Maximal term of Dirichlet series converging in half-plane: stability theorem

Abstract

We consider a problem on equivalence of logarithms of maximal terms in the Hadamard composition (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 convergence domain of which is a half-plane. A similar problem for entire Dirichlet series was first studied by A.M. Gaisin in 2003 and there was obtained a criterion of 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 asymptotic properties of the Dirichlet series on arbitrary curves going to infinity, namely, in the proof of the famous Pólya conjecture.Both in the case of entire Dirichlet series and ones converging only in the half-plane, a key role in such problems is played by Leontiev formulae for the coefficients. The functions of the corresponding biorthogonal system contains a factor, which the derivative of a characteristic function at the points $\lambda_n$, $n\geq 1$. This fact naturally leads to the considered here problem on the stability of the maximal term.We obtain a criterion ensuring the equivalence of logarithm of the maximal term in the Dirichlet series, the convergence domain of which is a half-plane, to the logarithm of the maximal term of the modified series on an asymptotic set.