Scaling a theorem of Harald Bohr

Abstract

A outstanding result of H. Bohr shows that under a certain condition on the frequency λ=(λn) (preventing the λn's from getting too close too fast) every λ-Dirichlet series D=∑ane−λns converges uniformly on all half-planes [Re>σ] with σ>0, provided D is pointwise convergent on some half-plane and has a limit function extending to a bounded holomorphic function f on [Re>0]. Riesz summability plays an important role within the classical literature of general Dirichlet series - recall that a Dirichlet series D=∑ane−λns is said to be Riesz summable of order ℓ≥0 in s∈C, whenever the sequence of all Riesz means ∑λn<xan(1−λnx)ℓe−λns converges as x tends to infinity. Extending classical work of M. Riesz, we for a given ℓ≥0 isolate and study the class of those frequencies λ, which have the following property: If the Dirichlet series D=∑ane−λns is pointwise Riesz summable for some order on some half-plane and has a limit function extending to a holomorphic function f on [Re>0] with growth of order O((1+s)ℓ), then uniformly on all half-planes [Re>σ],σ>0, we have f(s)(1+s)ℓ=1(1+s)ℓlimx→∞⁡∑λn<xan(1−λnx)ℓe−λns. Among others, we show that this holds true if λ satisfies a well-known condition isolated by E. Landau, which recovers the classical bounded case ℓ=0.