Composition Theorems on Dirichlet Series

Abstract

When two uniform functions , are given, each with a finite radius of absolute convergence R1 R2 respectively, and {λn}, {μν} are real positive increasing sequences tending to infinity, a theorem due to Eggleston [1], which is a generalisation of Hurwitz1 s composition theorem, gives information about the position of the singularities of a composition function h(z), which is assumed to be uniform, in terms of the position of the singularities of f(z) and g(z). This result can be extended to Dirichlet series with real exponents by use of the transformation z = es.