## Analogs of the Wiman-Valiron theorem for Dirichlet series whose indicators have a positive step.(English. Russian original)Zbl 0604.30003

Sov. Math. 30, No. 2, 79-82 (1986); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1986, No. 2(285), 57-59 (1986).
Suppose F is a function analytic in the half plane $$\{\sigma <0\}$$ which is represented by an absolutely convergent (for $$\sigma <0)$$ Dirichlet series $F(\sigma +it)=\sum^{\infty}_{n=1}a_ n\exp (\lambda_ n(\sigma +it)),\quad 0<\lambda_ i\to \infty.$ The author extends the Wiman-Valiron method to estimate $M(\sigma)=\sum^{\infty}_{n=1}| a_ n| \exp \lambda_ n\sigma$ in terms of the maximal term $$\mu$$ ($$\sigma)$$. For example $M(\sigma)\leq (\mu (\sigma)/| \sigma |^{1+\epsilon})[\ell n(\mu (\sigma)/| \sigma |)]^{+\epsilon},\quad \sigma \to 0$ outside an exceptional set of finite logarithmic measure.
Reviewer: A.E.Eremenko

### MSC:

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

### Keywords:

Dirichlet series; Wiman-Valiron method; maximal term