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

The article deals with the behavior of the sum of the Dirichlet series , with , converging absolutely in the left half-plane 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 . The essence of the questions we consider is as follows: Given a curve starting from the half-plane and ending asymptotically approaching on the boundary of , what are the conditions for the existence of a sequence , with , such that , where ? 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 is the finer asymptotic relation valid in the case that the argument tends to the imaginary axis along over a sufficiently massive set?