On simultaneous diophantine approximations to $\zeta(2)$ and $\zeta(3)$

Simon Dauguet (1) , Wadim Zudilin (2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 School of Mathematical and Physical Sciences

The authors present a hypergeometric construction of rational approximations to $\zeta(2)$ and $\zeta(3)$ which allows one to demonstrate simultaneously the irrationality of each of the zeta values, as well as to estimate from below certain linear forms in 1, $\zeta(2)$ and $\zeta(3)$ with rational coefficients. A new notion of (simultaneous) diophantine exponent is introduced to formalise the arithmetic structure of these specific linear forms. Finally, the properties of this newer concept are studied and linked to the classical irrationality exponent and its generalisations given recently by S.Fischler.

Mots clés: irrationality; diophantine approximation; irrationality exponent; value of Riemann's zeta function; hypergeometric series; hypergeometric integral; recurrence equation; algorithm of creative telescoping

Domaines: Théorie des nombres [math.NT]

Simon Dauguet, Wadim Zudilin. On simultaneous diophantine approximations to $\zeta(2)$ and $\zeta(3)$. 2013. ⟨hal-00933967⟩