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On the root extraction from an exponential sum. (Russian) Zbl 0624.30004

A \(G_{\rho}\)-quasipolynomial is a finite sum \[ (1)\quad \sum a_ k(z)\exp (\lambda_ kz^{\rho}), \] where \(a_ k(z)\) are functions of at most minimal type of order \(\rho\). The main result is as follows. Theorem 1. If an entire function f(z) is such, that \((f(z))^ n\) is a \(G_{\rho}\)-quasipolynomial, then f(z) is a \(G_{\rho}\)- quasipolynomial, too.
This theorem is a natural generalization of a statement by Selberg, who considered finite sums in the form (1), where \(a_ k(z)\) are entire functions of order less than \(\rho\). For the proof results by B. Ya. Levin and the author [Dokl. Akad. Nauk SSSR 280, 288-291 (1985; Zbl 0596.30039)] were used concerning special asymptotic series.

MSC:

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

Citations:

Zbl 0596.30039
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