Sargos, Patrick Croissance de certaines séries de Dirichlet et applications. (Growth of certain Dirichlet series and applications). (French) Zbl 0575.10035 J. Reine Angew. Math. 367, 139-154 (1986). 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 \(\lambda \to +\infty\). Cited in 1 ReviewCited in 7 Documents 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 Keywords:Dirichlet series; order of growth; meromorphic continuation; number of lattice points; asymptotic expansion PDFBibTeX XMLCite \textit{P. Sargos}, J. Reine Angew. Math. 367, 139--154 (1986; Zbl 0575.10035) Full Text: DOI Crelle EuDML