Estimate for deviation of logarithm of maximal term in Dirichlet series under Hadamard composition

Abstract

We study the problem of estimating the equivalence degree of the logarithms of the maximal terms in the Hadamard composition (the modified series)
$ \sum \limits_{n} a_nb_ne^{\lambda_nz}$ of the Dirichlet series $\sum \limits_{n} a_ne^{\lambda_nz} $ and $\sum \limits_{n} b_ne^{\lambda_nz}$ with positive
exponents, the domains of convergence of which are a half--plane. A similar problem for entire Dirichlet series was first studied by A.M.~Gaisin in 2003; he obtained a criterion for the stability of the maximal term $\mu(\sigma)=\max \limits_{n\geq 1}\{{\vert a_n\vert} e^{\lambda_n\sigma}\}. $ This result turned out to be useful in studying the asymptotic properties of a Dirichlet series on arbitrary curves going to infinity, namely in the proof of the well-known P´olya conjecture.

As in the case of entire Dirichlet series, and in the case of series converging only in a half-plane, in problems of this type Leontiev's formulas for the coefficients play a key role. The functions of the corresponding biorthogonal system contain a factor, which is the reciprocal of the derivative of the characteristic function at the points $\lambda_n,$ $n\geq 1.$ This circumstance naturally leads to the formulation of the considered problem on a quantitative characteristic of the stability of the maximal term. This question is relevant from the point of view of its applications to the problem of the minimum of modulus of an entire transcendental function and to other related problems of analysis and complex dynamics.

In terms of the deviation estimate, we obtain a condition for the logarithm of the maximal term of a Dirichlet series on an asymptotic set to be equivalent to the logarithm of the maximal term of the modified series.