Geodesic
Irrationality of values of the Riemann zeta function
Izvestiya. Mathematics , Tome 66 (2002) no. 3, pp. 489-542

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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 $\zeta(s)$. 
@article{IM2_2002_66_3_a2,
     author = {W. V. Zudilin},
     title = {Irrationality of values of the {Riemann} zeta function},
     journal = {Izvestiya. Mathematics },
     pages = {489--542},
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     volume = {66},
     number = {3},
     year = {2002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2002_66_3_a2/}
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W. V. Zudilin. Irrationality of values of the Riemann zeta function. Izvestiya. Mathematics , Tome 66 (2002) no. 3, pp. 489-542. http://geodesic.mathdoc.fr/item/IM2_2002_66_3_a2/