On the irrationality measure of ζ(2)
Annales de l'Institut Fourier, Tome 43 (1993) no. 1, pp. 85-109.

On démontre que 7. 398 537 est une mesure d’irrationalité de ζ(2)=π 2 /6. On utilise des intégrales doubles de fonctions rationnelles stables par un groupe de transformations birationnelles de ℂ 2 . Les résultats numériques sont obtenus à l’aide d’une méthode de programmation linéaire semi-infinie.

We prove that 7. 398 537 is an irrationality measure of ζ(2)=π 2 /6. We employ double integrals of suitable rational functions invariant under a group of birational transformations of ℂ 2 . The numerical results are obtained with the aid of a semi-infinite linear programming method.

@article{AIF_1993__43_1_85_0,
     author = {Rhin, Georges and Viola, Carlo},
     title = {On the irrationality measure of $\zeta (2)$},
     journal = {Annales de l'Institut Fourier},
     pages = {85--109},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {43},
     number = {1},
     year = {1993},
     doi = {10.5802/aif.1322},
     zbl = {0776.11036},
     mrnumber = {1209696},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.1322/}
}
TY  - JOUR
AU  - Rhin, Georges
AU  - Viola, Carlo
TI  - On the irrationality measure of $\zeta (2)$
JO  - Annales de l'Institut Fourier
PY  - 1993
SP  - 85
EP  - 109
VL  - 43
IS  - 1
PB  - Institut Fourier
PP  - Grenoble
UR  - http://archive.numdam.org/articles/10.5802/aif.1322/
DO  - 10.5802/aif.1322
LA  - en
ID  - AIF_1993__43_1_85_0
ER  - 
%0 Journal Article
%A Rhin, Georges
%A Viola, Carlo
%T On the irrationality measure of $\zeta (2)$
%J Annales de l'Institut Fourier
%D 1993
%P 85-109
%V 43
%N 1
%I Institut Fourier
%C Grenoble
%U http://archive.numdam.org/articles/10.5802/aif.1322/
%R 10.5802/aif.1322
%G en
%F AIF_1993__43_1_85_0
Rhin, Georges; Viola, Carlo. On the irrationality measure of $\zeta (2)$. Annales de l'Institut Fourier, Tome 43 (1993) no. 1, pp. 85-109. doi : 10.5802/aif.1322. http://archive.numdam.org/articles/10.5802/aif.1322/

[1] K. Alladi and M.L. Robinson, Legendre polynomials and irrationality, J. reine angew. Math., 318 (1980), 137-155. | MR | Zbl

[2] E.J. Anderson and P. Nash, Linear Programming in infinite-dimensional spaces, Wiley-Interscience, 1987. | MR | Zbl

[3] R. Apéry, Irrationalité de ζ ( 2 ) et ζ ( 3 ) , Astérisque, 61 (1979), 11-13. | Numdam | MR | Zbl

[4] F. Beukers, A note on the irrationality of ζ ( 2 ) and ζ ( 3 ) , Bull. London Math. Soc., 11 (1979), 268-272. | MR | Zbl

[5] D. V. Chudnovsky and G. V. Chudnovsky, Padé and rational approximations to systems of functions and their arithmetic applications, Lect. Notes in Math., 1052 (1984), 37-84. | MR | Zbl

[6] D. V. Chudnovsky and G. V. Chudnovsky, Transcendental methods and Theta-functions, Proc. Symp. Pure Math., 49 (1989), part 2, 167-232. | MR | Zbl

[7] G. V. Chudnovsky, Hermite-Padé approximations to exponential functions and elementary estimates of the measure of irrationality of π, Lect. Notes in Math., 925 (1982), 299-322. | MR | Zbl

[8] R. Dvornicich and C. Viola, Some remarks on Beukers' integrals, Colloquia Math. Soc. János Bolyai, 51 (1987), 637-657. | MR | Zbl

[9] M. Hata, Legendre type polynomials and irrationality measures, J. reine angew. Math., 407 (1990), 99-125. | MR | Zbl

[10] K. Mahler, On the approximation of π, Proc. K. Ned. Akad. Wet. Amsterdam, A 56 (1953), 30-42. | MR | Zbl

[11] M. Mignotte, Approximations rationnelles de π et quelques autres nombres, Bull. Soc. Math. France, Mémoire 37 (1974), 121-132. | Numdam | MR | Zbl

[12] J.A. Nelder and R. Mead, A simplex method for function minimization, Computer J., 7 (1965), 308-313. | Zbl

[13] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes. The art of scientific computing, Cambridge University Press, 1986. | Zbl

[14] G. Rhin, Approximants de Padé et mesures effectives d'irrationalité, Progr. in Math., 71 (1987), 155-164. | MR | Zbl

[15] E.A. Rukhadze, A lower bound for the approximation of ln 2 by rational numbers (Russian), Vestnik Moskov Univ., Ser 1 Math. Mekh., 6 (1987), 25-29. | MR | Zbl

Cité par Sources :