On the irrationality measure of ζ ( 2 )

Georges Rhin; Carlo Viola

Annales de l'institut Fourier (1993)

  • Volume: 43, Issue: 1, page 85-109
  • ISSN: 0373-0956

Abstract

top
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.

How to cite

top

Rhin, Georges, and Viola, Carlo. "On the irrationality measure of $\zeta (2)$." Annales de l'institut Fourier 43.1 (1993): 85-109. <http://eudml.org/doc/74995>.

@article{Rhin1993,
abstract = {We prove that 7. 398 537 is an irrationality measure of $\zeta (2)=\pi ^2/6$. We employ double integrals of suitable rational functions invariant under a group of birational transformations of $\{\Bbb C\}^2$. The numerical results are obtained with the aid of a semi-infinite linear programming method.},
author = {Rhin, Georges, Viola, Carlo},
journal = {Annales de l'institut Fourier},
keywords = {irrationality measure},
language = {eng},
number = {1},
pages = {85-109},
publisher = {Association des Annales de l'Institut Fourier},
title = {On the irrationality measure of $\zeta (2)$},
url = {http://eudml.org/doc/74995},
volume = {43},
year = {1993},
}

TY - JOUR
AU - Rhin, Georges
AU - Viola, Carlo
TI - On the irrationality measure of $\zeta (2)$
JO - Annales de l'institut Fourier
PY - 1993
PB - Association des Annales de l'Institut Fourier
VL - 43
IS - 1
SP - 85
EP - 109
AB - We prove that 7. 398 537 is an irrationality measure of $\zeta (2)=\pi ^2/6$. We employ double integrals of suitable rational functions invariant under a group of birational transformations of ${\Bbb C}^2$. The numerical results are obtained with the aid of a semi-infinite linear programming method.
LA - eng
KW - irrationality measure
UR - http://eudml.org/doc/74995
ER -

References

top
  1. [1] K. ALLADI and M.L. ROBINSON, Legendre polynomials and irrationality, J. reine angew. Math., 318 (1980), 137-155. Zbl0425.10039MR81i:10036
  2. [2] E.J. ANDERSON and P. NASH, Linear Programming in infinite-dimensional spaces, Wiley-Interscience, 1987. Zbl0632.90038MR88f:90180
  3. [3] R. APÉRY, Irrationalité de &ζ(2) et &ζ(3), Astérisque, 61 (1979), 11-13. Zbl0401.10049
  4. [4] F. BEUKERS, A note on the irrationality of &ζ(2) and &ζ(3), Bull. London Math. Soc., 11 (1979), 268-272. Zbl0421.10023MR81j:10045
  5. [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. Zbl0536.10028MR86a:11029
  6. [6] D. V. CHUDNOVSKY and G. V. CHUDNOVSKY, Transcendental methods and Theta-functions, Proc. Symp. Pure Math., 49 (1989), part 2, 167-232. Zbl0679.10026MR91e:11080
  7. [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. Zbl0518.41014MR83m:10051
  8. [8] R. DVORNICICH and C. VIOLA, Some remarks on Beukers' integrals, Colloquia Math. Soc. János Bolyai, 51 (1987), 637-657. Zbl0755.11019MR91f:11050
  9. [9] M. HATA, Legendre type polynomials and irrationality measures, J. reine angew. Math., 407 (1990), 99-125. Zbl0692.10034MR91i:11081
  10. [10] K. MAHLER, On the approximation of π, Proc. K. Ned. Akad. Wet. Amsterdam, A 56 (1953), 30-42. Zbl0053.36105MR14,957a
  11. [11] M. MIGNOTTE, Approximations rationnelles de π et quelques autres nombres, Bull. Soc. Math. France, Mémoire 37 (1974), 121-132. Zbl0286.10017MR51 #375
  12. [12] J.A. NELDER and R. MEAD, A simplex method for function minimization, Computer J., 7 (1965), 308-313. Zbl0229.65053
  13. [13] W.H. PRESS, B.P. FLANNERY, S.A. TEUKOLSKY, W.T. VETTERLING, Numerical Recipes. The art of scientific computing, Cambridge University Press, 1986. Zbl0587.65003
  14. [14] G. RHIN, Approximants de Padé et mesures effectives d'irrationalité, Progr. in Math., 71 (1987), 155-164. Zbl0632.10034MR90k:11089
  15. [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. Zbl0635.10025MR89b:11064

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.