# Birational transformations and values of the Riemann zeta-function

Journal de théorie des nombres de Bordeaux (2003)

- Volume: 15, Issue: 2, page 561-592
- ISSN: 1246-7405

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topViola, Carlo. "Birational transformations and values of the Riemann zeta-function." Journal de théorie des nombres de Bordeaux 15.2 (2003): 561-592. <http://eudml.org/doc/249114>.

@article{Viola2003,

abstract = {In his proof of Apery’s theorem on the irrationality of $\zeta (3)$, Beukers [B] introduced double and triple integrals of suitable rational functions yielding good sequences of rational approximations to $\zeta (2)$ and $\zeta (3)$. Beukers’ method was subsequently improved by Dvornicich and Viola, by Hata, and by Rhin and Viola. We give here a survey of our recent results ([RV2] and [RV3]) on the irrationality measures of $\zeta (2)$ and $\zeta (3)$ based upon a new algebraic method involving birational transformations and permutation groups acting on double and triple integrals of Beukers’ type. In the last two sections we give a constructive method to obtain the relevant birational transformations for triple integrals from the analogous transformations for double integrals, and we also apply such a method to get birational transformations acting on quadruple integrals of Vasilyev’s type.},

author = {Viola, Carlo},

journal = {Journal de théorie des nombres de Bordeaux},

language = {eng},

number = {2},

pages = {561-592},

publisher = {Université Bordeaux I},

title = {Birational transformations and values of the Riemann zeta-function},

url = {http://eudml.org/doc/249114},

volume = {15},

year = {2003},

}

TY - JOUR

AU - Viola, Carlo

TI - Birational transformations and values of the Riemann zeta-function

JO - Journal de théorie des nombres de Bordeaux

PY - 2003

PB - Université Bordeaux I

VL - 15

IS - 2

SP - 561

EP - 592

AB - In his proof of Apery’s theorem on the irrationality of $\zeta (3)$, Beukers [B] introduced double and triple integrals of suitable rational functions yielding good sequences of rational approximations to $\zeta (2)$ and $\zeta (3)$. Beukers’ method was subsequently improved by Dvornicich and Viola, by Hata, and by Rhin and Viola. We give here a survey of our recent results ([RV2] and [RV3]) on the irrationality measures of $\zeta (2)$ and $\zeta (3)$ based upon a new algebraic method involving birational transformations and permutation groups acting on double and triple integrals of Beukers’ type. In the last two sections we give a constructive method to obtain the relevant birational transformations for triple integrals from the analogous transformations for double integrals, and we also apply such a method to get birational transformations acting on quadruple integrals of Vasilyev’s type.

LA - eng

UR - http://eudml.org/doc/249114

ER -

## References

top- [A] R. Apéry, Irrationalité de ζ(2) et ζ(3). Astérisque61 (1979), 11-13. Zbl0401.10049
- [B] F. Beukers, A note on the irrationality of ζ(2) and ζ(3). Bull. London Math. Soc.11 (1979), 268-272. Zbl0421.10023
- [DV] R. Dvornicich, C. Viola, Some remarks on Beukers' integrals, in: Colloq. Math. Soc. János Bolyai51, Budapest (1987), 637-657. Zbl0755.11019MR1058238
- [H1] M. Hata, Legendre type polynomials and irrationality measures. J. reine angew. Math.407 (1990), 99-125. Zbl0692.10034MR1048530
- [H2] M. Hata, A note on Beukers' integral. J. Austral. Math. Soc. (A) 58 (1995), 143-153. Zbl0830.11026MR1323987
- [H3] M. Hata, A new irrationality measure for ζ(3). Acta Arith.92 (2000), 47-57. Zbl0955.11023
- [RV1] G. Rhin, C. Viola, On the irrationality measure of ζ(2). Ann. Inst. Fourier43 (1993), 85-109. Zbl0776.11036
- [RV2] G. Rhin, C. Viola, On a permutation group related to ζ(2). Acta Arith.77 (1996), 23-56. Zbl0864.11037
- [RV3] G. Rhin, C. Viola, The group structure for ζ(3). Acta Arith.97 (2001), 269-293. Zbl1004.11042
- [Va] D.V. Vasilyev, Approximations of zero by linear forms in values of the Riemann zeta-function. Dokl. Belarus Acad. Sci. 45 no. 5 (2001), 36-40 (Russian). Zbl1178.11059MR1983707
- [Vi] C. Viola, On Siegel's method in diophantine approximation to transcendental numbers. Rend. Sem. Mat. Univ. Pol. Torino53 (1995), 455-469. Zbl0873.11043MR1452398

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