Gram Points in the Universality of the Dirichlet Series with Periodic Coefficients

Let a={am:m∈N} be a periodic multiplicative sequence of complex numbers and L(s;a), s=σ+it a Dirichlet series with coefficients am. In the paper, we obtain a theorem on the approximation of non-vanishing analytic functions defined in the strip 1/2<σ<1 via discrete shifts L(s+ihtk;a), h>0, k...

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Published in:Mathematics (Basel) Vol. 11; no. 22; p. 4615
Main Authors: Šiaučiūnas, Darius, Tekorė, Monika
Format: Journal Article
Language:English
Published: Basel MDPI AG 01.11.2023
Subjects:
Analytic functions
Approximation
approximation of analytic functions
Complex numbers
Dirichlet problem
Mathematical analysis
Mathematical research
Prime numbers
Series, Dirichlet
space of analytic functions
Theorems
universality
weak convergence
Lithuania
ISSN:2227-7390, 2227-7390
Online Access:Get full text
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Summary:Let a={am:m∈N} be a periodic multiplicative sequence of complex numbers and L(s;a), s=σ+it a Dirichlet series with coefficients am. In the paper, we obtain a theorem on the approximation of non-vanishing analytic functions defined in the strip 1/2<σ<1 via discrete shifts L(s+ihtk;a), h>0, k∈N, where {tk:k∈N} is the sequence of Gram points. We prove that the set of such shifts approximating a given analytic function is infinite. This result extends and covers that of [Korolev, M.; Laurinčikas, A. A new application of the Gram points. Aequat. Math. 2019, 93, 859–873]. For the proof, a limit theorem on weakly convergent probability measures in the space of analytic functions is applied.
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ISSN:2227-7390
2227-7390
DOI:10.3390/math11224615