Some growth properties of a class of entire Dirichlet series. (English) Zbl 0885.30004

Summary: A class \(F\) of entire Dirichlet series, defined as \[ F=\left\{ f(s)= \sum^\infty_{n=1} a_n^.e^{s.\lambda_n}: (\lambda_n/e)^{\lambda_n} .|a_n |\text{ is bounded} \right\} \] where \(\{\lambda_n\}\) is a pre-assigned real sequence has been studied. The elements of \(F\) have been shown to possess very interesting relations among their types and orders. Relations between \(F\) and various sets of entire Dirichlet series with different growth properties have been established and have been depicted by a Venn Diagram. \(F\) has been provided with a ring structure. Finitely generated ideals in \(F\) have been characterized. Every finitely generated ideal in \(F\) is shown to be a principal ideal. Few results connecting elements of \(F\) and their product have been obtained.


30B50 Dirichlet series, exponential series and other series in one complex variable