Hypergeometrie et Fonction Zeta de Riemann
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The authors prove Rivoal's `'denominator conjecture'' concerning the common denominators of coefficients of certain linear forms in zeta values. These forms were recently constructed to obtain lower bounds for the dimension of the vector space over '\mathbb Q' spanned by '1,\zeta(m),\zeta(m+2),\dots,\zeta(m+2h)', where 'm' and 'h' are integers such that 'm\ge2' and 'h\ge0'. In particular, the authors immediately get the following results as corollaries: at least one of the eightnumbers '\zeta(5),\zeta(7),\dots,\zeta(19)' is irrational, and there exists an odd integer 'j' between '5' and '165' ...
The authors prove Rivoal's `'denominator conjecture'' concerning the common denominators of coefficients of certain linear forms in zeta values. These forms were recently constructed to obtain lower bounds for the dimension of the vector space over '\mathbb Q' spanned by '1,\zeta(m),\zeta(m+2),\dots,\zeta(m+2h)', where 'm' and 'h' are integers such that 'm\ge2' and 'h\ge0'. In particular, the authors immediately get the following results as corollaries: at least one of the eight
numbers '\zeta(5),\zeta(7),\dots,\zeta(19)' is irrational, and there exists an odd integer 'j' between '5' and '165' such that '1', '\zeta(3)' and '\zeta(j)' are linearly independent over '\mathbb{Q '. This strengthens some recent results. The authors also prove a related conjecture, due to Vasilyev, and as well a conjecture, due to Zudilin, on certain rational approximations of '\zeta(4)'. The proofs are based on a hypergeometric identity between a single sum and a multiple sum due to Andrews. The authors hope that it will.
Table of Contents:
Introduction et plan de l'article
Arriere plan
Les resultats principaux
Consequences diophantiennes du Theoreme $1$
Le principe des demonstrations des Theoremes $1$ a $6$
Deux identites entre une somme simple et une somme multiple
Quelques explications
Des identites hypergeometrico-harmoniques
Corollaires au Theoreme $8$
Corollaires au Theoreme $9$
Lemmes arithmetiques
Demonstration du Theoreme $1$, partie i)
Demonstration du Theoreme $1$, partie ii)
Demonstration du Theoreme $3$, partie i) et des Theoremes $4$ et $5$
Demonstration du Theoreme $3$, partie ii) et du Theoreme $6$
Encore un peu d'hypergeometrie
Perspectives
Bibliographie
numbers '\zeta(5),\zeta(7),\dots,\zeta(19)' is irrational, and there exists an odd integer 'j' between '5' and '165' such that '1', '\zeta(3)' and '\zeta(j)' are linearly independent over '\mathbb{Q '. This strengthens some recent results. The authors also prove a related conjecture, due to Vasilyev, and as well a conjecture, due to Zudilin, on certain rational approximations of '\zeta(4)'. The proofs are based on a hypergeometric identity between a single sum and a multiple sum due to Andrews. The authors hope that it will.
Table of Contents:
Introduction et plan de l'article
Arriere plan
Les resultats principaux
Consequences diophantiennes du Theoreme $1$
Le principe des demonstrations des Theoremes $1$ a $6$
Deux identites entre une somme simple et une somme multiple
Quelques explications
Des identites hypergeometrico-harmoniques
Corollaires au Theoreme $8$
Corollaires au Theoreme $9$
Lemmes arithmetiques
Demonstration du Theoreme $1$, partie i)
Demonstration du Theoreme $1$, partie ii)
Demonstration du Theoreme $3$, partie i) et des Theoremes $4$ et $5$
Demonstration du Theoreme $3$, partie ii) et du Theoreme $6$
Encore un peu d'hypergeometrie
Perspectives
Bibliographie