Well-poised hypergeometric service for diophantine problems of zeta values

Zudilin, Wadim

Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 2, pp. 593-626.

On montre comment les concepts classiques de séries et intégrales hypergéométriques bien équilibrées devient crucial dans l’étude des propriétés arithmétiques des valeurs de la fonction zêta de Riemann. Par ces arguments, on obtient (1) un groupe de permutations pour les formes linéaires en $1$ et $\zeta \left(4\right)={\pi }^4/90$ donnant une majoration conditionnelle de la mesure d’irrationalité de $\zeta \left(4\right)$ ; (2) une récurrence d’ordre deux pour $\zeta \left(4\right)$ semblable à celles introduites par Apéry pour $\zeta \left(2\right)$ et $\zeta \left(3\right)$, ainsi que des récurrences d’ordre réduit pour les formes linéaires en des valeurs de la fonction zêta aux entiers impairs ; (3) un gros groupe de permutations pour une famille d’intégrales multiples généralisant les intégrales dites de Beukers pour $\zeta \left(2\right)$ et $\zeta \left(3\right)$.

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 $\zeta \left(4\right)={\pi }^4/90$ yielding a conditional upper bound for the irrationality measure of $\zeta \left(4\right)$; (2) a second-order Apéry-like recursion for $\zeta \left(4\right)$ and some low-order recursions for linear forms in odd zeta values; (3) a rich permutation group for a family of certain Euler-type multiple integrals that generalize so-called Beukers’ integrals for $\zeta \left(2\right)$ and $\zeta \left(3\right)$.

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