TITLE:
Geometric Aspects of Quasi-Periodic Property of Dirichlet Functions
AUTHORS:
Dorin Ghisa, Andrei Horvat-Marc
KEYWORDS:
Dirichlet Functions, Analytic Continuation, Fundamental Domains, Quasi-periodic Functions
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.8 No.8,
August
6,
2018
ABSTRACT: The concept of quasi-periodic
property of a function has been introduced by Harald Bohr in 1921 and it
roughly means that the function comes (quasi)-periodically as close as we want
on every vertical line to the value taken by it at any point belonging to that
line and a bounded domain Ω. He proved that the functions defined by ordinary
Dirichlet series are quasi-periodic in their half plane of uniform convergence.
We realized that the existence of the domain Ω is not
necessary and that the quasi-periodicity is related to the denseness property
of those functions which we have studied in a previous paper. Hence, the
purpose of our research was to prove these two facts. We succeeded to fulfill
this task and more. Namely, we dealt with the quasi-periodicity of general
Dirichlet series by using geometric tools perfected by us in a series of
previous projects. The concept has been applied to the whole complex plane (not
only to the half plane of uniform convergence) for series which can be
continued to meromorphic functions in that plane. The question arise: in what
conditions such a continuation is possible? There are known examples of
Dirichlet series which cannot be continued across the convergence line, yet
there are no simple conditions under which such a continuation is possible. We
succeeded to find a very natural one.