# values of $\zeta$ function are linearly independent?

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

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

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: