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... the tangent to k,j at a point z v k,j and the positive real half axis coincide with that made by the position vector of L 1 s k,j s k,j1 and the positive real half axis. Since lim 0 z and lim 1 z exist, the equality (5.4) holds also for the ends of k,j and k,j . The hypothetical configuration shown in Fig. 10.a below implies that the two half-tangents to k,j at z 0 point one to the upper half plane and the other to the lower half plane. We have denoted by 1, 2, 3, . . . some corresponding points by the two functions. The arguments of L 1 s k,j s k,j1 at the ends of k,j are both between 0 and , which contradicts the ...
Citations
... Hence, each one of the left and the right half-planes is mapped conformally onto the whole complex plane with a slit alongside the real axis from −9 to − 1 9 . We dealt in [6] and [7] with the Blaschke products of the form An elementary computation shows that the equation B a (z) = λ n , where 0 ≤ λ ≤ 1 has the solutions ...
... Since the interval (1, ∞) of the real axis is included in the slit of every fundamental domain Ω k of ζ A,Λ (s), the pre-images by ζ A,Λ (s) of the Apollonius circles around z = 1, of the Steiner net of the Möbius transformation (7), are orthogonal to the pre-image of that interval. Some of them may cut also under different angles the curve γ k . ...
There are known conformal self-mappings of the fundamental domains of analytic functions via Möbius transformations. When two adjacent fundamental domains have a straight line or an arc of a circle as a common boundary, the Schwarz symmetry principle can be applied for one of those mappings and what we obtain is a conformal self-mapping of the union of those domains in which each one of the domains is mapped onto itself. Repeating this operation until the whole plane is exhausted, we obtain a conformal self-mapping of the complex plane in which every fundamental domain is conformally mapped onto itself. We prove in this paper that this is true for any analytic function. Since the self-mappings of fundamental domains have each one at least one fixed point, ultimately, for the self-mapping of the complex plane, we obtain at least as many fixed points as is the number of fundamental domains. When dealing with a rational function, this number is finite, otherwise we obtain infinitely many fixed points. Computer experimentation allows the illustration of these concepts for most of the familiar classes of analytic functions. There are known applications of the Möbius transformations in physics via the Lorentz group. Relating those application to the present work may contribute to the advancement of the knowledge in that field.
... The conformal mappings by Dirichlet functions have been studied in [10], [11], [12], [13] and the way the fundamental domains of these functions can be revealed has been explained. However, for a better understanding of this topic we will repeat some of the findings in those papers. ...
... We have shown in [10] that such a construction is always possible. ...
... The pre-image of the real axis by ζ(s) shows infinite strips, Fig. 7, which can be divided into substrips representing fundamental domains of this function. The way it can be done is described in [10]. ...
Conformal self mappings of a given domain of the complex plane can be obtained by using the Riemann Mapping Theorem in the following way. Two different conformal mappings φ and ψ of that domain onto one of the standard domains: the unit disc, the complex plane or the Riemann sphere are taken and then ψ −1 ◦ φ is what we are looking for. Yet, this is just a theoretical construction, since the Riemann Mapping Theorem does not offer any concrete expression of those functions. The Möbius transformations are concrete, but they can be used only for particular circular domains. We are proving in this paper that conformal self mappings of any fundamental domain of an arbitrary analytic function can be obtained via Möbius transformations as long as we allow that domain to have slits. Moreover, those mappings enjoy group properties. This is a totally new topic. Although fundamental domains of some elementary functions are well known, the existence of such domains for arbitrary analytic functions has been proved only in our previous publications mentioned in the References section. No other publication exists on this topic and the reference list is complete. We deal here with conformal self mappings of fundamental domains in its whole generality and present sustaining illustrations. Those related to the case of Dirichlet functions represent a real achievement. Computer experimentation with these mappings are made for the most familiar analytic functions.
... We illustrate this case by dealing with the function cos w z = (see [9], page 51). for which the fundamental domains are vertical half strips from −π to π . ...
... He also emphasized the importance of these domains by saying: Whatever the advantage of such a representation may be, the clearest picture of the Riemann surface is obtained by direct consideration of the fundamental regions in the z-plane. However, until recently (see [20], p 7) fundamental domains were known only for automorphic functions. But once discovered for some other classes of functions, we realized ( [2], [3], [4], [7], [15]) that they help finding groups of cover transformations with respect to 2 Dorin Ghisa which these functions are automorphic. ...
... An important part of the monograph [20] was devoted to Blaschke products since for these functions the illustrations are elegant and the case of finite Blaschke products is an appropriate introduction to arbitrary rational functions, while the case of infinite Blaschke products is as well an appropriate introduction to transcendental functions. Among the transcendental functions a special place was reserved to Euler Gamma function and to Riemann Zeta function. ...
The concept of fundamental domain, as defined by Ahlfors, plays an important role in the study of different classes of analytic functions. For more than a century the Dirichlet functions have been intensely studied by mathematicians working in the field of number theory as well as by those interested in their analytic properties. The fundamental domains pertain to the last field, yet we found a lot of theoretic aspects which can be dealt with by knowing in detail those domains. We gathered together in this survey paper some recent advances in this field. Proofs are provided for some of the theorems, so that the reader can navigate easily through it. MSC : 30C35, 11M26
... , which are all on the unit circle, are simple roots no matter if the zeros of ( ) B z are simple or not. On the other hand the equation1 m − roots (counted with multiplicities) inside the unit disc and they are instrumental in proving the following theorem (see[4], page 11):Theorem 3. For any Blaschke product ( ) B z of degree m there is a partition of the complex plane into m sets whose interior are fundamental domains of ( )w B z = .These domains are symmetric with respect to the unit circle. ...
... For any Blaschke product ( ) B z of degree m there is a partition of the complex plane into m sets whose interior are fundamental domains of ( )w B z = .These domains are symmetric with respect to the unit circle. Proof: There is a constructive proof of this theorem (see[4], page 10). < and there is a non self intersecting polygonal line L connecting the points ( )j B b and 1 w = . ...
... All of these functions are combinations of simple transformations of e z and therefore it will be enough to deal with just one of them. In [5], page 99 as well as in [4], page 51 it is the function ( ) cos ...
We are dealing with domains of the complex plane which are not symmetric in the 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.
... How big can they get? The answer is: they become the whole complex plane with some slits (see [33] [34] [35] [36]). A region with this property is called fundamental domain of ( ) ...
... The continuation over r C from each one of these points can be made clockwise and counter clockwise into k S , respectively Since a point turning around the origin in the same direction on an arbitrary circle r C centred at the origin will meet consecutively the positive and the negative real half axis, the components of the pre-image of r C (including r γ when 1 r > ) should meet consecutively the pre-image of the positive and the negative half axis (coloured differently). This is [33] Figure 4 as ( ) ...
... It is known that the Dirichlet L-functions are meromorphic continuations of ordinary Dirichlet series defined by Dirichlet characters and these series can be expressed as Euler products. This property is a corollary of the fact that the Dirichlet characters are totally multiplicative functions (see for example [33]. Yet the property of being total multiplicative can be extended to general Dirichlet series, as done in [35], and therefore some of the general Dirichlet series ...
... How big can they get? The answer is: they can become the whole complex plane with some slits (see Ghisa, 2013Ghisa, , 2014Ghisa, , 2016aGhisa, , 2016b. A region with this property is called fundamental domain of ζ A,Λ (s). ...
... Three kinds of intertwining curves have been distinguished (Ghisa, 2013(Ghisa, , 2014(Ghisa, , 2016a(Ghisa, , 2016b, namely: ...
We dealt in a series of previous publications with some geometric aspects of the mappings by functions obtained as analytic continuations to the whole complex plane of general Dirichlet series. Pictures illustrating those aspects contain a lot of other information which has been waiting for a rigorous proof. Such a task is partially fulfilled in this paper, where we succeeded among other things, to prove a theorem about general Dirichlet series having as corollary the Speiser's theorem. We have also proved that those functions do not possess multiple zeros of order higher than $2$ and the double zeros have very particular locations. Moreover, their derivatives have only simple zeros.
... 9438 with the abscissa 0. 51591 and respectively 0. 48409) gives us an idea of the configuration we should obtain in the case of a double zero of Davenport and Heilbronn function. Indeed, those zeros are enough close to each other and, as a consequence, the configuration of pre-images of rays and circles (see [4]) is similar to that we should obtain for a double zero. The first remark to make is that double zeros for this function, if they exist, should necessarily be located on the critical line. ...
... In what follows, we are using techniques perfected by D. Ghisa in his monograph [4] and a series of papers (see [2], [5], [6], [7]). For any function obtained by analytic continuation of a series 1 the complex plane can be partitioned into infinitely many strip S k , k , 0 S 0 and S k is below S k1 for every k (Ghisa's strips) such that every strip S k is mapped by the function (not necessarily one to one) onto the whole complex plane with a slit alongside the interval 1, of the real axis. ...
By using an analogy with the case of very close zeros symmetric with respect to the critical line of the Davenport and Heilbronn function, we study the conformal mapping of L-functions in a neighborhood of a hypothetical double zero and conclude that such a zero cannot exist. Keywords: Riemann Hypothesis, non trivial zeros, Ghisa's strips, Ghisa's fundamental domains, Ghisa's intertwined curves 2010 Mathematics Subject Classification: 30C35; 11M26
... The concept of fundamental domain coming from the automorphic functions theory and to which Ahlfors [1] gave an independent meaning helped us to fulfill this task and to advance the research in the field of Dirichlet series. The Utzinger and Speiser reellen Züge were used only to put into evidence in [3], [15], [12], [13] the boundaries of fundamental domains of the class of functions we were studying. This class of functions includes the Selberg class, for which the most general RH has been formulated. ...
... The regularity of the curves k,j and k,j (see [12] and [13]) expressed by two elementary topological properties, the color alternating rule and the color matching rule is subject also to a variety of configurations which were impossible to imagine at the beginning. Embracing curves came as a surprise for high values of t, only to notice later that it is a common phenomenon and a curve k,j , j 0 can embrace several such curves, but never a curve k,0 . ...
... However, we are reluctant to say that those zeros represent counterexamples to RH, since they are zeros of a function which is not a Dirichlet L-function neither is it a member of the Selberg's class of functions. Now we need to prove that the unwanted behavior of k,0 we have noticed in this case cannot happen in the case of Dirichlet L-functions and to complete in this way the proof of the Theorem 9 from [12], the description given in [7] for the location of the zeros of the derivative of Dirichlet L-functions, as well as the Section 9.3 from [13] . ...
Some of my previous publications were incomplete in the sense that non trivial zeros belonging to a particular type of fundamental domain have been inadvertently ignored. Due to this fact, I was brought to believe that computations done by some authors in order to show counterexamples to RH were affected of approximation errors. In this paper I illustrate graphically the correctness of those computations and I fill the gaps in my publications.
... It is known that a branch point s 0 of multiplicity q 2 of A, s is either a zero of order q 1 of A, s or a pole of order q of A, s. In a neighborhood of s 0 we have A, s s s 0 q s, in the first case and A, s s s 0 q s in the second case, where is analytic at s 0 and s 0 0 (See [1], p.133 and [9], p.7). ...
... The effect of this is that every k,0 is mapped bijectively by A, s onto the positive real half axis, every k is mapped onto the negative real half axis and every k,j , j 0 is mapped bijectively onto the whole real axis. If we use four different colors denoted a, b, c, d for the pre-image by A, s and by A, s of the negative and respectively positive real half axis, we realize that the intertwining curves must have specific colors (see [9], p. 102) and this is another simple topological fact which has been called the color matching rule. Namely, the color b meets always c and if j 0, or 1/2, the color a meets always d. Notice that for 1/2 a curve k,0 can intersect k,0 and therefore the color b will meet d. ...
... A corollary of the two rules is the fact that the zeros of A, s and those of A, s are all simple zeros. The proof of this affirmation for an arbitrary general Dirichlet series verifying the hypothesis of Theorems 1 and 2 is similar to that presented for the Riemann Zeta function in [9] and we will omit it. ...
Fundamental domains are found for functions defined by general Dirichlet series and using basic properties of conformal mappings the Great Riemann Hypothesis is studied.
