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A generalization of a theorem of Schinzel

Georges Rhin — 2004

Colloquium Mathematicae

We give lower bounds for the Mahler measure of totally positive algebraic integers. These bounds depend on the degree and the discriminant. Our results improve earlier ones due to A. Schinzel. The proof uses an explicit auxiliary function in two variables.

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.

Periodic Jacobi-Perron expansions associated with a unit

Brigitte AdamGeorges Rhin — 2011

Journal de Théorie des Nombres de Bordeaux

We prove that, for any unit ϵ in a real number field K of degree n + 1 , there exits only a finite number of n-tuples in  K n which have a purely periodic expansion by the Jacobi-Perron algorithm. This generalizes the case of continued fractions for n = 1 . For n = 2 we give an explicit algorithm to compute all these pairs.

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