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'?



When does the zeta function take on integer values?
Regarding 3), this "Big Picard" stuff is serious overkill.


Think like an undergraduate real analysis student:


The p-series $\zeta(p)$ converges for real $p > 1$, whereas $\zeta(1)$ = sum of the harmonic series = oo.


An easy argument using (e.g.) the integral test shows that


$$lim\_{p \rightarrow \infty} \zeta(p) = 1$$


The function $\zeta(p)$ is continuous in p [the convergence is uniform on right half-planes, hence on compact subsets], so by the intermediate value theorem it takes on every positive integer value $n \ge 2$ at least once -- and, since it is a decreasing function of p, exactly once -- on the real line.


Thus $A\_n$ is nonempty for all $n > 1$.


EDIT: Let me show that zeta(s) takes on all real values infinitely many times on the negative real axis. 


For this, note that for all $n > 0$, 


$$\zeta(-(2n-1)) = - \frac{B\_{2n}}{(2n)}$$, 


where $B\_{2n}$ is the $(2n)$th Bernoulli number. It is known that the $B\_{2n}$'s alternate in sign and grow rapidly in absolute value:


$$|B\_{2n}| \sim 4 \sqrt{\pi n} \left(\frac{n}{(\pi e)^{2n}}\right)$$


The claim follows from this and the Intermediate Value Theorem.


