Let $a >0$ a real number. I want to compute the cardinality of the set :
$$ \lbrace x \in \mathbb{Q}^{*} \; | \; |x| \leq a \; \text{and } \; v_p(x) \geq 0,\; \forall p \in \mathcal{P} \rbrace $$
For context, this set appears when I consider an Archimedean Arakelov divisor, of the form $D = \ln(a) ( \infty)$ on $\mathrm{spec}(\mathbb{Z})$. And we have : $$ H^0(D) = \lbrace x \in \mathbb{Q}^{*} \; | \; (x) + D \geq 0 \rbrace $$