Stability degree of maximal term of Dirichlet series

Abstract

We study the stability of a maximal term of a Dirichlet series with positive exponents, the sum of which is an entire function. For a class of entire Dirichlet series defined by a certain convex growth majorant, we prove a theorem on the quantitative estimate of the equivalence degree (outside of some exceptional $c_{q}$-set) of the logarithms of maximal terms in the original series and the modified Dirichlet series. A similar problem for entire Dirichlet series of an arbitrary rapid growth, but with no quantitative estimate of the stability degree for the maximal term, was first studied by A.M. Gaisin in the late 1990s and early 2000s. He then obtained a stability criterion, which was the equivalence of the logarithms of the maximal terms of the original and modified series on the asymptotic set. This result, as well as the corresponding stability statements for Dirichlet series converging only in a certain half–plane obtained by A.M. Gaisin and T.I. Belous, found useful applications in the theory of asymptotic properties of Dirichlet series, specifically in proving Pólya type identities. The formulation of the stability problem considered in this paper is relevant for its applications to the minimum modulus problem, as well as to other related problems in analysis and complex dynamics.