Birational transformations and values of the Riemann zeta-function

@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 -