Séries de Dirichlet et intégrales associées à un polynôme à deux indéterminées. (Dirichlet series and integrals associated with a polynomial of two variables). (French) Zbl 0584.10022

Cassou-Nogués, Pierrette

J. Number Theory 23, 1-54 (1986).

The author studies the meromorphic continuation on \({\mathbb{C}}\) of the three following functions: \[ \sum^{\infty}_{m_ 1=1}\sum^{\infty}_{m_ 2=1}m_ 1^{\beta_ 1-1} m_ 2^{\beta_ 2-1} P(m_ 1,m_ 2)^{-s}, \]
\[ \int^{\infty}_{1}\int^{\infty}_{1}x_ 1^{\beta_ 1-1} x_ 2^{\beta_ 2-1} P(x_ 1,x_ 2)^{-x} dx_ 1 dx_ 2, \]
\[ \int^{1}_{0}\int^{1}_{0}x_ 1^{\beta_ 1-1} x_ 2^{\beta_ 2-1} P(x_ 1,x_ 2)^ s dx_ 1 dx_ 2, \] which are defined in a half plane \(Re(s)>\sigma\). It is shown that the poles are contained in a given set defined from the Newton polygon at infinity for the first two functions and from the Newton polygon at zero for the last one. A condition is given for the poles to be simple and the residue in this case. This can be used to calculate some roots of the Bernstein polynomial of P. Finally some examples are given.
{For the case of several \((>2)\) variables see the following review.}

MSC:

11M35	Hurwitz and Lerch zeta functions
32A05	Power series, series of functions of several complex variables
30B50	Dirichlet series, exponential series and other series in one complex variable

Keywords:

Dirichlet series; meromorphic continuation; poles; Newton polygon at infinity; Newton polygon at zero; residue; Bernstein polynomial; examples

Citations:

Zbl 0584.10023

References:

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