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Birational transformations and values of the Riemann zeta-function

Carlo Viola — 2003

Journal de théorie des nombres de Bordeaux

In his proof of Apery’s theorem on the irrationality of ζ ( 3 ) , Beukers [B] introduced double and triple integrals of suitable rational functions yielding good sequences of rational approximations to ζ ( 2 ) and ζ ( 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 ζ ( 2 ) and ζ ( 3 ) based upon a new algebraic method involving birational transformations and permutation groups...

Approssimazione diofantea, frazioni continue e misure d’irrazionalità

Carlo Viola — 2004

Bollettino dell'Unione Matematica Italiana

Nella sua accezione classica, l’approssimazione diofantea ad un dato numero irrazionale α è la ricerca degli interi positivi s tali che la distanza di s α dall’insieme dei numeri interi sia eccezionalmente piccola; cioè, detto r l’intero più vicino a s α , tali che | s α - r | = s | α - r / s | sia piccolo. Dunque interessano le approssimazioni razionali r / s ad α che rendano piccola la distanza | α - r / s | pur avendo denominatore s non eccessivamente grande. In questo articolo richiamiamo alcune nozioni fondamentali in approssimazione diofantea,...

Gruppi di permutazioni e risultati di irrazionalità

Carlo Viola — 2008

Bollettino dell'Unione Matematica Italiana

We recall some basic concepts in diophantine approximation, in particular the notion of irrationality measure. We describe the main aspects of the permutation group method due to G. Rhin and the author, with some arithmetical applications.

On the irrationality measure of ζ ( 2 )

Georges RhinCarlo Viola — 1993

Annales de l'institut Fourier

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 2 . The numerical results are obtained with the aid of a semi-infinite linear programming method.

The permutation group method for the dilogarithm

Georges RhinCarlo Viola — 2005

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We give qualitative and quantitative improvements on all the best previously known irrationality results for dilogarithms of positive rational numbers. We obtain such improvements by applying our permutation group method to the diophantine study of double integrals of rational functions related to the dilogarithm.

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