Geometric Aspects of Quasi-Periodic Property of Dirichlet Functions

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.


Introduction
The theory of Dirichlet series started at the end of the 19-th Century with works of celebrated mathematicians as Hadamard, Landau, Bohr etc.These series are natural generalizations of the Riemann Zeta series.From the beginning questions were asked of what are those Dirichlet series which can be continued as meromorphic functions in the whole complex plane and satisfy there similar properties with those of the Riemann Zeta function, as for example a Riemann type of functional equation, similar display of non trivial zeros (the famous Riemann Hypothesis) etc.We devoted a lot of studies to these questions by using geometric methods.We perfected an idea of Speiser (1934) of studying the pre-image of the real axis by functions obtained as meromorphic continuations to the whole complex plane of general Dirichlet series.The key result was a way to identify the fundamental domains of these functions.These are domains represented conformally (hence injectively) by the functions onto the whole complex plane with some slits.
As Ahlfors [1] noticed, this is the most natural way to proceed when studying different classes of functions.The results are promising and there are a lot of followers mainly in the field of Blaschke products but also in that of Dirichlet functions.
By a general Dirichlet series we understand an expression of the form ( ) , 1 e , series converge for all the points of that line, or only for some points.
When (1) does not converge for If (1) converges for The abscissa a σ of absolute convergence of the series (1) is defined in an analogous way and it is obvious that We followed the tradition of this monograph by using the notation "log" for the principal branch of the multivalued function logarithm.Obviously, when the argument is positive, it simply means natural logarithm.is an analytic function in that half plane and sometimes it can be continued analytically to the whole complex plane except possibly for some poles.We will deal with this problem in Section 3.
We keep the notation ( ) for this extended function and we call it Dirichlet function.Since 1 1 a = and 1 0 λ = in the series (1) we have that ( ) Λ →+∞ + = and it can be easily seen [3] that this limit is uniform with respect to t.In other words, for every 0 ε > there is ε σ such that for σ can be taken by u σ for particular series (1).
Studying Dirichlet L-functions ( ) ) Harald Bohr (see [4]) discovered that they display on vertical lines a quasi-periodic behavior, namely for every bounded domain Ω of uniform convergence of the series and for every ) such that for every s ∈Ω we have This roughly means that the function comes (quasi)-periodically on a vertical line as close as we want to the value of it at any point of Ω belonging to that line.
We study in this paper the quasi-periodic property of functions defined by general Dirichlet series and show that this is a geometric property of the image by ( ) of vertical lines related to the fundamental domains of these functions.These fundamental domains are obtained as shown in [3] and [5].

The Quasi-Periodicity on Vertical Lines of General Dirichlet Series
Let us give first to the concept of quasi-periodicity a slightly different definition.
We will say that ( ) We notice that this definition is no more attached to bounded domains, hence it appears less restrictive than that given by Bohr, yet the inequality refers only to the points of a given vertical line and not to the points of any vertical line intersecting the domain Ω , which is a restriction.This new definition serves Advances in Pure Mathematics Re s σ = , when 0 ε > is given, there is a rank n such that ( ) ( ) for every real number τ .On the other hand By Diophantine approximation, a sequence (4) exists such that for every m τ of that sequence e k m iλ τ − is as close to 1 on the unit circle as we whish.Since the set { } = , which proves the theorem.
Remark For ordinary Dirichlet series we have log and these are linearly independent in the field of rational numbers, therefore these series are quasi-periodic on every vertical line from the half plane of convergence.
It is known (see [5]) that for every series (1) which can be continued analytically to a the whole complex plane except possibly for a simple pole at 1 s = , the complex plane is divided into infinitely many horizontal strips , k S k ∈  bounded by components of the pre-image of the real axis which are mapped bijectively by ( ) counted with multiplicities, then it will contain 1 k j − zeros of ( ) . The strip 0 S can contain infinitely many zeros of such a function.

′
are all simple zeros (see [6]) and are all located on the boundaries of the fundamental domains.
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 .At the points which are zeros of ( ) the corresponding arcs are tangent to each other (see [7]).
When the analytic continuation to the whole complex plane of the series (1) is possible the arcs

Analytic Continuation of General Dirichlet Series
It is known that some functions defined by Dirichlet series cannot be extended across the line Re c s σ = since all the points of the abscissa of convergence are singular points.Examples of such series can be easily found as seen in [8] and [3].On the other hand all the Dirichlet L-functions are analytic continuations to the whole complex plane, except for some poles of particular Dirichlet series.
These continuations have been performed by using the Riemann technique of contour integration.In that follow, we will show that a similar technique is applicable also to general Dirichlet series.
We recall that the Gamma function can be expressed as ( ) and this is a meromorphic function in the complex plane.
Here we have denoted by e Λ the sequence e ,e , λ λ  and we have interchanged the integration and the summation, which is allowed, since the integrals of the terms are absolutely convergent at both ends.We notice that Hardy and Riesz [2] have found (in Theorem 11) a similar formula to (8) in the case a Dirichlet series convergent for Re 0 s > , yet they did not use it to extend the function ( ) across the imaginary axis.
For the Riemann Zeta function we have 1 c σ = (see [1]) and after summation under integral in (8) one obtains ( ) ( ) Riemann has shown that the integral from ( 9) is equivalent to a contour integral of ( ) on a curve C formed with a part of a circle of radius r centered at the origin and two half lines parallel to the real axis ([1], p. 216).The integrand is well defined on the respective curve and as 0 r → the integral approaches that in (9).We cannot do the same thing with (8) as long as we don't make sure that the circle is in the half plane of convergence of ( ) . There is however a way to circumvent this difficulty by noticing that: Theorem 2 If the series (1) has a finite abscissa of convergence c σ , then the abscissa of convergence of ( ) Proof Suppose that 0 c σ ≥ .Then , which should be also less than or equal to zero, is given by the formula ,e 1 γ is the blue contour, r γ the red one and γ the green contour, then as both of these integrals are obviously equal to the integral on r γ does not really depend on r.We can take r such that no singular point of ( )  In order to study the image of vertical lines by the series (1) the condition that the exponents n λ are linearly independent in the field of rational numbers has been assumed in [9].As we suppose that 1 0 λ = , we cannot use the results in [9]   for the series ( 1), yet we can study the series .By [9], under the condition of linear independence of the exponents of ( ) f s , the closure of the image by ( ) f s of the line 0 Re s σ = , where 0 σ is greater than the abscissa of absolute convergence of the series (1) (which obviously the same as that of ( ) For the Riemann series, the term 1 2 σ is leading term for

Conclusion
A Dirichlet function is defined by an arbitrary sequence of complex numbers (the coefficients) and a sequence of increasing positive numbers (the exponents), otherwise also arbitrary.It is intriguing how two such arbitrariness can involve a strong property as that of quasi-periodicity.We have shown that this is in fact a geometric property related to the fundamental domains of the respective function.The domains are infinite strips which are mapped conformally by the function onto the whole complex plane with slits.A vertical line intersects all those strips and the values of the on each one of the segments obtained come quasi-periodically close to every given value on that line as illustrated in Figure 3 and How to cite this paper: Ghisa, D. and Horvat-Marc, A. (2018) Geometric Aspects of Quasi-Periodic Property of Dirichlet D. Ghisa, A. Horvat-Marc DOI: 10.4236/apm.2018.88042700 Advances in Pure Mathematics sequence Λ of non negative numbers with lim n n λ →∞ = ∞ .There is no loss of generality by considering only normalized series (1) in which 1 1 a = and 1 0 λ = .It is known [2] that if the series (1) converges for exists, is called the abscissa of convergence of the series (1).When the series does not converge for any s ∈  we denote c σ = +∞ and if it converges for every s we put c σ = −∞ .The line Re c s σ = is called the line of convergence of (1), although there are examples ofDirichlet series (see[2]) which do not converge for any s with Re that the series (1) converges uniformly on that half plane.Harald Bohr defined the abscissa of uniform convergence of (1) as being the infimum u σ of the abscisas σ such that (1) converges uniformly for ( ) Re s σ > .It has been found that c u a σ σ σ ≥ ≥ and every value between c σ and a s generated by ordinary Dirichlet series (the case where log n n λ =

A
better the purpose of studying the denseness properties of Dirichlet functions.Theorem 1 If n λ with 2,3, n = are linearly independent in the field of rational numbers then the series (1) is quasi-periodic on every vertical line of the half plane of the first n terms and the rest n R .Since the series converges uniformly on 0

Figure 1 . 2 S + − and , 3 S
Figure 1.The pre-image of the real axis by two Dirichlet L-functions.

γ
are defined for any vertical line, not only for the lines included in the half plane of convergence of this series.Then, expressing the quasi-periodic property in terms of these arcs, we can extend this concept to any vertical line.The extension can be performed by noticing that if ( ) where m τ is any term of the sequence (4).This means that to every term m τ of this sequence corresponds an arc , belonging to the half plane of uniform convergence of this series) if for every s with On replacing x by e n x λ in(7), we obtain D. Ghisa, A. Horvat-Marc DOI: 10.4236/apm.
half plane.Once we know this half plane of convergence, we can try to use the Riemann technique, but taking care to choose the integration curve in the right half plane.Fortunately such a choice is possible and we can prove: DOI: 10.4236/apm.2018.singularities on the imaginary axis, then the series (1) can be continued across the line Re the whole complex plane.Proof Let us form a contour r γ with a half circle r isolated, we can chose r such that the half circle r C does not contain anyone of them.We study the line integral integrand is a continuous function on r γ and therefore it is bounded on r the interval of the imaginary axis between ir − and ir .Now, we can let 0 r → and show that the integral on r C tends to zero.Indeed, with e

Figure 2 .
Figure 2. The integrals on r γ and R γ are equal.

≥ 2 nρ=
has a leading (vorhanden) term or not.Consequently, under the same condition, this image by Bohr are in agreement with the fact that the function (1) tends uniformly with respect to t to 1 as σ → +∞ .Indeed, for is a dense set in this domain.Here we have denoted by A the sequence ( ) n a .As 0 σ gets smaller, some other term can become leading term and then the image by no leading term appears, then the respective ring domain will evolve into a disc.At the limit, as 0 a σ σ = we have R = ∞ and the image of the line Re a s σ = is a dense set either in the whole complex plane or outside of an open disc.Hence we can statewe have R = ∞ and these domains become the exterior of an open disc or respectively the whole complex plane.

..
Let us denote by σ * the solution of the equation ( ) By inspecting a table of values of ( ) ζ σ we noticed that

Figure 4 .
Figure 4.The image by ( ) s ζ of