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

Gal’, Yu. M.

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