zbMATH — the first resource for mathematics

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 $$L(s, \chi)$$
Zbl 0973.11072
Full Text: