Geometric Aspects of Denseness Theorems for Dirichlet Functions

Ghişa, Dorin; Horvat-Marc, Andrei

doi:10.9734/jamcs/2017/37947

Cited by 2 publications

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“…Figure 1 illustrates the pre-image of the real axis for t between −20 and 20 by two Dirichlet L-functions defined by Dirichlet characters modulo 13 studied in [3], the first one by a complex character and the second by a real one. On both arcs are tangent to each other (see [7]). …”

Section: The Quasi-periodicity On Vertical Lines Of General Dirichletmentioning

confidence: 99%

Geometric Aspects of Quasi-Periodic Property of Dirichlet Functions

Ghişa¹,

Horvat-Marc²

2018

APM

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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.

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Section: The Quasi-periodicity On Vertical Lines Of General Dirichletmentioning

confidence: 99%

Geometric Aspects of Quasi-Periodic Property of Dirichlet Functions

Ghişa¹,

Horvat-Marc²

2018

APM

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The Extension of Cauchy Integral Formula to the Boundaries of Fundamental Domains

Ghişa¹

2020

APM

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The Cauchy integral formula expresses the value of a function () f z , which is analytic in a simply connected domain D, at any point 0 z interior to a simple closed contour C situated in D in terms of the values of () 0 f z z z − on C. We deal in this paper with the question whether C can be the boundary ∂Ω of a fundamental domain Ω of () f z. At the first look the answer appears to be negative since ∂Ω contains singular points of the function and it can be unbounded. However, the extension of Cauchy integral formula to some of these unbounded curves, respectively arcs ending in singular points of () f z is possible due to the fact that they can be obtained at the limit as r → ∞ of some bounded curves contained in the pre-image of the circle z r = and of some circles 1 z a r − = for which the formula is valid.

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Geometric Aspects of Denseness Theorems for Dirichlet Functions

Cited by 2 publications

References 0 publications

Geometric Aspects of Quasi-Periodic Property of Dirichlet Functions

Geometric Aspects of Quasi-Periodic Property of Dirichlet Functions

The Extension of Cauchy Integral Formula to the Boundaries of Fundamental Domains

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