TITLE: Does $\pi_1(Spec(\mathbb{Z}[1/p]))$ depend on p?
QUESTION [14 upvotes]: How difficult is it to know what $\pi_1(Spec(\mathbb{Z}[1/(p_1...p_r)]))$ is? Is it independent of the choice of $p_1,...,p_r$? When is it known, and what is known about it?

REPLY [13 votes]: To add on to Pete's answer, let me comment that the differences are even more pronounced if we look at the maximal pro-$p$ quotient $\pi_1(\operatorname{Spec}(\mathbb{Z}\left[\frac{1}{p_1p_2\cdots p_r}\right])^{(p)}$ of this etale fundamental group.  For example, if $r\geq 4$, then this group is infinite, in fact non-$p$-adic-analytic, if each $p_i\equiv 1\pmod{p}$ and is trivial if each $p_i\not\equiv1\pmod{p}$.  The latter is basically for stupid reasons (only primes which are 1 mod p can ramify in a $p$-extension).  But even ignoring stupid cases, there's a lot of fantastic arithmetic going on here.  For example, $\pi_1(\operatorname{Spec}(\mathbb{Z}\left[\frac{1}{19\cdot 103}\right])^{(2)}$ is finite whereas $\pi_1(\operatorname{Spec}(\mathbb{Z}\left[\frac{1}{17\cdot 103}\right])^{(2)}$ is infinite, results which stem from simple quadratic residue calculations.  Figuring out to generate these kinds of results more generally is an active and difficult area of research.