Let $X = \Bbb P^n(\Bbb Z) = \mathrm{Proj}(\Bbb Z[X_0, ..., X_n])$ be the projective space over the integers. What is the set of its closed points? Even for $n=1$, I'm not sure. I saw this question, but it doesn't look very easy to determine homogeneous maximal ideals, which give closed points in $X$ – I'm not sure to obtain all closed points like this, though.
My goal is to determined the zeta function of $X$ (what is its relation with the usual Riemann zeta). I'm not even sure what happens for $Y = \Bbb A^n(\Bbb Z)$, when $n >1$.