When $\frac{\pi ^{x}}{\zeta (x)}$ is rational?

Question

When $n$ is a positive integer, we know

$$\zeta (2n)=\frac{(-1)^{n+1}B_{2n}(2\pi )^{2n}}{2(2n)!}$$

Now let's say $x>1$ is a real number. Can we say if $\frac{\pi ^{x}}{\zeta (x)}$ is a rational number then x must be an even integer?

Answer 1

Definitely not. $\frac{\pi^x}{\zeta(x)}$ is continuous on at least $x\in[3,4)$ (just giving a counterexample here), and non-constant, thus, it'll have rational values for some $x\in[3,4)$ (and so $x$ is not an even integer)