Convergence of $\zeta(k)$ to arbitrary rational number

Question

I have a question about Riemann's hypothesis for $s>1$, we know that $\displaystyle\zeta(s)=\sum_{n\ge 1}\dfrac{1}{n^s}$ converges, yet $\zeta(2)=\dfrac{\pi^2}{6}$ and $\zeta(4)$ and so forth are irrational. I wonder that there exists $s>1$ for which $\zeta(s)\in\mathbb{Q}$? Or even more general, is there exists $s>1$ for arbitrary rational number $q$ that $\displaystyle q=\zeta(s)$.

Answer 1

$\zeta$ is continuous on $(1,+\infty)$, strictly decreasing, $\lim\limits_{s\rightarrow 1}\zeta(s)=+\infty$ and $\lim\limits_{s\rightarrow +\infty}\zeta(s)=1$, thus for all rationnals $q>1$, there exists a unique $s_q>1$ such that $\zeta(s_q)\in\mathbb{Q}$.