On Bohr's theorem for general Dirichlet series

I. Schoolmann

Mathematische Nachrichten, 2020, vol. 293, issue 8, 1591-1612

Abstract: We present quantitative versions of Bohr's theorem on general Dirichlet series D=∑ane−λns assuming different assumptions on the frequency λ=(λn), including the conditions introduced by Bohr and Landau. Therefore, using the summation method by typical (first) means invented by M. Riesz, without any condition on λ, we give upper bounds for the norm of the partial sum operator SN(D):=∑n=1Nan(D)e−λns of length N on the space D∞ext(λ) of all somewhere convergent λ‐Dirichlet series, which allow a holomorphic and bounded extension to the open right half plane [Re>0]. As a consequence for some classes of λ's we obtain a Montel theorem in D∞(λ); the space of all D∈D∞ext(λ) which converge on [Re>0]. Moreover, following the ideas of Neder we give a construction of frequencies λ for which D∞(λ) fails to be complete.

Date: 2020
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