Notes 66, 395–403 (1999). https://doi.org/10.1007/BF02679086

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  • DOI: https://doi.org/10.1007/BF02679086

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