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):
$$h^0(\widehat{D})=\log(\#H^0(\widehat{D}))$$




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Now in [this paper (van der Geer, Schoof- Effectivity of Arakelov Divisors and
the Theta Divisor of a Number Field)](https://arxiv.org/pdf/math/9802121v3.pdf), 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.



The notions of $H^0(\widehat{ D})$ and $h^0(\widehat{D})$ are not satisfactory
The Arakelov-theoretical analogue of Riemann's inequality is Minkowski's theorem. As Felipe Voloch's answer indicates, Tate's approach makes the Riemann-Roch theorem a consequence of the Poisson summation formula, and the self duality of the characteristic functions of (ultrametric) balls is crucial. Over archimedean fields, the characteristic functions of balls are no longer self-dual, but the Gaussian kernel is. It is thus natural, in the Arakelov-theory framework, to consider a mollified version of the $H^0$, as has been done by van der Geer and Schoof and, apparently independently, by Damian Rössler; see also Groenwegen.
This has also various advantages:


* A better Minkowski-type lower bound;
* A clean version of Serre duality;
* Subadditivity properties in exact sequences.


This is amply explained in recent preprint of J-B. Bost, *Theta invariants of euclidean lattices and infinite-dimensional hermitian vector bundles over arithmetic curves*, <https://arxiv.org/abs/1512.08946>, where he uses them to study infinite dimensional analogues (that will crucially appear in forthcoming work).


