Memoirs of the American Mathematical Society
2007;
87 pp;
Softcover
MSC: Primary 11;
Secondary 33
Print ISBN: 978-0-8218-3961-4
Product Code: MEMO/186/875
List Price: $66.00
AMS Member Price: $39.60
MAA Member Price: $59.40
Electronic ISBN: 978-1-4704-0479-6
Product Code: MEMO/186/875.E
List Price: $66.00
AMS Member Price: $39.60
MAA Member Price: $59.40
Hypergéométrie et Fonction Zêta de Riemann
Share this pageC. Krattenthaler; T. Rivoal
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
Table of Contents
Hypergeometrie et Fonction Zeta de Riemann
- Table des matières v6 free
- Remerciements ix10 free
- Chapitre 1. Introduction et plan de l'article 112 free
- Chapitre 2. Arrière plan 314 free
- Chapitre 3. Les résultats principaux 1122
- Chapitre 4. Conséquences diophantiennes du Théorème 1 1324
- Chapitre 5. Le principe des demonstrations des Théorèmes 1 à 6 1526
- Chapitre 6. Deux identités entre une somme simple et une somme multiple 1930
- Chapitre 7. Quelques explications 2334
- Chapitre 8. Des identités hypergéométrico-harmoniques 2738
- Chapitre 9. Corollaires au Théorème 8 3748
- Chapitre 10. Corollaires au Théorème 9 3950
- Chapitre 11. Lemmes arithmétiques 4354
- Chapitre 12. Démonstration du Théorème 1, partie i) 5566
- Chapitre 13. Démonstration du Théorème 1, partie ii) 5970
- Chapitre 14. Démonstration du Théorème 3, partie i), et des Théorèmes 4 et 5 6374
- Chapitre 15. Démonstration du Théorème 3, partie ii), et du Théorème 6 6778
- Chapitre 16. Encore un peu d'hypérgéometrie 7586
- Chapitre 17. Perspectives 7990
- Bibliographie 8596