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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.

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 [

By a general Dirichlet series we understand an expression of the form

ζ A , Λ ( s ) = ∑ n = 1 ∞ a n e − λ n s , (1)

where A = ( a n ) is an arbitrary sequence of complex numbers and λ 1 < λ 2 < ⋯ is an increasing sequence Λ of non negative numbers with lim n → ∞ λ n = ∞ . There is no loss of generality by considering only normalized series (1) in which a 1 = 1 and λ 1 = 0 . It is known [

When (1) does not converge for s = 0 , then (see [

σ c = lim sup n → ∞ 1 λ n log | ∑ k = 1 n a k | ≥ 0. (2)

If (1) converges for s = 0 , then

σ c = lim sup n → ∞ 1 λ n + 1 log | ζ A , Λ ( 0 ) − ∑ k = 1 n a k | ≤ 0. (3)

The abscissa σ a of absolute convergence of the series (1) is defined in an analogous way and it is obvious that − ∞ ≤ σ c ≤ σ a ≤ + ∞ . For the Riemann Zeta function σ c = σ a = 1 , while for the alternate Zeta function σ c = 0 and σ a = 1 . When σ c < + ∞ then ζ A , Λ ( s ) converges uniformly on compact sets of the half plane Re s > σ c and ζ A , Λ ( s ) 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 ζ A , Λ ( s ) for this extended function and we call it Dirichlet function. Since a 1 = 1 and λ 1 = 0 in the series (1) we have that lim σ → + ∞ ζ A , Λ ( σ + i t ) = 1 and it can be easily seen [

This fact suggests that the series (1) converges uniformly on that half plane. Harald Bohr defined the abscissa of uniform convergence of (1) as being the infimum σ u of the abscisas σ such that (1) converges uniformly for Re ( s ) > σ . It has been found that σ c ≥ σ u ≥ σ a and every value between σ c and σ a can be taken by σ u for particular series (1).

Studying Dirichlet L-functions f ( s ) generated by ordinary Dirichlet series (the case where λ n = log n ) Harald Bohr (see [

⋯ < τ − 2 < τ − 1 < 0 < τ 1 < τ 2 < ⋯ lim inf n → ± ∞ ( τ n + 1 − τ n ) > 0 , lim sup n → ± ∞ τ n n < ∞ (4)

such that for every s ∈ Ω we have | f ( s + i τ n ) − f ( s ) | < ε .

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 ζ A , Λ ( s ) of vertical lines related to the fundamental domains of these functions. These fundamental domains are obtained as shown in [

Let us give first to the concept of quasi-periodicity a slightly different definition. We will say that f ( s ) is quasi-periodic on a line Re s = σ 0 if for every ε > 0 and for every s = σ 0 + i t a sequence (4) exists such that | f ( s + i τ n ) − f ( s ) | < ε . 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 better the purpose of studying the denseness properties of Dirichlet functions.

Theorem 1 If λ n with n = 2 , 3 , ⋯ are linearly independent in the field of rational numbers then the series (1) is quasi-periodic on every vertical line of the half plane R e s > σ u .

Proof Let s be arbitrary with Re s = σ 0 > σ u and divide ζ A , Λ ( s ) and ζ A , Λ ( s + i τ ) into the sum A n of the first n terms and the rest R n . Since the series converges uniformly on Re s = σ 0 , when ε > 0 is given, there is a rank n such that | R n ( s ) | < ε 3 and | R n ( s + i τ ) | < ε 3 for every real number τ . On the other hand

| A n ( s + i τ ) − A n ( s ) | = | ∑ k = 1 n a k e − λ k s ( e − i λ k τ − 1 ) | (5)

By Diophantine approximation, a sequence (4) exists such that for every τ m of that sequence e − i λ k τ m is as close to 1 on the unit circle as we whish. Since the set { a k e − λ k s } is bounded, we have | A n ( s + i τ ) − A n ( s ) | < ε 3 for every τ = τ m and then | ζ A , Λ ( s + i τ ) − ζ A , Λ ( s ) | < ε for every τ = τ m , which proves the theorem.

Remark For ordinary Dirichlet series we have λ n = log n , for n = 2 , 3 , ⋯ 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 [

Every fundamental domain contains either a simple zero or no zero and in this last case a double zero belongs to the boundary of two adjacent fundamental domains. The zeros of

of them the strips

The image of the whole line by

The image of an interval determined by

When the analytic continuation to the whole complex plane of the series (1) is possible the arcs

for

It is known that some functions defined by Dirichlet series cannot be extended across the line

We recall that the Gamma function can be expressed as

and this is a meromorphic function in the complex plane.

On replacing x by

which multiplied by

Here we have denoted by

For the Riemann Zeta function we have

Riemann has shown that the integral from (9) is equivalent to a contour integral of

Theorem 2 If the series (1) has a finite abscissa of convergence

Proof Suppose that

and the abscissa of convergence of

If

Consequently, the half plane of convergence of

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:

Theorem 3 If

Proof Let us form a contour

and the half lines

Since the singularities of

as both of these integrals are obviously equal to

We can take r such that no singular point of

Since the function under the integral is bounded between

The right hand side in (10) is defined for every complex value s and represents a meromorphic function in the whole complex plane.

The connection between the quasi-periodic property and the denseness property of the image of vertical lines by Dirichlet functions appears clearly when we interpret the first one in terms of the arcs

Theorem 4 The necessary and sufficient condition for

Proof The condition is necessary, since if for every

Hence, if

No two arcs

In order to study the image of vertical lines by the series (1) the condition that the exponents

The results of Bohr are in agreement with the fact that the function (1) tends uniformly with respect to t to 1 as

Theorem 5 For any Dirichlet function

For the Riemann series, the term

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 some slits. A vertical line intersects all those strips and the values of the function on each one of the segments obtained come quasi-periodically close to every given value on that line as illustrated in

the same role as in the denseness property and this is the reason why the two properties come simultaneously. We brought in this paper some light into these two complex phenomena.

The authors declare no conflicts of interest regarding the publication of this paper.

Ghisa, D. and Horvat-Marc, A. (2018) Geometric Aspects of Quasi-Periodic Property of Dirichlet Functions. Advances in Pure Mathematics, 8, 699-710. https://doi.org/10.4236/apm.2018.88042