Here $\zeta(s)$ is the usual Riemann zeta function, defined as $\sum_{n=1}^\infty n^{-s}$ for $\Re(s)>1$.

Let $A_n=${$s\;:\;\zeta(s)=n$}. The behaviour of $A_0$ is basically just the Riemann hypothesis; my question concerns $A_n$ for $n\neq0$.

1) Is determining this just as hard as the Riemann hypothesis?

2) If we know the behaviour of some $A_n$, does it help in deducing the behaviour of other $A_m$?

3) For which $n$ is $A_n$ non-empty?

Question 3 has now been answered for all strictly positive $n$ - it is non-empty, and has points on the real line to the left of $s=1$. For $n=0$, it is known to be non-empty. Any idea for negative $n$? (the same answer won't work, since $\zeta(s)$ is strictly positive on the real line to left of $s=1$. Big Picard gives it non-empty for all but at most one $n$. How can we remove the 'at most one'?