nt.number theory - 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.