On Hadamard compositions of entire Dirichlet series and Dirichlet series absolutely converging in half-plane

Анотація

For an entire Dirichlet series $F\left(s\right)=\sum _{k=0}^{\infty }{f}_{k}exp\left\{s{\lambda }_k\right\}$ and a Dirichlet series $G\left(s\right)=\sum _{k=0}^{\infty }{g}_{k}exp\left\{s{\lambda }_k\right\}$ with  finite abscissa of the absolute convergence the Dirichlet series $(F*G)\left(s\right)=\sum _{k=0}^{\infty }{f}_k{g}_{k}exp\left\{s{\lambda }_k\right\}$ is called the Hadamard composition. In terms of generalized orders the growth of  this composition and their derivatives is investigated. A relation between the  behavior of the maximal terms  of the Hadamard composition of the derivatives  and of the  derivative of the Hadamard composition is established.