Singularités de séries de Dirichlet associées à des polynômes de plusieurs variables et applications en théorie analytique des nombres. (Singularities of Dirichlet series associated to polynomials of several variables and applications to analytic number theory). (French) Zbl 0882.11051

Summary: The author considers the Dirichlet series \(s\mapsto Z(P;s) =\sum\limits_{m\in{\mathbb N}^{*n}}P(m)^{-s} (s\in{\mathbb C})\) where \(P\in{\mathbb R}[X_1,...,X_n]\). Say that \(Z(P;s)\) exists if this multiple series is absolutely convergent. In this paper he studies meromorphic continuations of such series, under the assumptions that there exists a constant \(B\in]0,1[\) such that: i) \(P(x)\to +\infty\) when \(|x|\to + \infty\) and \(x \in [B,+\infty[^n\) and ii) \(d(Z(P),\;[B,+\infty[^n)>0\) where \(Z(P)=\{ z \in{\mathbb C}^n\mid P(z)=0\}\). This assumption is probably optimal, and in any way strictly includes all classes of polynomials previously treated. Under this assumption, he proves the existence of meromorphic continuations of Dirichlet series and gives a set of candidate poles and an upper bound to the orders of these poles. Moreover the author obtains bounds for these meromorphic continuations on vertical bands. As an application, he shows the existence of a finite asymptotic expansion of the counting function: \[ N_P(t)=\#\{ m \in {\mathbb N}^{*n}\mid P(m)\leq t \} \text{ when } t\to + \infty. \]


11M41 Other Dirichlet series and zeta functions
11P21 Lattice points in specified regions
14P10 Semialgebraic sets and related spaces
32A20 Meromorphic functions of several complex variables
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
14B05 Singularities in algebraic geometry
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