### A generalization of a result of André-Jeannin concerning summation of reciprocals.

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Let $F$ be a class of entire functions represented by Dirichlet series with complex frequencies $\sum {a}_{k}{\mathrm{e}}^{\langle {\lambda}^{k},z\rangle}$ for which $\left(\right|{\lambda}^{k}{|/\mathrm{e})}^{|{\lambda}^{k}|}k!\left|{a}_{k}\right|$ is bounded. Then $F$ is proved to be a commutative Banach algebra with identity and it fails to become a division algebra. $F$ is also proved to be a total set. Conditions for the existence of inverse, topological zero divisor and continuous linear functional for any element belonging to $F$ have also been established.

The questions considered in this paper arose from the study [KS] of I. Fredholm's (insufficient) proof that the gap series Σ0∞ an ζn2 (where 0 < |a| < 1) is nowhere continuable across {|ζ| = 1}. The interest of Fredholm's method ([F],[ML]) is not so much its efficacy in proving gap theorems (indeed, much more general results can be got by other means, cf. the Fabry gap theorem in [Di]) as in the connection it made between certain special gap series and partial differential equations...

Nous étudions les fonctions $p$-adiques associées à des séries du type$$Z(P,Q,\xi )\left(s\right)=\sum _{n\in {\mathbf{N}}^{r}}Q\left(n\right){\xi}^{n}P(n{)}^{-s}$$dans certains cas, où elles admettent un prolongement méromorphe à $C$ avec un nombre fini de pôles et des valeurs aux entiers négatifs algébriques. On retrouve comme cas particulier les fonctions $L$$p$-adiques des corps totalement réels et les fonctions $\Gamma $-multiples $p$-adiques.

The spaces of entire functions represented by Dirichlet series have been studied by Hussein and Kamthan and others. In the present paper we consider the space $X$ of all entire functions defined by vector-valued Dirichlet series and study the properties of a sequence space which is defined using the type of an entire function represented by vector-valued Dirichlet series. The main result concerns with obtaining the nature of the dual space of this sequence space and coefficient multipliers for some...

In this work, we begin with a survey of composition operators on the Hardy space H² and on the Wiener algebra A⁺ of absolutely convergent Taylor series, with special emphasis on their compactness, or invertibility, or isometric character. The main results are due respectively to J. Shapiro and D.~Newman. In a second part, we present more recent results, due to Gordon and Hedenmalm on the one hand, and to Bayart, the author et al. on the other hand, concerning the analogues of H² and A⁺ in the setting...