TITLE: When does the zeta function take on integer values?
QUESTION [22 upvotes]: 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'?

REPLY [7 votes]: There are in fact rather precise conjectures as to the location of roots $s$ of the equation $\zeta(s) = a$ for any $a$ complex (this includes $a = n$ of course). 
For $a = 0$ this is the Riemann Hypothesis. 
For $a \neq 0$ it is conjectured that half of the points cluster slightly to the left of the half-line (precisely $\sqrt{\log\log t}/\log t$ to the left, if $t$ is the imaginary part) while the other half clusters very closely to the half-line (possibly $o(1/\log t)$). Around each of these lines the zeros spread uniformly to the left and right. In particular this led Selberg to conjecture that $3/4$ of the $a$-points lie to the right of the half-line while $1/4$ lies to the right. Selberg complemented this with the conjecture that on any given line, there are at most two $a$-points. Selberg has in addition obtained a central limit theorem for the distribution of the $a$-points around $1/2 - c \sqrt{\log\log t}/\log t$ with $c$ varying. A good reference for these questions are the paper of Selberg mentionned above, the thesis of Kai-Man Tsang (written under Selberg), which can be found on the internet, and finally for some more recent developments I would refer you to the papers: 
http://arxiv.org/abs/1402.0169
http://arxiv.org/abs/1402.6682
http://www.ams.org/journals/proc/2012-140-12/S0002-9939-2012-11275-4/home.html