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Croissance de certaines séries de Dirichlet et applications. (Growth of certain Dirichlet series and applications). (French) Zbl 0575.10035

Let \(P(x)=P(x_ 1,...,x_ n)\) be a polynomial with positive coefficients. Let \(Z_ p(s)=\sum^{\infty}_{\nu_ 1,...,\nu_ n=1}P(\nu)^{-s}\) be the Dirichlet series associated with P. We study the order of growth of the meromorphic continuation of \(Z_ p(s)\) in vertical strips. From it, we deduce that the number of lattice points: \[ N_ P(\lambda)=\#\{\nu \in {\mathbb{N}}^{*n}: P(\nu)\leq \lambda \} \] admits an asymptotic expansion as \(\lambda \to +\infty\).

MSC:

11M35 Hurwitz and Lerch zeta functions
11P21 Lattice points in specified regions
30B50 Dirichlet series, exponential series and other series in one complex variable
30B40 Analytic continuation of functions of one complex variable
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