# zbMATH — the first resource for mathematics

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
Full Text: