The Vortex-like Behavior of the Riemann Zeta Function to the Right of the Critical Strip

Based on an equivalence relation that was established recently on exponential sums, in this paper we study the class of functions that are equivalent to the Riemann zeta function in the half-plane {s∈C:Res>1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{s\in {\mathbb {C}}:\mathrm{Re}\, s>1\}$$\end{document}. In connection with this class of functions, we first determine the value of the maximum abscissa from which the images of any function in it cannot take a prefixed argument. The main result shows that each of these functions experiments a vortex-like behavior in the sense that the main argument of its images varies indefinitely near the vertical line Res=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{Re}\, s=1$$\end{document}. In particular, regarding the Riemann zeta function ζ(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\zeta (s)$$\end{document}, for every σ0>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _0>1$$\end{document} we can assure the existence of a relatively dense set of real numbers {tm}m≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{t_m\}_{m\ge 1}$$\end{document} such that the parametrized curve traced by the points (Re(ζ(σ+itm)),Im(ζ(σ+itm)))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathrm{Re} (\zeta (\sigma +it_m)),\mathrm{Im}(\zeta (\sigma +it_m)))$$\end{document}, with σ∈(1,σ0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma \in (1,\sigma _0)$$\end{document}, makes a prefixed finite number of turns around the origin.


Introduction
General Dirichlet series consist of those exponential sums that take the form n≥1 a n e −λns , a n ∈ C, s = σ + it, where {λ n } is a strictly increasing sequence of positive numbers tending to infinity. In particular, it is widely known that the Riemann zeta function ζ(s), which plays a pivotal role in analytic number theory, is defined as the analytic continuation of the function defined for σ > 1 by the sum ∞ n=1 1 n s , which constitutes a classical Dirichlet series.
In the beginnings of the 20th century, H. Bohr gave important steps in the understanding of Dirichlet series and their regions of convergence, uniform convergence and absolute convergence. As a result of his investigations on these functions, Bohr introduced an equivalence relation among them that led to the so-called Bohr's equivalence theorem, which shows that equivalent Dirichlet series take the same values in certain vertical lines or strips in the complex plane (e.g. see [1,2,6,10]).
Precisely, by analogy with Bohr's theory, we established in [7] an equivalence relation ∼ on the classes S Λ consisting of exponential sums of the form j≥1 a j e λj p , a j ∈ C, λ j ∈ Λ, where Λ = {λ 1 , λ 2 , . . . , λ j , . . .} is an arbitrary countable set of distinct real numbers, and p is a parameter (in our case, it will be changed by s = σ + it in the complex case, or by t in the real case). In the context of almost periodic functions, to which this equivalence relation can also be extended, the main result of [7] refined Bochner's result in the sense that it was proved that the condition of almost periodicity is equivalent to that fact that every sequence of (vertical) translates has a subsequence that converges uniformly to an equivalent function (see also [8]). Throughout this work, we will use a generalization of Bohr's equivalence relation, defined in Section 2, which was used in [9] to get a a result like Bohr's equivalence theorem extended to certain classes of almost periodic functions in vertical strips {s ∈ C : α < Re s < β}.
Regarding the Riemann zeta function ζ(s), the connection between it and prime numbers was discovered by L. Euler, who proved the identity where the product on the right hand extends over all prime numbers p k . In view of the Euler product, it is easily seen that ζ(s) has no zeros in the half-plane σ > 1. It is also known that the Dirichlet series and the Euler product of ζ(s) converge absolutely in the half-plane σ > 1 and uniformly in σ ≥ 1 + δ for any δ > 0. A more advanced introduction to the theory surrounding the Riemann hypothesis can be found for example in [3].
In general terms, let {a 2 , a 3 , a 5 , . . . , a pj , . . .} be an arbitrary sequence of complex numbers such that |a pj | = 1 for each j = 1, 2, . . .. Take a 1 = 1 and define a n = a α1 p1 a α2 p2 · · · a α kn p kn when n = p α1 is not a prime number. Throughout this paper, associated with such a sequence {a 2 , a 3 , a 5 , . . .}, we will consider the generic exponential sum ∞ n=1 a n n s which also converges absolutely in σ > 1 and uniformly in σ ≥ 1 + δ for any δ > 0. For example, the choices a pj = 1 for each j = 1, 2, . . . and a pj = −1 for each j = 1, 2, . . . provide respectively the Riemann zeta function and the Dirichlet series of the Liouville function λ(n) = (−1) Ω(n) , where Ω(n) is the number of prime factors of n (counted with multiplicities). Precisely, we show in Proposition 2.3 that the image of these two functions on the real axis provides the above and below bounds for the absolute value of the image of each exponential sum of type (2) throughout every vertical line or closed half-strip in {s ∈ C : Re s > 1}.
With respect to the arguments of all exponential sums of type (2), the special choices a pj = i for each j = 1, 2, . . . and a pj = −i for each j = 1, 2, . . . provide bounds for the change of these arguments (see Lemma 3.1) and, in fact, they allows us to determine the maximum abscissa from which the images of any exponential sum of type (2) cannot take a prefixed argument (see Lemma 3.4). In particular, this yields that the images of any function equivalent to the Riemann zeta function cannot take negative real values on a certain half-plane of the form {s ∈ C : Re s > σ π }, with σ π > 1.
Likewise, from Euler-type product formula for these sums S(s) of type (2) (see Lemma 2.2), the main result of our paper shows that each one of these functions, and in particular the Riemann zeta function, experiments a vortex-like behavior in the sense that, given σ 0 > 1 and n ∈ N, there exists a relatively dense set of real numbers {t n,m } m≥1 such that, for each m = 1, 2, . . ., the image of the vector-valued function (Re S(σ + it n,m ), Im S(σ + it n,m )), for σ in the interval (1, σ 0 ), traces a curve in the plane which makes at least n turns around the origin (see Theorem 3.8 in this paper, and related results in Lemma 3.2 and propositions 3.6 and 3.7). To the best of our knowledge, this result has not been reported in the literature.

The class of functions equivalent to the Riemann zeta function
Based on the Bohr's equivalence relation, which was considered in [1, p.173] for general Dirichlet series, we defined in [7,8,9] new equivalence relations in the more general context of the classes S Λ of exponential sums of type (1). In this paper, we will use the following definition which constitutes the same equivalence relation as that of [9, Definition 2]. Definition 2.1. Given Λ = {λ 1 , λ 2 , . . . , λ j , . . .} a set of distinct real numbers, consider A 1 (p) and A 2 (p) two exponential sums in the class S Λ , say A 1 (p) = j≥1 a j e λj p and A 2 (p) = j≥1 b j e λj p . We will say that . .} be a basis of the vector space over the rational numbers generated by a set Λ = {λ 1 , λ 2 , . . . , λ j , . . .}, which implies that G Λ is linearly independent over the rational numbers and each λ j is expressible as a finite linear combination of terms of G Λ , say λ j = qj k=1 r j,k g k , for some r j,k ∈ Q.
By abuse of notation, we will say that G Λ is a basis for Λ. Moreover, we will say that G Λ is an integral basis for Λ when r j,k ∈ Z for each j, k, i.e. Λ ⊂ span Z (G Λ ) (it is worth noting that all the results of [7] which can be formulated in terms of an integral basis are also valid under Definition 2.1).
Although the following preliminary results are reasonably simple, we next provide their proof for the sake of completeness. We first obtain an Euler-type product for ζ x (s) on Re s > 1 and we prove that the convergence of this Euler-type product is uniform in every half-plane Re s ≥ 1 + δ, δ > 0. In this way, given δ > 0, for any s = σ + it such that σ ≥ 1 + δ we get which tends to 0 as m → ∞. Hence the result holds.
The following result, which is a clear consequence of the Euler product representation, shows that the functions ζ x (s) are bounded throughout every vertical line or closed strip in {s ∈ C : Re s > 1}. Proof. Let K be a strip in {s ∈ C : Re s > 1} with σ 0 = inf{Re s : s ∈ K} > 1 and take s = σ + it ∈ K. By the Euler-type product formula, we have

Moreover, it is clear that
Thus the result holds.

On the values of the arguments of the functions that are equivalent to the Riemann zeta function
Let ζ x (s) = n≥1 e <rj,x>i e −s log n , with Re s > 1 and x ∈ R ∞ , be an exponential sum which is equivalent to the Riemann zeta function. By Lemma 2.2 we know that ζ x (s) can be expressed in terms of the Eulertype product In our case, given x ∈ R ∞ , we will handle the mapping A ζx (s) : Notice that lim σ→∞ ζ(σ + it) = 1 for any t ∈ R (and hence lim σ→∞ Arg(ζ(σ + it)) = 0 for any t ∈ R). So, thanks to [9, Theorem 18], we state that every function ζ x (s) satisfies lim σ→∞ ζ x (σ + it) = 1 for any t ∈ R (and hence lim σ→∞ Arg(ζ x (σ+it)) = 0 for any t ∈ R). In particular, we have that lim σ→∞ A ζx (σ+it) = 0 for any t ∈ R.
As a consequence of the lemma above, given x ∈ R ∞ , the mapping A ζx (s) : U → R + considered in (4) leads to a function well defined. Furthermore, it is clear that the case ζ π 2 (s) is particularly significant in our context. We next prove the following lemma regarding this function.
In particular, this result yields the existence of a real number σ π > 1 such that the images of any function equivalent to the Riemann zeta function on the half-plane {s ∈ C : Re s > σ π } cannot take negative real values.

By Lemma 3.1, it is accomplished that
Finally, the result follows from Lemma 3.4.
We next focus our attention on the vortex-like behaviour of the Riemann zeta function, and of every function that is equivalent to it. For this reason, we first show the following two preliminary results. Proposition 3.6. Let x ∈ R ∞ , σ 0 > 1, θ ∈ (−π, π] and n ∈ N. Then there exists a real number ρ n with 1 < ρ n < σ 0 such that Arg(ζ x (s)) = θ is satisfied for at least n distinct values in the vertical strip {s ∈ C : ρ n < Re s < σ 0 }.
Then Arg(ζ x (s)) = θ is satisfied for at least n distinct values in the vertical strip {s ∈ C : ρ n < Re s < σ 0 } and the result follows.
Moreover, Theorem 3.8 (and other results as Lemma 3.4 and Proposition 3.5) can also be immediately extended to its reciprocal sum (and all exponential sums included in its equivalence class) which is expressed as a Dirichlet series over the Möbius function µ(n) in the following terms (see [11, p.3]