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A note on Beukers' integral

  • Masayoshi Hata (a1)

Abstract

The aim of this note is to give a sharp lower bound for rational approximations to ζ(2) = π2/6 by using a specific Beukers' integral. Indeed, we will show that π2 has an irrationality measure less than 6.3489, which improves the earlier result 7.325 announced by D. V. Chudnovsky and G. V. Chudnovsky.

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References

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[1]Beukers, F., ‘A note on the irrationality of ζ(2) and ζ(3)’, Bull. London Math. Soc. 11 (1979), 268272.
[2]Chudnovsky, D. V. and Chudnovsky, G. V., Padé and rational approximations to systems of functions and their arithmetic applications Lecture Notes in Mathematics 1052 (Springer, Berlin, 1984) pp. 3784.
[3]Chudnovsky, D. V., ‘Transcendental methods and theta-functions’, in: Proc. Sympos. Pure Math. 49 (Amer. Math. Soc., Providence, 1989), pp. 167232.
[4]Dvornicich, R. and Viola, C., ‘Some remarks on Beukers' integrals’, Colloq. Math. Soc. János Bolyai 51 (1987), 637657.
[5]Erdélyi, A. et al. , Higher transcendental functions, volume 1 (McGraw-Hill, New York, 1953).
[6]Hata, M., ‘Legendre type polynomials and irrationality measures’, J. Reine Angew. Math. 407 (1990), 99125.
[7]Hata, M., ‘Rational approximations to π and some other numbers’, Acta Arithmetica 63 (1993), 335349.
[8]Rhin, G. and Viola, C., ‘On the irrationality measure of ζ(2)’, Ann. Inst. Fourier (Grenoble) 43 (1993), 85109.
[9]Rukhadze, E. A., ‘A lower bound for the approximation of In 2 by rational numbers’, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 6 (1987), 2529 in Russian.
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A note on Beukers' integral

  • Masayoshi Hata (a1)

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