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Uniqueness theorems for entire Dirichlet series. (Russian) Zbl 0634.30002

One of the typical results on uniqueness of Dirichlet series is as follows. Let \(\Phi\) be a positive convex function on R increasing to \(+\infty\) and let \(\phi\) be the inverse function. If an entire function \[ F(z)=\sum_{n\geq 1}a_ n \exp z\lambda_ n\not\equiv 0,\quad 0=\lambda_ 1<...<\lambda_ n\to \infty \] is such that \(| F(x)| \leq const\), for all \(x\in {\mathbb{R}}\) and \[ \sum_{n\geq 1}| a_ n| \exp x\lambda_ n\leq \Phi (x), \] then \[ \liminf_{R\to \infty} \phi^{-1}(R)\sum_{\lambda_ n<R}1/\lambda_ n\geq 1/2. \]
Reviewer: V.A.Tkachenko

MSC:

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