Some properties of universal Dirichlet series

• Published in: Monatshefte für Mathematik Vol. 189; no. 3; pp. 487 - 506
• Main Author: Mouze, A.
• Format: Journal Article
• Language: English
• Published: Vienna Springer Vienna 01.07.2019; Springer Nature B.V; Springer Verlag [1948-....]
• Subjects: Dirichlet problem; Entire functions; Mathematics; Mathematics and Statistics; Polynomials; 30K10; Universal Dirichlet series; 30E10; Approximation in the complex domain; Boundary behavior of Dirichlet series; Over-convergence
• ISSN: 0026-9255; 1436-5081
• DOI: 10.1007/s00605-018-1208-5

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 α > - 1 , the sequence of Riesz means ∑ k = 1 n k α - 1 ∑ k = 1 n k α D k ( f ) 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 ∈ C : σ ≤ R ( z ) ≤ 0 } are not necessarily universal in the left half-plane { z ∈ C : R ( z ) ≤ 0 } .