Irrationality of values of the Riemann zeta function

W. V. Zudilin

Izvestiya. Mathematics , Tome 66 (2002) no. 3, pp. 489-542.

Résumé

The paper deals with a generalization of Rivoal's construction, which enables one to construct linear approximating forms in 1 and the values of the zeta function $\zeta(s)$ only at odd points. We prove theorems on the irrationality of the number $\zeta(s)$ for some odd integers $s$ in a given segment of the set of positive integers. Using certain refined arithmetical estimates, we strengthen Rivoal's original results on the linear independence of the $\zeta(s)$.

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