Regularity of the Growth of Dirichlet Series with respect to a Strongly Incomplete Exponential System

Abstract

The article deals with the behavior of the sum of the Dirichlet series $F(s)=\sum\nolimits_{n}a_{n}e^{\lambda_{n}s}$, 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 $\log M_{F}(\operatorname{Re}\xi_{n})\sim\log|F(\xi_{n})|$, where $M_{F}(\sigma)=\sup\nolimits_{|t|<\infty}|F(\sigma+it)|$? 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?