Let $K$ be a number field with ring of integers $O_K$. Moreover consider an Arakelov divisor $\widehat{D}\in\overline{\operatorname{Div }(\operatorname {Spec }O_K)}$, namely $$D=\sum_{\mathfrak p\;\text{prime}}n_\mathfrak p\mathfrak p+\sum_\sigma\lambda_\sigma \sigma$$ where $\sigma$ is an archimedian place i.e. a complex embedding of $K$ up to conjugation. Clearly we have the proper notion of principal Arakelov divisor $\widehat{(f)}$ for any $f\in K^\ast$. The usual (and most natural) definition of $H^0(\widehat{D})$ is the following: $$H^0(\widehat{D})=\{f\in K^\ast:v_\mathfrak p(f)\ge -n_\mathfrak p \;\forall \mathfrak p,\; v_\sigma(f)\ge-\lambda_\sigma\; \forall \sigma\}$$ where $v_\sigma$ is the adequate normalized real valuation at archimedian places. The above definition of $H^0(\widehat{D})$ in my opinion emulates perfectly the notion of $H^0(X,\mathcal O_X(D))$ in the usual geometric setting. One can show that $H^0(\widehat{D})$ is a finite set and its "dimension" is defined simply as (a multiple of): $$h^0(\widehat{D})=\log(\#H^0(\widehat{D}))$$

Now in this paper (van der Geer, Schoof- Effectivity of Arakelov Divisors and the Theta Divisor of a Number Field), the authors claim that

**the above notions of $H^0(\widehat{D})$ and $h^0(\widehat{D})$ are not satisfactory**,

and they suggest new definitions which lead to a "Riemann-Roch theorem". The point is that they don't explain why the usual objects $H^0(\widehat{D})$ and $h^0(\widehat{D})$ don't work, so I'd like to understand the motivation of their strong claim.

On the contrary it seems to me that the above objects are very useful to get the usual Riemann-Roch theorem for number fields which is due to Lang.