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Behavior of the Dirichlet series minimum modulus of a system of segments. (Behavior of the Dirichlet series modulus minimum of a system of segments.) (Russian. English summary) Zbl 1240.30011

Let \(\Lambda=\{\lambda_{n}\}\) be an infinite increasing sequence of positive numbers and assume that the Dirichlet series \(F(s)=\sum_{n=1}^{\infty}a_{n}e^{\lambda_{n}s}\), \(s=\sigma+it\), is absolutely convergent in the whole complex plane. Assume further that \(F\) has Fejer gaps (i.e., \(S_{\Lambda}=\sum_{n=1}^{\infty}\frac{1}{\lambda_{n}}<\infty\)) and denote \( M_{F}(\sigma)=\sup_{| t| <\infty}| F(\sigma+it)| \), \( m_{F}(\sigma, h)=\min_{| t| \leq h}| F(\sigma+it)| \), \(0<h<\infty\), and \(\mu(\sigma)=\max_{n}\{| a_{n}| e^{\lambda_{n}\sigma}\}\). The main result of the article (Theorem 1) is as follows:
Let \( \int_{0}^{\infty}\frac{c(t)}{t^{2}}dt<\infty\), where \( c(t)=\max_{\lambda_{n}\leq t}q_{n}\), \(q_{n}=-\ln | q'(\lambda_{n})|\), \( q(z)=\prod_{n=1}^{\infty}(1-\frac{z^{2}}{\lambda_{n}^{2}})\). Then there exists a set \(E\subset[0, \infty)\) of finite measure, such that for all vertical segments \(I_{H}(\sigma)=\{s=\sigma+it: | t-t_{0}| \leq H\}\), \(H=\mathrm{constant}\), and for all sufficiently large \(\sigma\) one can find a modified segment \(I^{\ast}_{\mu}(\sigma)\) outside \(E\) with the following properties:
(1) \(\operatorname{mes}(I_{H}(\sigma)\cap I^{\ast}_{H}(\sigma))\rightarrow 2H\) for \(\sigma\rightarrow\infty\);
(2) \(\ln M_{F}(\sigma+d(\sigma))<\ln M_{F}(\sigma)+o(1)\) for \(\sigma\rightarrow\infty\) outside \(E\), where \( d(\sigma)=\max_{s\in I^{\ast}_{H}(\sigma)}| \operatorname{Re} s-\sigma|\);
(3) \(\ln M_{F}(\sigma)=(1+o(1))\ln m^{\ast}_{F}(\sigma)\) for \(\sigma\rightarrow\infty\) outside \(E\), where \( m^{\ast}_{F}(\sigma)=\min _{s\in I^{\ast}_{H}(\sigma)}| F(s)|\).

MSC:

30B50 Dirichlet series, exponential series and other series in one complex variable
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