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Growth of coefficients of universal Dirichlet series. (English) Zbl 1121.30002
The paper deals with the class \({\mathcal{D}}_a({\mathbb{C}}_{+})\) of all Dirichlet series \(f(s)=\sum_{n\geq 1}a_n n^{-s},\) which are absolutely convergent on \({\mathbb{C}}_{+}=\{s:\;\operatorname{Re} s >0 \}.\)
Definition. The compact set \(K\subset {\mathbb{C}}\) is admissible for Dirichlet series if \({\mathbb{C}}\backslash K\) is connected and \(K\) can be written as \(K=K_1 \cup K_2 \cup \ldots \cup K_d,\) where each \(K_i\) contains in a strip \(S_i=\{s\in {\mathbb{C}}: \;a_i \;\leq \;\operatorname{Re} s \;\leq b_i \}\) with \(b_i-a_i < \frac{1}{2}, \) and the strips \(S_i\) are disjoint.
The authors consider two classes of Dirichlet series \(f \in {\mathcal{D}}_a({\mathbb{C}}_{+})\). The class \({\mathcal{W}}_a\) consists of all Dirichlet series \(f \in {\mathcal{D}}_a({\mathbb{C}}_{+}), \;f(s)=\sum_{k=1}a_k k^{-s},\) satisfying the following condition: for every admissible compact set \(K \subset \{s:\;\operatorname{Re} s \leq 0 \} \) and every function \(g,\) continuous on \(K\) and analytic in \(\overset{\circ}{K},\) there exists a sequence \((\lambda _n)_{n\geq 0}\) of integers such that \(\sup_{s\in K}| \sum_{k=1}^{\lambda _n}a_k k^{-s} - g(s)| \to 0, \;\text{when} \;n\to \infty.\) The class \({\mathcal{W}}_1\) consists of all Dirichlet series \(f \in {\mathcal{D}}_a({\mathbb{C}}_{+}), \;f(s)=\sum_{k=1}a_k k^{-s},\) satisfying the following condition: for every admissible compact set \(K \subset \{s:\;\operatorname{Re} s < 0 \} \) and every function \(g,\) continuous on \(K\) and analytic in \(\overset{\circ}{K},\) there exists a sequence \((\lambda _n)_{n\geq 0}\) of integers such that \(\sup_{s\in K}| \sum_{k=1}^{\lambda _n}a_k k^{-s} - g(s)| \to 0, \;\text{when} \;n\to \infty.\)
The authors obtain the following estimates on the growth of coefficients of \(f \in {\mathcal{W}}_a:\) \[ \limsup_{n \in {\mathbb{N}}} \frac{n| a_n| }{e^{\sqrt{b_n \log n}}}\;= \;\infty, \] where \((b_n)_{n \in {\mathbb{N}}}\) is a decreasing sequence such that \(\sum _{n=2}^{\infty}b_n/n \log n < \infty.\) Besides this the following decomposition theorem is proved.
Theorem. Let \(f(s)=\sum_{n\geq 1}d_n n^{-s}\) be a Dirichlet series in \({\mathcal{D}}_a({\mathbb{C_{+}}}).\) Then there exist \(g_1(s)=\sum_{n\geq 1}a_n n^{-s}\) and \(g_2(s)=\sum_{n\geq 1}b_n n^{-s}\) in \({\mathcal{W}}_1\) such that \(f=g_1+g_2 \) on \({\mathbb{C_{+}}}\) with \(\limsup n | a_n| = \limsup n | b_n| = \limsup n | d_n| .\)
In the paper there is also obtained the relationship between several classes of Dirichlet series.

MSC:
30B50 Dirichlet series, exponential series and other series in one complex variable
41A28 Simultaneous approximation
46B25 Classical Banach spaces in the general theory
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