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Regularity of the growth of Dirichlet series with respect to a strongly incomplete exponential system
A. M. Gaisina,
R. A. Gaisina,
T. I. Belousb a Institute of Mathematics with Computing Centre — Subdivision of the Ufa Federal Research Centre of the Russian Academy of Sciences, Ufa
b Ufa University of Science and Technology
Abstract:
The article deals with the behavior of the sum of the Dirichlet series
$F(s)=\sum\nolimits_{n} a_ne^{\lambda_ns}$, with
$0<\lambda_{n}\uparrow\infty$, converging absolutely in the left half-plane
$\Pi_0=\{ s=\sigma+it: \sigma<0\}$ along a curve arbitrarily approaching the imaginary axis, the boundary of this half-plane. We assume that the maximal term of the series satisfies some lower estimate on some sequence of points
$ \sigma_n \uparrow 0-$. The essence of the questions we consider is as follows: Given a curve
$\gamma$ starting from the half-plane
$\Pi_0$ and ending asymptotically approaching on the boundary of
$\Pi_0$, what are the conditions for the existence of a sequence
$ \{ \xi_n\} \subset\gamma$, with
$\operatorname{Re} \xi_n \to 0-$, such that $ \ln M_F(\operatorname{Re} \xi_n) \sim \ln\vert F(\xi_n)\vert $, where $M_F(\sigma)=\sup\nolimits_{\vert t\vert <\infty}\vert F(\sigma+it) \vert $? A.M. Gaisin obtained the answer to this question in 2003. In the present article, we solve the following problem: Under what additional conditions on
$\gamma$ is the finer asymptotic relation valid in the case that the argument
$s$ tends to the imaginary axis along
$\gamma$ over a sufficiently massive set?
Keywords:
Dirichlet series, lacunar power series, maximal term, curve of bounded slope, convergence half-plane.
UDC:
517.53
MSC: 35R30 Received: 03.03.2023
Revised: 20.04.2023
Accepted: 16.05.2023
DOI:
10.33048/smzh.2023.64.407