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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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