The Critical Strips of the Sums 1+2^𝑧+⋯+𝑛^𝑧

Abstract

We give a partition of the critical strip, associated with each partial sum 1+2^𝑧+⋯+𝑛^𝑧 of the Riemann zeta function for Re 𝑧<−1, formed by infinitely many rectangles for which a formula allows us to count the number of its zeros inside each of them with an error, at most, of two zeros. A generalization of this formula is also given to a large class of almost-periodic functions with bounded spectrum.

1. Introduction

Some industrial processes can be modeled [1] by functional equations of the form 𝑓(𝑥)+𝑓(2𝑥)=0 or 𝑓(𝑥)+𝑓(2𝑥)+𝑓(3𝑥)=0, 𝑥>0. The generalization of these functional equations to the complex plane is formally given by𝑓(𝑧)+𝑓(2𝑧)+⋯+𝑓(𝑛𝑧)=0,𝑛≥2, (1.1) which admits analytic solutions of the form 𝑧^𝛼 on the open set Ω=ℂ⧵(−∞,0] if and only if 𝛼 is a zero of𝐺_𝑛(𝑧)≡1+2^𝑧+⋯+𝑛^𝑧. (1.2) For each integer 𝑛≥2, each function 𝐺_𝑛(𝑧) represents the 𝑛th partial sum of the Riemann zeta function 𝜁(𝑧) on the half-plane Re𝑧<−1, and it belongs to the class of the entire almost-periodic functions of exponential type. In [2], we can see a complete introduction devoted to the study of such class of functions. There, we can also find a theorem of Bohr [2, p. 270] that identifies the functions of the above class having their zeros in a strip (the critical strip) parallel to the real axis with those functions for which the upper and lower bounds of their spectra enter into the spectrum. That is, for instance, the case of the functions defined from (1.2) by means of a rotation of angle 𝜋/2𝐻_𝑛(𝑧)≡𝐺_𝑛(𝑖𝑧)=1+2^𝑖𝑧+⋯+𝑛^𝑖𝑧, (1.3) whose spectra, for each integer 𝑛≥2, are the finite sets{𝑖ln𝑘∶𝑘=1,…,𝑛}. (1.4) Furthermore, the functions 𝐻_𝑛(𝑧) have the property consisting on the existence of some value of 𝑥=Re𝑧, say 𝑥₀, such that either Re𝐻_𝑛(𝑥₀,𝑦)≠0 or Im𝐻_𝑛(𝑥₀,𝑦)≠0 for all 𝑦∈ℝ. Indeed, 𝑥₀=0 satisfies such property. Therefore, the 𝐻_𝑛(𝑧)’s belong to a very special class of almost-periodic functions whose study will be our main objective to determine a nonasymptotic formula that allows us to count the amount of zeros that they have inside the rectangles of a certain partition of their critical strips.

One of the most important formulae [2, p. 277] on the number of roots of an almost-periodic function, with closed and bounded spectrum, say 𝑓(𝑧), inside a rectangle in the strip where the zeros of 𝑓(𝑧) are located, is given by2𝜋lim_{𝑥2−𝑥1→∞}𝑁𝑥₁,𝑥₂,𝑦₁,𝑦₂𝑥₂−𝑥₁=𝜑^′𝑦₂−𝜑^′𝑦₁, (1.5) where 𝑁(𝑥₁,𝑥₂,𝑦₁,𝑦₂) denotes the number of zeros of 𝑓(𝑧) in the rectangle𝑥₁<Re𝑧<𝑥₂,𝑦₁<Im𝑧<𝑦₂, (1.6) and 𝜑(𝑦) is the mean function associated with ln|𝑓(𝑥+𝑖𝑦)| defined as𝜑𝑦=lim_𝑇→∞12𝑇^𝑇+𝛼_−𝑇+𝛼ln𝑓𝑥+𝑖𝑦𝑑𝑥. (1.7)

The formula (1.5) is of asymptotic type, and it is based on the assumption of the existence of derivative of 𝜑(𝑦). However, if the spectrum of 𝑓(𝑧) is contained in the boundary of a bounded convex polygon of the complex plane and all the vertices of the polygon enter into the spectrum, there exists a formula [2, p. 298] much more explicit than (1.5). For instance, if the polygon is reduced to a segment of the imaginary axis, the formula is, for sufficiently large values of |𝑦₁| and |𝑦₂|,𝑁𝑥₁,𝑥₂,𝑦₁,𝑦₂=𝑑𝑥2𝜋²−𝑥₁+𝑂(1), (1.8) where 𝑑 is the length of the segment.

Formula (1.8) could be used, for instance, to estimate the number of zeros, of our functions 𝐻_𝑛(𝑧), inside the rectangle defined by the intersection of its critical strip with the strip 𝑥₁≤Re𝑧≤𝑥₂. Indeed, since the spectrum of 𝐻_𝑛(𝑧) is contained in the line segment [0,𝑖ln𝑛] of the imaginary axis, a simple application of (1.8) leads to the formula𝑁𝑥₁,𝑥₂,𝑦₁,𝑦₂=ln𝑛𝑥2𝜋²−𝑥₁+𝑂(1). (1.9) Nevertheless, it is well known that the term 𝑂(1) is an “obscure” function which we only know to represent a bounded quantity. In general, the term 𝑂(1) that appears in formula (1.8) depends on the function and the rectangle where we are counting the number of its zeros. Our aim is to give much more precise information about the expression 𝑂(1) when the function belongs to that special class of almost-periodic functions which contains, in particular, to our functions 𝐻_𝑛(𝑧). In fact, on this subject, we find in [3] the following result.

“There exist infinitely many rectangles 𝑥₁<Re𝑧<𝑥₂, 𝑦₁<Im𝑧<𝑦₂ in the critical strip of the function 𝐻_𝑛(𝑧) for which the number of zeros of 𝐻_𝑛(𝑧)  is given by the formula 𝑁𝑥₁,𝑥₂,𝑦₁,𝑦₂=ln𝑛𝑥2𝜋²−𝑥₁+Ω_𝑛, (1.10)where Ω_𝑛 is a real number with |Ω_𝑛|<1.”

Now, by following the ideas exhibited in [3], our aim is to demonstrate that for the functions of that special class of almost-periodic functions, there exists a formula similar to that of (1.10) to determine the number of its zeros inside infinitely many rectangles in their critical strips with an error, at most, of two zeros.

In particular, our main result will also prove that the bound 𝑛−1 which appears in the formula that determines the number of the zeros of an exponential polynomial of degree 𝑛 inside certain rectangles of its critical strip can be substituted by a universal bound, namely, 2. In fact, to illustrate the scope of our result, we will start recalling an old theorem of Polya [4]:

“if 𝑧=𝑥+𝑖𝑦𝜇₁<𝜇₂<⋯<𝜇_𝑙, (1.11)and 𝑃_𝜈(𝑧) is for 𝜈=1,2,…,𝑙 a polynomial of degree ≤𝑚_𝜈−1 with 𝑚₁+𝑚₂+⋯+𝑚_𝑙𝑃=𝑛,₁(𝑧)𝑃_𝑙(𝑧)≠0, (1.12)then the number 𝑁(𝑔_𝑙,𝑎,𝑎+𝑏) of the zeros (according to multiplicity) of the function 𝑔_𝑙(𝑧)=^𝑙_𝜈=1𝑃_𝜈(𝑧)𝑒^{𝑖𝜇𝜈𝑧} (1.13)in the infinite vertical strip 𝑎≤𝑥≤𝑎+𝑏 (1.14)satisfies the inequality 𝑁𝑔_𝑙−𝑏,𝑎,𝑎+𝑏𝜇2𝜋^𝑙−𝜇₁≤𝑛−1.” (1.15)

Then, under the same hypotheses of the above theorem, our result could be stated as follows.

Under the hypotheses of theorem of Polya with frequencies 𝜇₁<𝜇₂<⋯<𝜇_𝑙 linearly independent and 𝑚₁=𝑚₂=⋯=𝑚_𝑙=1, there exist infinitely many values for 𝑎, 𝑏 such that, independently of 𝑙, the inequality 𝑁𝑔_𝑙−𝑏,𝑎,𝑎+𝑏𝜇2𝜋^𝑙−𝜇₁≤2 (1.16)holds.

This result will be an immediate consequence of Lemma 2.6 and Theorem 3.1 of the present paper.

2. Preliminaries

To prove our main theorem, we will use some elementary concepts and results such as the following.

Definition 2.1. A set {𝑎₁,𝑎₂,…,𝑎_𝑘} of real numbers is said to be linearly independent if and only if any linear combination ^𝑘_𝑗=1𝑛_𝑗𝑎_𝑗=0, (2.1) with integers 𝑛_𝑗, implies that 𝑛_𝑗=0 for all 𝑗=1,…,𝑘.

For example, the setln𝑝₁,ln𝑝₂,…,ln𝑝_𝑘, (2.2) where 𝑝₁,𝑝₂,…,𝑝_𝑘 are different prime numbers, is linearly independent. Nevertheless, for a given set of real numbers {𝑥₁,𝑥₂,…𝑥_𝑙}, we can always suppose the existence of a basis {𝑎₁,𝑎₂,…,𝑎_𝑘}. That is, on the one hand, {𝑎₁,𝑎₂,…,𝑎_𝑘} is linearly independent and, on the other hand, for each 𝑚=1,…,𝑙, we can write𝑥_𝑚=^𝑘_𝑗=1𝑛_𝑚𝑗𝑎_𝑗, (2.3) where the 𝑛_𝑚𝑗 are integers.

An important result on linearly independent sets of real numbers is the famous theorem of Kronecker [5, p. 382] which will be used in the following form.

Theorem 2.2 (Kronecker). Let {𝑎₁,𝑎₂,…,𝑎_𝑘} be a linearly independent set of nonnull real numbers. For arbitrary numbers 𝑏₁,𝑏₂,…,𝑏_𝑘 and 𝑇, 𝜖>0, there exists a real number 𝑡>𝑇 and integers 𝑛₁,𝑛₂,…,𝑛_𝑘 such that 𝑡𝑎_𝑗−𝑛_𝑗−𝑏_𝑗<𝜖,∀𝑗=1,…,𝑘. (2.4)

Given an entire almost-periodic function 𝑓 with closed and bounded spectrum, a rectangle in its critical strip will be defined as the intersection of the rectangle 𝑥<Re𝑧<𝑥+𝑇, 𝑦₁<Im𝑧<𝑦₂, for some 𝑇>0 and sufficiently large values of |𝑦₁| and |𝑦₂|, with the strip where the zeros of 𝑓 are situated, of course, by assuming that the critical strip is parallel to the real axis. Then, the number of zeros of 𝑓(𝑧) in a rectangle in its critical strip will merely be denoted by 𝑁(𝑓(𝑧);𝑥,𝑥+𝑇). Similarly, 𝑁(𝑓(𝑧);𝑦,𝑦+𝑇) will denote the number of zeros of 𝑓(𝑧) in a rectangle in its critical strip, provided that the critical strip of 𝑓(𝑧) to be parallel to the imaginary axis. Nevertheless, noticing the change 𝑧 by −𝑖𝑧 transforms the zeros of a strip parallel to the real axis onto the zeros of a strip parallel to the imaginary axis and conversely, from now on, we will do our study on those functions by assuming that their critical strips are parallel to the imaginary axis.

Because our aim is to study the number of zeros of almost-periodic functions, and noticing these functions, from Bochner’s theorem [2, p. 266], are characterized as uniform limits of exponential polynomials, we will start by demonstrating a formula of the type (1.10) assuming that they adopt the normalized form𝑃(𝑧)=1+^𝑛_𝑗=1𝑤_𝑗𝑒^𝜇𝑗𝑧, (2.5) where the coefficients 𝑤_𝑗 are nonnull complex numbers and the frequencies 𝜇_𝑗 are positive real numbers so that𝜇₁<𝜇₂<⋯<𝜇_𝑛. (2.6) Then, a normalized exponential polynomial of the form (2.5), not affecting the zeros, will be considered as a prototype of an almost-periodic function whose definition [6, p. 101] we recall.

Definition 2.3. An entire function 𝑓 is said to be almost periodic if and only if for every 𝜖>0 there exists a length 𝑙=𝑙(𝜖) such that every interval 𝑏<𝑦<𝑏+𝑙 of length 𝑙 on the imaginary axis contains at least one translation number 𝜏 associated with 𝜖 satisfying the inequality 𝑓(𝑧+𝑖𝜏)−𝑓(𝑧)≤𝜖,∀𝑧∈ℂ. (2.7)

From (2.7), we derive the notion of interval of almost periodicity.

Definition 2.4. Let 𝑓 be an almost-periodic entire function on ℂ and 𝜖>0. Then, any interval of length 𝑙, 𝑙=𝑙(𝜖,𝑓), will be called an 𝜖-interval of almost periodicity of 𝑓.

In each interval of almost periodicity of an exponential polynomial 𝑃(𝑧), the solutions of the equations Re𝑃(𝑧)=0, Im𝑃(𝑧)=0 have a very special form, as we will prove in the following result.

Lemma 2.5. Let 𝑃(𝑧)=1+^𝑛_𝑗=1𝑤_𝑗𝑒^𝜇𝑗𝑧,𝑤_𝑗∈ℂ⧵{0} (2.8) be an exponential polynomial with increasing positive frequencies 𝜇₁<⋯<𝜇_𝑛. Then, there exist two real numbers 𝑥₁, 𝑥₂ such that all the zeros of 𝑃(𝑧) are in the strip 𝑆_𝑃(𝑧)=𝑧∶𝑥₁<Re𝑧<𝑥₂. (2.9) Furthermore, for 𝑛=1,2, there exists a value for 𝑦, say 𝑦₀, such that either {𝑧∶Re𝑃(𝑧)=0}∩𝑧∶Im𝑧=𝑦₀=∅ (2.10) or {𝑧∶Im𝑃(𝑧)=0}∩𝑧∶Im𝑧=𝑦₀=∅. (2.11)

Proof. Since lim_{𝑥→−∞}𝑃𝑥+𝑖𝑦=1,lim_𝑥→+∞𝑃𝑥+𝑖𝑦𝑤_𝑛𝑒^{𝜇𝑛(𝑥+𝑖𝑦)}=1, (2.12) for any value of 𝑦, there exist 𝑥₁<0<𝑥₂ such that |𝑃(𝑧)−1|<1,∀𝑧withRe𝑧≤𝑥₁,𝑃(𝑧)𝑤_𝑛𝑒^𝜇𝑛𝑧−1<1,∀𝑧withRe𝑧≥𝑥₂. (2.13) Hence, 𝑃(𝑧) has no zero neither in the half-plane Re𝑧≤𝑥₁ nor in the half-plane Re𝑧≥𝑥₂. Consequently, all the zeros of 𝑃(𝑧) are situated in the strip 𝑥₁<Re𝑧<𝑥₂. (2.14) To prove the second part of the lemma, we will only consider the case Im𝑃(𝑧)=0 (the case Re𝑃(𝑧)=0 is completely similar). In this case, for any positive integer 𝑛, the equation Im𝑃(𝑧)=0 can be explicitly written as ^𝑛_𝑗=1𝑒^𝜇𝑗𝑥𝛼_𝑗sin𝜇_𝑗𝑦+𝛽_𝑗cos𝜇_𝑗𝑦=0, (2.15) where 𝛼_𝑗=Re𝑤_𝑗 and 𝛽_𝑗=Im𝑤_𝑗. By defining 𝐴_𝑗𝑦≡𝛼_𝑗sin𝜇_𝑗𝑦+𝛽_𝑗cos𝜇_𝑗𝑦,foreach𝑗=1,…,𝑛, (2.16) equation (2.15) becomes ^𝑛_𝑗=1𝑒^𝜇𝑗𝑥𝐴_𝑗𝑦=0. (2.17) On the other hand, it is plain that the set of the zeros of each function 𝐴_𝑗(𝑦), denoted by 𝐵_𝑗, is given by 𝐵_𝑗=1𝜇_𝑗𝜋𝑘_𝑗𝛽−arctan^𝑗𝛼_𝑗∶𝑘_𝑗∈ℤ, (2.18) where arctan(𝛽_𝑗/𝛼_𝑗) is taken as 𝜋/2 when 𝛼_𝑗=0. Since 𝑒^𝜇1𝑥>0 for all real 𝑥, the case 𝑛=1 easily follows by taking 𝑦=𝑦₀, for arbitrary 𝑦₀∉𝐵₁.
Now, assume that 𝑛=2. If the sets 𝐵₁ and 𝐵₂ are distinct, suppose that there exists some 𝑦₀∈𝐵₁ such that 𝑦₀∉𝐵₂. Then, the right-line of equation 𝑦=𝑦₀ does not meet Im𝑃(𝑧)=0. Indeed, if for some real 𝑥 the point (𝑥,𝑦₀) satisfies the equation Im𝑃(𝑧)=0, then, from (2.17) and taking into account that 𝑦₀∈𝐵₁, it necessarily would have 𝐴₂(𝑦₀)=0 and, therefore, 𝑦₀∈𝐵₂, which is a contradiction. Consequently, the lemma follows for the value 𝑦₀. Finally, we analyse the case 𝐵₁=𝐵₂. This case means that for each integer 𝑘₁ there exists another integer 𝑘₂ such that 1𝜇₁𝜋𝑘₁𝛽−arctan¹𝛼₁=1𝜇₂𝜋𝑘₂𝛽−arctan²𝛼₂, (2.19) and reciprocally. By defining the numbers 𝜇𝜇≡¹𝜇₂,𝛽𝑏≡arctan₁/𝛼₁−𝜇arctan𝛽₂/𝛼₂𝜋, (2.20) equality (2.19) can be written as 𝑘₁−𝜇𝑘₂=𝑏, (2.21) which represents an equation with infinitely many solutions for integers 𝑘₁ and 𝑘₂. Let 𝑘₁, 𝑘₂ and 𝑘^′₁, 𝑘^′₂ be integers verifying (2.21). Then, by subtracting in (2.21), one has 𝑘₁−𝑘^′₁−𝜇𝑘₂−𝑘^′₂=0, (2.22) which implies that 𝜇 must be necessarily a rational number (observe that it means, in particular, that the frequencies 𝜇₁, 𝜇₂ are linearly dependent) and, because of 0<𝜇₁<𝜇₂, the number 𝜇 is a positive rational less than 1. On the other hand, since 𝛽arctan¹𝛼₁,𝛽arctan²𝛼₂≤𝜋2, (2.23)𝑏 is a rational number verifying |𝑏|<1. (2.24) Now, suppose the value 𝑘₂=0 is given. Then, there exists an integer 𝑘₁ satisfying (2.19) and, according to (2.21), it follows that 𝑘₁=𝑏. Hence, 𝑏 is an integer and then, noticing (2.24), 𝑏=0. Consequently, 𝑘₁=0. Since 𝐵₁=𝐵₂, let 𝑦₁ be the point of 𝐵₁=𝐵₂ corresponding to the values 𝑘₁=𝑘₂=0. Then, from (2.18), one has 𝑦₁1=−𝜇₁𝛽arctan¹𝛼₁1=−𝜇₂𝛽arctan²𝛼₂. (2.25) Now, assume that, for any real number 𝑦, there exists a value of 𝑥 such that 𝑧=𝑥+𝑖𝑦∶Im𝑃𝑥,𝑦=0∩𝑧∶Im𝑧=𝑦≠∅. (2.26) Thus, in particular, given 𝑦₁, there exists 𝑎₁ such that Im𝑃(𝑎₁,𝑦₁)=0. On the other hand, as the set 𝐵₁=𝐵₂ is discrete, there exists an open interval (𝑣,𝑦₁) such that one has 𝑦∉𝐵₁=𝐵₂ for any 𝑦∈(𝑣,𝑦₁) and, therefore, 𝐴_𝑗(𝑦)≠0 for 𝑗=1,2. Then, by assuming (2.26), if we divide (2.17) by 𝑒^𝜇1𝑥𝐴₁(𝑦) one has the following property.
For each 𝑦∈(𝑣,𝑦₁), there exists 𝑥 such that the relation 𝐴1+²𝑦𝐴₁𝑦𝑒^{(𝜇2−𝜇1)𝑥}=0, (2.27)holds.
Now, by taking the limit in (2.27), when 𝑦→𝑦₁, it follows that the point (𝑎₁,𝑦₁) satisfies 1+lim_𝑦→𝑦1𝐴₂𝑦𝐴₁𝑦𝑒^{(𝜇2−𝜇1)𝑎1}=0. (2.28) However, since lim_𝑦→𝑦1𝐴₂𝑦𝐴₁𝑦=𝜇₂𝑤₂𝜇₁𝑤₁ (2.29) is positive, by substituting in (2.28), we are led to a contradiction. Consequently, the case 𝐵₁=𝐵₂ follows, and the proof of the lemma is now completed.

When the frequencies are linearly independent, the preceding lemma is valid for arbitrary 𝑛.

Lemma 2.6. Let 𝑛 be an arbitrary positive integer and 𝑃(𝑧)=1+^𝑛_𝑗=1𝑤_𝑗𝑒^𝜇𝑗𝑧,𝑤_𝑗∈ℂ⧵{0} (2.30) an exponential polynomial with increasing positive frequencies 𝜇₁<⋯<𝜇_𝑛 forming a linearly independent set. Then, there exists a value for 𝑦, say 𝑦₀, such that either {𝑧∶Re𝑃(𝑧)=0}∩𝑧∶Im𝑧=𝑦₀=∅ (2.31) or {𝑧∶Im𝑃(𝑧)=0}∩𝑧∶Imz=𝑦₀=∅. (2.32)

Proof. For the sake of brevity, we will prove the lemma in the case Im𝑃(𝑧)=0 (the case Re𝑃(𝑧)=0 is completely similar). Consider the coefficients 𝑤_𝑗 of the exponential polynomial 𝑃(𝑧), since all them are nonnull, the set 𝐽={1,2,…,𝑛} can be partitioned in the following four disjoint sets (some of them could be eventually empty) 𝐽₁=𝑗∈𝐽∶𝛼_𝑗≥0,𝛽_𝑗,𝐽>0₂=𝑗∈𝐽∶𝛼_𝑗>0,𝛽_𝑗,𝐽≤0₃=𝑗∈𝐽∶𝛼_𝑗≤0,𝛽_𝑗,𝐽<0₄=𝑗∈𝐽∶𝛼_𝑗<0,𝛽_𝑗≥0. (2.33) Now, define the numbers 𝑎_𝑗≡𝜇_𝑗𝑏2𝜋,∀𝑗∈𝐽,_𝑗≡18,if𝑗∈𝐽₁,38,if𝑗∈𝐽₂,58,if𝑗∈𝐽₃,78,if𝑗∈𝐽₄. (2.34) Let us pick an arbitrary real number 𝑇 and a positive 𝜖 such that 𝜖<1/4𝜋. Then, by applying Theorem 2.2, there exists 𝑡>𝑇 and integers 𝑛_𝑗 such that 𝑡𝑎_𝑗−𝑛_𝑗−𝑏_𝑗<𝜖,∀𝑗∈𝐽. (2.35) Hence, by substituting the values of 𝑎_𝑗 and 𝑏_𝑗 in the preceding inequality and multiplying by 2𝜋, one has 𝑡𝜇_𝑗=𝜋4+𝜂_𝑗+2𝜋𝑛_𝑗,if𝑗∈𝐽₁,3𝜋4+𝜂_𝑗+2𝜋𝑛_𝑗,if𝑗∈𝐽₂,5𝜋4+𝜂_𝑗+2𝜋𝑛_𝑗,if𝑗∈𝐽₃,7𝜋4+𝜂_𝑗+2𝜋𝑛_𝑗,if𝑗∈𝐽₄. (2.36) where the 𝜂_𝑗’s are real numbers such that |𝜂_𝑗|<1/2. Then, according to the definition of the 𝐽_𝑘’s, it is clear that 𝛼_𝑗sin𝑡𝜇_𝑗+𝛽_𝑗cos𝑡𝜇_𝑗>0,∀𝑗∈𝐽. (2.37) Consequently, Im𝑃(𝑥,𝑡)=^𝑛_𝑗=1𝑒^𝜇𝑗𝑥𝛼_𝑗sin𝜇_𝑗𝑡+𝛽_𝑗cos𝜇_𝑗𝑡>0,∀𝑥∈ℝ, (2.38) and then the lemma follows by taking 𝑦₀=𝑡.

Corollary 2.7. Let 𝑃(𝑧)=1+^𝑛_𝑗=1𝑤_𝑗𝑒^𝜇𝑗𝑧,𝑤_𝑗∈ℂ⧵{0} (2.39) be an exponential polynomial with increasing positive frequencies 𝜇₁<⋯<𝜇_𝑛 forming a linearly independent set. Then, there exist infinitely many rectangles {𝑅_𝑘} in the critical strip of 𝑃(𝑧) such that either Re𝑃(𝑧) or Im𝑃(𝑧) is always positive at every point of the sides of each 𝑅_𝑘 that are parallel to the real axis.

Proof. By applying Lemma 2.5, determine the right lines of equations 𝑥=𝑥₁, 𝑥=𝑥₂ that define the strip 𝑥₁<Re𝑧<𝑥₂ where all the zeros of 𝑃(𝑧) are comprised. Let 𝑚 be an arbitrary integer, by taking 𝑇₁=𝑚 and by applying Theorem 2.2, just as we have done in Lemma 2.6, we determine a value 𝑡_𝑚>𝑇₁ such that Im𝑃(𝑥,𝑡_𝑚)>0 for all 𝑥∈ℝ. Now, again from Theorem 2.2, for 𝑇₂=𝑡_𝑚 there exists a value 𝑡_𝑚+1>𝑇₂ such that Im𝑃(𝑥,𝑡_𝑚+1)>0. Then, the four right lines of equations 𝑥=𝑥₁, 𝑦=𝑡_𝑚, 𝑥=𝑥₂, 𝑦=𝑡_𝑚+1 define a rectangle, say 𝑅_𝑚, such that Im𝑃(𝑧) is positive when 𝑧 lies on any of the sides of 𝑅_𝑚 that are parallel to the real axis. By reiterating this process, we will obtain the infinitely many rectangles {𝑅_𝑘∶𝑘≥𝑚} desired. A completely analogous result we would have obtained if we had considered Re𝑃(𝑧) instead of Im𝑃(𝑧). Our corollary is then proved.

3. A Class of Almost-Periodic Functions with Bounded Spectrum Containing the Partial Sums of the Riemann Zeta Function

In this section, we are going to generalize the preceding results to the class of almost-periodic functions 𝑓(𝑧) with bounded spectrum having the property of the existence of some value of 𝑦=Im𝑧, say 𝑦₀, such that either Re𝑓(𝑥,𝑦₀)≠0 or Im𝑓(𝑥,𝑦₀)≠0 for all 𝑥∈ℝ. By denoting this class by 𝒜_𝒮, it follows that 𝒜_𝒮 is nonvoid. Indeed, from lemmas 2.5 and 2.6, 𝒜_𝒮 contains all exponential polynomials of degree 𝑛=1,2 with increasing positive frequencies and all exponential polynomials of arbitrary degree with linearly independent positive frequencies. Then, in particular, 𝐺₂(𝑧)≡1+2^𝑧 belongs to 𝒜_𝒮, and, since the frequencies log2<log3 are linearly independent, the function 𝐺₃(𝑧)≡1+2^𝑧+3^𝑧 is also in the class 𝒜_𝒮. However, although for any 𝑛≥4, the frequencies log2<log3<⋯<log𝑛 are always linearly dependent, all the approximants𝐺_𝑛(𝑧)≡1+2^𝑧+⋯+𝑛^𝑧 (3.1) of the Riemann zeta function 𝜁(𝑧), in the half-plane Re𝑧<−1, belong to the class 𝒜_𝒮. Likewise, all the derivatives of 𝐺_𝑛(𝑧) are in the class 𝒜_𝒮. To see that it is enough to check that for the value 𝑦₀=0, Re𝐺_𝑛^(𝑘)(𝑥,0)>0, for all 𝑥∈ℝ and every 𝑘=0,1,2,…. Then, we are going to obtain our formula to count the zeros of the functions 𝐺_𝑛(𝑧) as a consequence of a general result on the functions of the class 𝒜_𝒮.

Theorem 3.1. Let 𝑓(𝑧) be a function of the class 𝒜_𝒮. Then, there exist infinitely many rectangles {𝑅_𝑘} in the critical strip of 𝑓(𝑧) such that the number of zeros inside each rectangle 𝑅_𝑘, 𝑁(𝑓(𝑧);𝑅_𝑘), satisfies𝑁𝑓(𝑧);𝑅_𝑘−𝜇ℎ_𝑘2𝜋<2, (3.2) where 𝜇 denotes the difference between the upper and the lower bounds of the spectrum of 𝑓(𝑧) and ℎ_𝑘 is the height of 𝑅_𝑘.

Proof. As 𝑓(𝑧) is an entire function of exponential type with bounded spectrum, from the Bohr theorem [2, p. 270], its zeros are all in a critical strip which we can suppose parallel to the imaginary axis (otherwise, we would consider the function 𝑔(𝑧)≡𝑓(𝑖𝑧)). Hence, without loss of generality, we can assume the existence of two real numbers 𝑥₁, 𝑥₂ such that all the zeros of 𝑓(𝑧) are located in the strip 𝑧∶𝑥₁<Re𝑧<𝑥₂. (3.3)
On the other hand, since 𝑓(𝑧) is an almost-periodic function, let ∑^∞_𝑗=1𝐴_𝑗𝑒^𝜇𝑗𝑧 be the Dirichlet series [7, p. 77] associated with 𝑓(𝑧), denoted by 𝑓(𝑧)∼^∞_𝑗=1𝐴_𝑗𝑒^𝜇𝑗𝑧. (3.4) Then, the set of the Fourier exponents of the above series, also called Dirichlet exponents, 𝜇_𝑗∶𝑗∈ℕ, (3.5) forms the spectrum of 𝑓(𝑧), say 𝑆_𝑓(𝑧). Now, because the lower and the upper bounds of the spectrum of 𝑓(𝑧) enter in the spectrum, let us define 𝜇₁≡min𝜇_𝑗,𝜇∶𝑗∈ℕ_∞≡max𝜇_𝑗∶𝑗∈ℕ,𝜇≡𝜇_∞−𝜇₁, (3.6) with 𝜇₁,𝜇_∞∈𝑆_𝑓. Then, by considering the two almost-periodic functions 𝑓(𝑧)𝐴₁𝑒^𝜇1𝑧𝑓(𝑧)−1,𝐴_∞𝑒^𝜇∞𝑧−1, (3.7) from (3.4), it follows that these functions have associated Dirichlet series ^∞_𝑗=1𝐴_𝑗+1𝐴₁𝑒^{(𝜇𝑗+1−𝜇1)𝑧},^∞_𝑗=1𝐴_𝑗+1𝐴_∞𝑒^{(𝜇𝑗+1−𝜇∞)𝑧}, (3.8) respectively. Now, as the Dirichlet exponents of the above series, 𝜇_𝑗+1−𝜇₁ and 𝜇_𝑗+1−𝜇_∞ are strictly positive and negative, respectively, because of [7, Theorem 3.21], the functions 𝑓(𝑧)𝐴₁𝑒^𝜇1𝑧𝑓(𝑧)−1,𝐴_∞𝑒^𝜇∞𝑧−1 (3.9) tend to zero as Re𝑧→−∞ and Re𝑧→+∞ uniformly with respect to 𝑦, respectively. Therefore, there exist two reals 𝑎, 𝑏 with 𝑎≤𝑥₁<𝑥₂≤𝑏 such that 𝑓(𝑧)𝐴₁𝑒^𝜇1𝑧−1<1,for𝑓(𝑧)Re𝑧≤𝑎,𝐴_∞𝑒^𝜇∞𝑧−1<1,forRe𝑧≥𝑏. (3.10) On the other hand, as 𝑓 is of class 𝒜_𝒮, let 𝑦₀ be a value of 𝑦=Im𝑧 such that, for instance, Im𝑓𝑥+𝑖𝑦₀>0,∀𝑥∈ℝ. (3.11) Then, from continuity, given the interval 𝐾≡[𝑎,𝑏] there exists 𝛿>0 such that Im𝑓𝑥+𝑖𝑦₀≥𝛿,∀𝑥∈𝐾. (3.12) Now, from Definition 2.4, by taking 𝜖₁=𝛿/2, let (0,𝑙₁) be the 𝜖₁-interval of almost periodicity of 𝑓(𝑧), with 𝑙₁>0. Then, Definition 2.3 involves the existence of a translation number 𝜏₁∈(0,𝑙₁) such that 𝑓𝑧+𝑖𝜏₁−𝑓(𝑧)≤𝜖₁,∀𝑧=𝑥+𝑖𝑦₀,with𝑥∈𝐾. (3.13) According to (3.12), inequality (3.13) implies Im𝑓𝑥+𝑖𝑦₀+𝜏₁≥𝜖₁,∀𝑥∈𝐾. (3.14) Then, the four right lines of equations 𝑥=𝑎, 𝑦=𝑦₀, 𝑥=𝑏, 𝑦=𝑦₀+𝜏₁ define a rectangle 𝑆₁ such that, from (3.12) and (3.14), Im𝑓(𝑧) is positive when 𝑧 lies on any of the sides of 𝑆₁ parallel to the real axis. Furthermore, from (3.10), one has 𝑓(𝑧)≠0 on the sides of 𝑆₁ parallel to the imaginary axis.
As above, by now taking 𝜖₂=𝛿/2², in the 𝜖₂-interval of almost periodicity of 𝑓, (0,𝑙₂), with 𝑙₂>0, there exists a translation number 𝜏₂∈(0,𝑙₂) such that 𝑓𝑧+𝑖𝜏₂−𝑓(𝑧)≤𝜖₂,∀𝑧=𝑥+𝑖𝑦₀+𝜏₁,with𝑥∈𝐾. (3.15) Then, inequalities (3.14) and (3.15) imply Im𝑓𝑥+𝑖𝑦₀+𝜏₁+𝜏₂≥𝜖₂,∀𝑥∈𝐾. (3.16) Therefore, the four right lines of equations 𝑥=𝑥₁, 𝑦=𝑦₀+𝜏₁, 𝑥=𝑥₂, 𝑦=𝑦₀+𝜏₁+𝜏₂ define a rectangle 𝑆₂ such that Im𝑓(𝑧) is positive on the sides of 𝑆₂ that are parallel to the real axis, and 𝑓(𝑧)≠0 on the sides of 𝑆₂ that are parallel to the imaginary one. Then, continuing in this way, we obtain a family of rectangles 𝑆_𝑘∶𝑘∈ℕ, (3.17) with the property consisting of Im𝑓(𝑧)>0 on the sides that are parallel to the real axis and 𝑓(𝑧)≠0 on the sides that are parallel to the imaginary one, for each rectangle 𝑆_𝑘. Now, the intersection of each 𝑆_𝑘 with the critical strip of 𝑓, defined by (3.3), is a new rectangle, say 𝑅_𝑘. Then, we claim that the set 𝑅_𝑘∶𝑘∈ℕ (3.18) is the desired family of rectangles. Indeed, firstly, we observe that the number of zeros of 𝑓(𝑧) inside each rectangle 𝑅_𝑘 is equal to the number of zeros inside each rectangle 𝑆_𝑘. Secondly, 𝑅_𝑘 and 𝑆_𝑘 have the same height ℎ_𝑘. Then, the variation of the argument of 𝑓(𝑧) on each rectangle 𝑆_𝑘 is held to the following considerations:(1)since Im𝑓(𝑧)>0 on the sides of 𝑆_𝑘 that are parallel to the real axis, the variation of the argument of 𝑓(𝑧) on these sides is less than 𝜋;(2)from (3.10), the variation of the argument of 𝑓(𝑧) on each side of 𝑆_𝑘 defined by the lines 𝑥=𝑎 and 𝑥=𝑏 is 𝜇₁ℎ_𝑘+Θ_𝑎,withΘ_𝑎𝜇<𝜋,_∞ℎ_𝑘+Θ_𝑏,withΘ_𝑏<𝜋, (3.19) respectively. Therefore, noticing the previous considerations, the total variation of the argument of 𝑓(𝑧) on each rectangle 𝑆_𝑘 is 𝜇_∞−𝜇₁ℎ_𝑘+Φ_{𝑓(𝑧),𝑘},withΦ_{𝑓(𝑧),𝑘}<4𝜋. (3.20) Consequently, the number of zeros of 𝑓(𝑧) inside each 𝒜_𝒮 satisfies the inequality 𝑁𝑓(𝑧);𝑅_𝑘−𝜇ℎ_𝑘2𝜋<2. (3.21) Therefore, the 𝑅_𝑘’s are the desired rectangles, as we claimed. Now, the proof of the theorem is completed.

Corollary 3.2. The critical strip associated with each partial sum of the Riemann zeta function on the half-plane Re𝑧<−1, 𝐺_𝑛(𝑧)≡1+2^𝑧+⋯+𝑛^𝑧, 𝑛≥2, can be partitioned in infinitely many rectangles {𝑅_𝑘∶𝑘∈ℤ} such that the number of its zeros inside each rectangle 𝑅_𝑘, 𝑁(𝐺_𝑛(𝑧);𝑅_𝑘), satisfies 𝑁𝐺_𝑛(𝑧);𝑅_𝑘−ℎ_𝑘ln𝑛2𝜋<2, (3.22) where ℎ_𝑘 is the height of 𝑅_𝑘.

Proof. Firstly, from Lemma 2.5, for each integer 𝑛≥2, there exist two real numbers 𝑎_𝑛, 𝑏_𝑛 such that the critical strip of the zeros of 𝐺_𝑛(𝑧) is defined by 𝑆_{𝐺𝑛(𝑧)}=𝑧∶𝑎_𝑛<Re𝑧<𝑏_𝑛. (3.23) Now, starting from 𝑦₀=0 and taking into account that 𝐺_𝑛(𝑧)∈𝒜_𝒮, determine the family of rectangles {𝑅_𝑘∶𝑘=0,1,2,…} whose existence is guaranteed from the preceding theorem. It is plain that this family of rectangles forms a partition of the upper critical strip 𝑧∈𝑆_{𝐺𝑛(𝑧)}∶0≤Im𝑧. (3.24) Now, defining the rectangle 𝑅_−𝑘 as the conjugate of 𝑅_𝑘−1 for 𝑘=1,2,…, the desired partition of 𝑆_{𝐺𝑛(𝑧)} is formed by the rectangles of the family {𝑅_𝑘∶𝑘∈ℤ}. Finally, noticing 𝐺_𝑛(𝑧)=0 if and only if 𝐺_𝑛(𝑧)=0, one has 𝑁𝐺_𝑛(𝑧);𝑅_−𝑘=𝑁𝐺_𝑛(𝑧);𝑅_𝑘−1,𝑘=1,2,…, (3.25) and, since the set {ln𝑘∶1≤𝑘≤𝑛} is the spectrum of each 𝐺_𝑛(𝑧), inequality (3.22) follows. The corollary is then proved.