Let $D$ a Weil divisor on $\mathrm{Spec} (\mathbb{Z})$, so we can write :
$$ D = \sum_{p} \mathrm{ord}_p(D) [p]$$
with coefficients $\mathrm{ord}_p(D) \in \mathbb{Z}$. We define :
$$ H^0(\mathrm{Spec}(\mathbb{Z}),\mathcal{O}_{\mathrm{Spec}(\mathbb{Z})}(D))= \lbrace f \in \mathbb{Q}^{*} \; : \; (f)+D \geq 0 \rbrace \cup \lbrace 0 \rbrace$$
where $(f)$ is the principal divisor associated with the rational number $f$.
$H^0(\mathrm{Spec}(\mathbb{Z}),\mathcal{O}_{\mathrm{Spec}(\mathbb{Z})}(D))$ is a free $\mathbb{Z}$-module of rank $1$.
For any $n \in \mathbb{N}$, $H^0(\mathrm{Spec}(\mathbb{Z}),\mathcal{O}_{\mathrm{Spec}(\mathbb{Z})}(nD))$ is also a free module of rank $1$ ?
Let $\|.\|$ be a norm on $H^0(\mathrm{Spec}(\mathbb{Z}),\mathcal{O}_{\mathrm{Spec}(\mathbb{Z})}(D)) \otimes_{\mathbb{Z}} \mathbb{R}$. We denote $B(D,\|.\|)$ the set : $$ B(D,\|.\|) = \lbrace f \in H^0(\mathrm{Spec}(\mathbb{Z}),\mathcal{O}_{\mathrm{Spec}(\mathbb{Z})}(D)) \otimes_{\mathbb{Z}} \mathbb{R} \; : \; \|f\| \leq 1\rbrace$$ the unit ball for the norm. I want to compute or etablish an explicit formula for $\log \mathrm{vol} (B(D,\|.\|))$
But I don't have any strategy, I just know that I have to put a Haar measure on $\mathbb{R}$, can I choose the Lebesgue measure ? And then, what's the formula for $\mathrm{vol} (B(D,\|.\|))$ ?
If you have a reference for this kind of theory I would like (article or book).