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Some properties of universal Dirichlet series

  • A. Mouze
Article
  • 27 Downloads

Abstract

We establish some properties of universal Dirichlet series. In particular we give a new estimate on the growth of their coefficients. As a consequence we obtain an information about the admissible size of coefficients of Dirichlet polynomials that approximate a given entire function on a compact set. Moreover we prove that, for all \(\alpha >-1\), the sequence of Riesz means \(\left( \left( \sum _{k=1}^nk^{\alpha }\right) ^{-1}\sum _{k=1}^n k^{\alpha }D_k(f)\right) \) of partial sums of an universal Dirichlet series f is automatically universal. Finally we show that the Dirichlet series satisfying the universal approximation property with respect to every compact set K (with connected complement) contained in a strip \(\{z\in {\mathbb {C}}:\sigma \le \mathfrak {R}(z)\le 0\}\) are not necessarily universal in the left half-plane \(\{z\in {\mathbb {C}}:\mathfrak {R}(z)\le 0\}\).

Keywords

Universal Dirichlet series Boundary behavior of Dirichlet series Over-convergence Approximation in the complex domain 

Mathematics Subject Classification

30K10 30E10 

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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire Paul Painlevé, UMR 8524Villeneuve d’Ascq CedexFrance
  2. 2.École Centrale de LilleVilleneuve d’Ascq CedexFrance

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