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Hypergéométrie et Fonction Zêta de Riemann
 
C. Krattenthaler Université Claude Bernard, Villeurbanne, France
T. Rivoal Université de Grenoble I, Saint-Martin d’Héres, France
Front Cover for Hypergeometrie et Fonction Zeta de Riemann
Available Formats:
Electronic ISBN: 978-1-4704-0479-6
Product Code: MEMO/186/875.E
87 pp 
List Price: $66.00
MAA Member Price: $59.40
AMS Member Price: $39.60
Front Cover for Hypergeometrie et Fonction Zeta de Riemann
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Hypergéométrie et Fonction Zêta de Riemann
C. Krattenthaler Université Claude Bernard, Villeurbanne, France
T. Rivoal Université de Grenoble I, Saint-Martin d’Héres, France
Available Formats:
Electronic ISBN:  978-1-4704-0479-6
Product Code:  MEMO/186/875.E
87 pp 
List Price: $66.00
MAA Member Price: $59.40
AMS Member Price: $39.60
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1862007
    MSC: Primary 11; Secondary 33;

    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 be possible to apply their construction to the more general linear forms constructed by Zudilin, with the ultimate goal of strengthening his result that one of the numbers \(\zeta(5),\zeta(7),\zeta(9),\zeta(11)\) is irrational.

  • Table of Contents
     
     
    • Chapters
    • 1. Introduction et plan de l’article
    • 2. Arrière plan
    • 3. Les résultats principaux
    • 4. Conséquences diophantiennes du Théorème 1
    • 5. Le principe des démonstrations des Théorèmes 1 à 6
    • 6. Deux identités entre une somme simple et une somme multiple
    • 7. Quelques explications
    • 8. Des identités hypergéométrico-harmoniques
    • 9. Corollaires au Théorème 8
    • 10. Corollaires au Théorème 9
    • 11. Lemmes arithmétiques
    • 12. Démonstration du Théorème 1, partie i)
    • 13. Démonstration du Théorème 1, partie ii)
    • 14. Démonstration du Théorème 3, partie i), et des Théorèmes 4 et 5
    • 15. Démonstration du Théorème 3, partie ii), et du Théorème 6
    • 16. Encore un peu d’hypérgéometrie
    • 17. Perspectives
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Volume: 1862007
MSC: Primary 11; Secondary 33;

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 be possible to apply their construction to the more general linear forms constructed by Zudilin, with the ultimate goal of strengthening his result that one of the numbers \(\zeta(5),\zeta(7),\zeta(9),\zeta(11)\) is irrational.

  • Chapters
  • 1. Introduction et plan de l’article
  • 2. Arrière plan
  • 3. Les résultats principaux
  • 4. Conséquences diophantiennes du Théorème 1
  • 5. Le principe des démonstrations des Théorèmes 1 à 6
  • 6. Deux identités entre une somme simple et une somme multiple
  • 7. Quelques explications
  • 8. Des identités hypergéométrico-harmoniques
  • 9. Corollaires au Théorème 8
  • 10. Corollaires au Théorème 9
  • 11. Lemmes arithmétiques
  • 12. Démonstration du Théorème 1, partie i)
  • 13. Démonstration du Théorème 1, partie ii)
  • 14. Démonstration du Théorème 3, partie i), et des Théorèmes 4 et 5
  • 15. Démonstration du Théorème 3, partie ii), et du Théorème 6
  • 16. Encore un peu d’hypérgéometrie
  • 17. Perspectives
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