Let $F$ be a number field such that $[F:\mathbb{Q}]=n$ and with ring of integers $O_F$. Let's put $B=\operatorname{Spec } O_F$, then an Arakelov divisor is an element of: $$Div(X)\times \bigoplus_\sigma \mathbb R[\sigma]$$ namely it can be written as $$\bigg(\sum_{\text{$\mathfrak p$ prime $\neq 0$}}n_{\mathfrak p}[\mathfrak p],\sum_{\sigma}x_{\sigma}[\sigma]\bigg)\quad \text{with $x_\sigma\in\mathbb R$, $n_{\mathfrak p}\in\mathbb Z$}$$

Now: some references say that $\sigma$ runs among the **archimedean places** of $F$ other say that $\sigma$ runs among the **complex embeddings** of $F$.

There is a substantial difference in the two cases, because in the former the dimension of the vector space of "divisors at infinity" is $r_1+r_2$, whereas in the latter it is $n=r_1+2r_2$ (as usual $r_1$ and $2r_2$ are the number of real and complex embeddings of $F$ ).

Which is the most natural approach?

In the first case the degree of an Arakelov divisor is $\sum_{\mathfrak p} n_{\mathfrak p}\log(\#(O_F/\mathfrak p))+\sum_\sigma x_\sigma$ whereas in the second case **I think** that we should put
$$\sum_{\mathfrak p} n_{\mathfrak p}\log(\#(O_F/\mathfrak p))+\sum_{\text{$\sigma$ real }} x_\sigma+\frac{1}{2}\sum_{\text{$\sigma$ complex }} x_\sigma$$

Is it correct?

**Edit:**
books/papers using places: Van De Geer,Schoof- Effectivity of Arakelov divisors of a number field,Neukirch ANT.

books/papers using embeddings: Moriwaki's book, De Jong thesis.

Moreover I'd like to understand what happens when you consider Arakelov theory on arithmetic surfaces $X\to B$. Here I think that would be meaningful to consider all the embeddings.