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Irrationality of values of the Riemann zeta function. (English. Russian original) Zbl 1114.11305
Izv. Math. 66, No. 3, 489-542 (2002); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 66, No. 3, 49-102 (2002).
The paper deals with a generalization of Rivoal’s construction [T. Rivoal, C. R. Acad. Sci., Paris, Sér. I Math. 331, No. 4, 267–270 (2000; Zbl 0973.11072)], which enables one to construct linear approximating forms in 1 and the values of the zeta function \(\zeta(s)\) only at odd points. The author proves 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, the author strengthens Rivoal’s original results on the linear independence of \(\zeta(s)\). Some examples:
Theorem: For any odd positive integer \(b\) at least one of \(\zeta(b+2), \zeta(b+4),\ldots,\zeta(8b-1)\) is irrational.
Theorem: Let \(a\geq3\) be an odd integer. Then the \(\mathbb Q\)-rank \(\delta(a)\) of the numbers \(1,\zeta(3),\zeta(5),\ldots,\zeta(a)\) is bounded below by \(\delta(a)> 0.395 \log a\).
The main ingredient in the proofs is a special case of Nesterenko’s theorem giving a criterion for linear independence.

MSC:
11J72 Irrationality; linear independence over a field
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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