Displaying similar documents to “Birational transformations and values of the Riemann zeta-function”

The permutation group method for the dilogarithm

Georges Rhin, Carlo Viola (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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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.

Arithmetic of linear forms involving odd zeta values

Wadim Zudilin (2004)

Journal de Théorie des Nombres de Bordeaux

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A general hypergeometric construction of linear forms in (odd) zeta values is presented. The construction allows to recover the records of Rhin and Viola for the irrationality measures of ζ ( 2 ) and ζ ( 3 ) , as well as to explain Rivoal’s recent result on infiniteness of irrational numbers in the set of odd zeta values, and to prove that at least one of the four numbers ζ ( 5 ) , ζ ( 7 ) , ζ ( 9 ) , and ζ ( 11 ) is irrational.

Well-poised hypergeometric service for diophantine problems of zeta values

Wadim Zudilin (2003)

Journal de théorie des nombres de Bordeaux

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It is explained how the classical concept of well-poised hypergeometric series and integrals becomes crucial in studying arithmetic properties of the values of Riemann’s zeta function. By these well-poised means we obtain: (1) a permutation group for linear forms in 1 and ζ ( 4 ) = π 4 / 90 yielding a conditional upper bound for the irrationality measure of ζ ( 4 ) ; (2) a second-order Apéry-like recursion for ζ ( 4 ) and some low-order recursions for linear forms in odd zeta values; (3) a rich permutation group...

On the irrationality measure of ζ ( 2 )

Georges Rhin, Carlo Viola (1993)

Annales de l'institut Fourier

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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.

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R. C. Entriger (1971)

Gaceta Matemática

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Bounds for double zeta-functions

Isao Kiuchi, Yoshio Tanigawa (2006)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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In this paper we shall derive the order of magnitude for the double zeta-functionof Euler-Zagier type in the region 0 ≤ ℜ s j < 1 ( j = 1 , 2 ) .First we prepare the Euler-Maclaurinsummation formula in a suitable form for our purpose, and then we apply the theory of doubleexponential sums of van der Corput’s type.