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

Abstract

For an entire Dirichlet series F ( s ) = ∑ k = 0 ∞ f k e x p { s λ k } and a Dirichlet series G ( s ) = ∑ k = 0 ∞ g k e x p { s λ k } with finite abscissa of the absolute convergence the Dirichlet series ( F * G ) ( s ) = ∑ k = 0 ∞ f k g k e x p { s λ k } 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.