An estimate for the sum of a Dirichlet series in terms of the minimum of its modulus on a vertical line segment. (English. Russian original) Zbl 1253.30009

Gaisin, A. M.; Rakhmatullina, Zh G.

Sb. Math. 202, No. 12, 1741-1773 (2011); translation from Mat. Sb. 202, No. 12, 23-56 (2011).

Let \(f\) be a transcendental entire function. Set \[ M(r, f)=\max_{|z|=r}|f(z)|\quad\text{and}\quad m(r, f)=\min_{|z|=r}|f(z)|. \] We know that the behaviour of a transcendental entire in terms of the minimum of its modulus is a topical problem in function theory. When we consider the asymptotic values of an entire function \[ f(z)=\sum_{n=1}^{\infty}a_{n}z^{p_{n}}, \quad z=x+i y, \] where \(p_{n}\) is an increasing sequence of positive numbers, how can we find conditions on the sequence \(\{p_{n}\}\) ensuring that for any curve \(\gamma\) going to infinity there exists a sequence \(\{\xi_{n}\}\), \(\xi_{n}\in \gamma\), such that \[ \ln M(|\xi_{n}, f|)=(1+o(1))\ln|f(\xi_{n})|,\quad \xi_{n}\rightarrow \infty. \] This problem is called the Pólya problem in the literature.
In this paper, the authors mainly deal with this problem, and obtain results on the behaviour of the sum of an entire Dirichlet series in terms of the minimum of its modulus on a system of vertical line segments.

Reviewer: Zhibo Huang (Joensuu)

MSC:

30B50	Dirichlet series, exponential series and other series in one complex variable
30D10	Representations of entire functions of one complex variable by series and integrals

Keywords:

Dirichlet series; minimum modulus; Fejér gaps