The Geometry of the Mappings by General Dirichlet Series

Ghişa, Dorin

doi:10.4236/apm.2017.71001

Cited by 7 publications

(8 citation statements)

References 22 publications

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“…These are strips unbounded to the right and to the left when boundaries of fundamental domains bounded to the right. It is known (see, for example [13]…”

Section: General Properties Of Dirichlet Functionsmentioning

confidence: 99%

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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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“…These are strips unbounded to the right and to the left when boundaries of fundamental domains bounded to the right. It is known (see, for example [13]…”

Section: General Properties Of Dirichlet Functionsmentioning

confidence: 99%

“…The 0 S -strip contains infinitely many fundamental domains. The way they are mapped conformally onto the complex plane with some slits by the Riemann Zeta function is illustrated in Figure 2 (see [13], Figure 6).…”

Section: General Properties Of Dirichlet Functionsmentioning

confidence: 99%

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

Ghişa¹

2020

APM

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“…The knowledge we need here about Dirichlet functions can be found in [7]. We are dealing first with series of the form ( ) It is known (see [8]) that for any Dirichlet function the complex plane admits a partition into infinitely many horizontal strips bounded by curves   is a hidden symmetry of ∆ . Figure 8 illustrates the symmetries in the case of real n a and lack of obvious symmetries when some n a are complex.…”

Section: Hidden Symmetries Of Dirichlet Functionsmentioning

confidence: 99%

Hidden Symmetries of Complex Analysis

Cao-Huu¹,

Ghişa²,

Muscutar³

2019

APM

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We are dealing with domains of the complex plane which are not symmetric in common sense, but support fixed point free antianalytic involutions. They are fundamental domains of different classes of analytic functions and the respective involutions are obtained by composing their canonical projections onto the complex plane with the simplest antianalytic involution of the Riemann sphere. What we obtain are hidden symmetries of the complex plane. The list given here of these domains is far from exhaustive.

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“…It is illustrated inFigure 3. The same rule applies also to the pre-image of the real axis by any derivative of 5: When represented in the same plane, the pre-image of the real axis by both ( ) in couples of curves ′ Γ k and ′ ϒ k , respectively , Γ k j and , ϒ k j (see[2], Theorem 4) which intersect two by two (the intertwining curves).…”

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confidence: 97%

On the Zeros of Euler Product Dirichlet Functions

Ghişa¹

2019

APM

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We study a class of Dirichlet functions obtained as analytic continuation across the line of convergence of Dirichlet series which can be written as Euler products. This class includes that of Dirichlet L-functions. The problem of the existence of multiple zeros for this last class is outstanding. It is tacitly accepted, yet not proved that the Riemann Zeta function, which belongs to this class, does not possess multiple zeros. In a previous study we provided an example of Dirichlet function having double zeros, but that function is not an Euler product function. In this paper we deal with Euler product functions and by using the geometric properties of the mapping realized by these functions, we tackle the problem of the multiplicity of their zeros.

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The Geometry of the Mappings by General Dirichlet Series

Cited by 7 publications

References 22 publications

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

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

Hidden Symmetries of Complex Analysis

On the Zeros of Euler Product Dirichlet Functions

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