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Are the elements of the set $\{\zeta(2n+1)| n\in \mathbb{N}\}$ $\mathbb{Q}$-linearly independent?

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    $\begingroup$ Not known but there are weaker results in this direction due to Rivoal and others. $\endgroup$ Dec 24, 2016 at 23:58
  • $\begingroup$ It is conjectured that the elements are even algebraically independent. $\endgroup$ Dec 26, 2016 at 20:05

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People have been exerting steady effort to prove/disprove irrationality of the Riemann zeta values in your list. Of course, $\zeta(3)$ is known to be irrational due to Roger Apery. Such investigations, among others, motivated the question of linear independence. As Felipe commented, however, not much is known, apart from the following article:

  • Rivoal, Tanguy. La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs. [There are infinitely many irrational values of the Riemann zeta function at odd integers] C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), no. 4, 267–270.

The article is available online here. Wadim Zudilin makes some improvement, see:

  • Zudilin. Arithmetic of linear forms involving odd zeta values. J. Théor. Nombres Bordeaux 16 (2004), no. 1, 251–291.
  • Zudilin. On the irrationality of the values of the Riemann zeta function. Izv. Ross. Akad. Nauk Ser. Mat. 66 (2002), no. 3, 49--102; translation in Izv. Math. 66 (2002), no. 3, 489–542.

That is the state-of-the-art for the present issue you raised.

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