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Analytic continuation of Dirichlet series.

J. Milne Anderson, Dimitry Khavinson, Harold S. Shapiro (1995)

Revista Matemática Iberoamericana

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...

Applications arithmétiques de l'étude des valeurs aux entiers négatifs des séries de Dirichlet associées à un polynôme

Philippe Cassou-Noguès (1981)

Annales de l'institut Fourier

Nous étudions les fonctions p -adiques associées à des séries du type Z ( P , Q , ξ ) ( s ) = n N r Q ( n ) ξ 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 Γ -multiples p -adiques.

Composition operators in the Dirichlet series setting

Hervé Queffélec (2007)

Banach Center Publications

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...

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