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Irrationality of infinitely many values of the zeta function at odd integers. (Irrationalité d’une infinité de valeurs de la fonction zêta aux entiers impairs.) (French) Zbl 1058.11051
The authors prove that the dimension over $${\mathbb Q}$$ of the vector space generated by the values of the zeta functions at odd integers $$\geq 3$$ is infinite. More precisely, let $$a$$ be an odd integer $$\geq 3$$ and let $$\delta(a) = \dim_{\mathbb Q} ({\mathbb Q}+{\mathbb Q}\zeta(3)+{\mathbb Q}\zeta(5)+\cdots+{\mathbb Q}\zeta(a))$$. Then: (i) $$\delta(a) \geq \frac 13 \log(a)$$ and (ii) for any $$\varepsilon > 0$$ there exists an integer $$A(\varepsilon)$$ such that if $$a> A(\varepsilon)$$ then $$\delta(a) \geq \frac{1-\varepsilon}{1+\log(2)} \cdot \log(a)$$. As announced in the paper T. Rivoal [C. R. Acad. Sci., Paris, Sér. I, Math. 331, No. 4, 267–270 (2000; Zbl 0973.11072)], the proof uses a clever modification of the method of Nikishin for the approximation of the values of polylogarithms at rational numbers to get small linear combinations of $$1,\zeta(3),\zeta(5),\dots$$ and a linear independence criterion of Nesterenko to conclude the proof.
Meanwhile further progress has been made on this subject. In particular, T. Rivoal [Acta Arith. 103, No. 2, 157–167 (2002; Zbl 1015.11033)] and W. Zudilin [Russ. Math. Surv. 56, No. 4, 774–776 (2001); translation from Usp. Mat. Nauk 56, No. 4, 149–150 (2001; Zbl 1047.11072)] have proved that least one of the nine numbers $$\zeta(5), \zeta(7),\dots,\zeta(21)$$ (resp. one of the four numbers $$\zeta(5), \zeta(7),\zeta(9),\zeta(11)$$) is irrational.

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