Singularities of Dirichlet series associated to polynomials of several variables and application to the analytic number theory

We consider the Dirichlet series [GRAPHICS] We will say that Z(P; s) exists if this multiple series is absolutely convergent. In this paper we study meromorphic continuations of such series, under the assumptions that there exists a constant B is an element of]0, 1[such that : i) P(x) --> + infinity when parallel to x parallel to --> +infinity and x is an element of [B, +infinity[n, and ii) d(Z(P), [B, +infinity[(n)) > 0 where Z(P) = {z is an element of C-n\P(z) = 0}. This assumption is probably optimal, and in any way strictly includes all classes of polynomials previously treated. Under this assumption, we prove the existence of meromorphic continuation of Dirichlet series, we give a set of candidate poles and an upper bound to the orders of these poles. Moreover we obtain bounds for these meromorphic continuation on vertical bands. As an application, we show the existence of a finite asymptotic expansion of the counting function : N-P(t)= #{m is an element of N-*n\P(m) less than or equal to t} when t --> +infinity.