Algebra of entire Dirichlet series with real frequencies

Abstract

We study the class E(Λ,C) of all entire Dirichlet series with real frequencies Λ (which are generalizations of classical Dirichlet series as well as power series). Our method, based on Number Theory, allows us to get a link between the set of real frequencies and the set of counting numbers. So the following problems posed by C.O. Kiselman have been solved: (i) the criteria for E(Λ,C) to be closed (invariant) under multiplication; (ii) the explicit description of the smallest subalgebra of entire functions that contains E(Λ,C). Some open questions are provided.