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Geometric Aspects of Quasi-Periodic Property of Dirichlet Functions

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Document pages: 12 pages

Abstract: The concept of quasi-periodicproperty of a function has been introduced by Harald Bohr in 1921 and itroughly means that the function comes (quasi)-periodically as close as we wanton every vertical line to the value taken by it at any point belonging to thatline and a bounded domain Ω. He proved that the functions defined by ordinaryDirichlet series are quasi-periodic in their half plane of uniform convergence.We realized that the existence of the domain Ω is notnecessary and that the quasi-periodicity is related to the denseness propertyof those functions which we have studied in a previous paper. Hence, thepurpose of our research was to prove these two facts. We succeeded to fulfillthis task and more. Namely, we dealt with the quasi-periodicity of generalDirichlet series by using geometric tools perfected by us in a series ofprevious projects. The concept has been applied to the whole complex plane (notonly to the half plane of uniform convergence) for series which can becontinued to meromorphic functions in that plane. The question arise: in whatconditions such a continuation is possible? There are known examples ofDirichlet series which cannot be continued across the convergence line, yetthere are no simple conditions under which such a continuation is possible. Wesucceeded to find a very natural one.

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