  We consider series of the form $\sum a_n \{n\cdot x\}$, where $n\in\Z^{d}$
and $\{x\}$ is the sawtooth function. They are the natural multivariate
extension of Davenport series. Their global (Sobolev) and pointwise regularity
are studied and their multifractal properties are derived. Finally, we list
some open problems which concern the study of these series.
