We study the question of estimating the degree of stability of the maximal term of the Dirichlet series with positive exponents, the sum of which is an entire function.This problem is of interest because the Leont'ev formulas for coefficients calculated using a biorthogonal system of functions play a key role in obtaining asymptotic estimates for the entire Dirichlet series on various continua going to infinity (for example, curves).This fact naturally leads to the need to study the behavior of the logarithm of the maximum term also for the Hadamard composition of the corresponding Dirichlet series.The issues discussed here have important applications in complex dynamics, namely, in problems related to the structure of the Fatou set of entire transcendental functions.For the wide class of entire Dirichlet series determined by a convex growth majorant, we establish an estimate of the degree for the equivalence of the logarithms of maximal terms of the original series and a modified Dirichlet series outside some exceptional -set.
Gaisin et al. (Mon,) studied this question.