STABILITY-PRESERVING PERTURBATION OF THE MAXIMAL TERMS OF DIRICHLET SERIES

Abstract

We study 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 the key role in obtaining asymptotic estimates for 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. For the wide class of entire Dirichlet series determined by a convex growth majorant, we establish a criterion for the equivalence of the logarithms of the moduli of the original series and a modified Dirichlet series outside some exceptional set.

Keywords

• dirichlet series
• hadamard composition
• stability of the maximal term
• borel–nevanlinna lemma
• convex function