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On the irrationality of the values of the zeta function at odd integer points. (English. Russian original) Zbl 1037.11048
Russ. Math. Surv. 56, No. 2, 423-424 (2001); translation from Usp. Mat. Nauk 56, No. 2, 215-216 (2001).
Generalizing the construction introduced by T. Rivoal [C. R. Acad. Sci., Paris, Sér. I, Math. 331, 267–270 (2000; Zbl 0973.11072)], the author establishes the following results:
Theorem 1. Each of the sets $$\{\zeta(k)$$, $$k=5,7, \dots, 21\}$$, $$\{\zeta(k)$$, $$k=7,9, \dots, 37\}$$, $$\{\zeta(k)$$, $$k= 9, 11, \dots, 53\}$$ contains at least one irrational number.
Theorem 2: There is an odd integer $$a\leq 145$$ such that 1, $$\zeta(3)$$ and $$\zeta(a)$$ are linearly independent over $$\mathbb{Q}$$.
Theorem 3: Let $$a\geq 3$$ be any odd integer, and let $$\delta(a)$$ be the dimension of the space spanned over $$\mathbb{Q}$$ by the numbers $$1, \zeta(3), \zeta(5), \dots, \zeta(a-2), \zeta(a)$$, then $\delta(a)> 0.395 \log a> 2\log a/3 (1+\log 2).$ Theorem 2 improves T. Rivoal’s result [Rapport de recherche SDAD, No. 2000-9, Univ. Caen (2000)], in which $$a\leq 169$$ is assumed. Proofs are only sketched.

##### MSC:
 11J72 Irrationality; linear independence over a field 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$
##### Keywords:
irrationality; linear independence; Riemann zeta-function
Zbl 0973.11072
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