We consider the Dirichlet series Z(P,A;s) = [sum ]$_m∈ A ∩ Z^n$P$^-s$(m) (s ∈ C) where P ∈ R[X$_1$, …, X$_n$] and A is an open semi-algebraic subset of R$^n$. We will say that Z(P,A;s) exists if this multiple series is absolutely convergent. In this paper we study the existence and several properties of meromorphic continuations of such series, under certain assumption on P and A. As an application, we show the existence of a finite asymptotic expansion of the counting function with support in A: N$_p$(A,t):= [sharp ] m ∈ A ∩ Z$^n$ | P(m) [les ] t} when t → +∞.
