Is there any value of zeta that is an integer?

Question

Is there any value which we can substitute for $s$ in $\zeta (s)$ such that $$\sum_{n=1}^{\infty }n^{-s}\in \mathbb{Z}$$

Answer 1

$\zeta(s)$ is a continuous function with $\lim_{s \rightarrow \infty} \zeta(s) = 1$ and $\lim_{s \rightarrow 1} \zeta(s) = \infty$ (on the real line), so it takes every positive integer at some $s > 1$. Just the intermediate value theorem.

Answer 2

When we analytically continue $\zeta$ to values of $s$ where the series doesn't converge, we get the famous Riemann zeta function, for which $\zeta(-2)=0$. That might be the easiest integer value to write down!