On the growth of a class of entire Dirichlet series

Abstract

In terms of generalized orders it is investigated a relation between the growth of an entire Dirichlet series $F(s)=\sum\limits_{n=1}^{\infty}a_n\exp\{s\lambda_n\}$ and the growth of entire Dirichlet series $F_j(s)=\sum\limits_{n=1}^{\infty}a_{n, j}\exp\{s\lambda_n\}$, $1\le j\le 2$, provided the coefficients $a_n$ are connected with the coefficients $a_{n, j}$ by some correlations.