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on Irrationality of odd positive integer values of the Riemann zeta function

The statement

Let $\zeta(s)$ denote the Riemann zeta function, defined for $\Re(s) > 1$ by $\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$ and extended by analytic continuation to $\mathbb{C} \setminus \{1\}$ . A real number is called irrational if it is not equal to any ratio of two integers. For each positive integer $n$ , determine whether there exists an irrational number $x$ such that $\zeta(2n+1) = x$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.