Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2011, Article ID 909674,15 pages doi:10.1155/2011/909674

Research Article

The Critical Strips of the Sums 1 + 2z + ••• + n

G. Mora and J. M. Sepulcre

Department of Mathematical Analysis, University of Alicante, 03080 Alicante, Spain Correspondence should be addressed to J. M. Sepulcre, jm.sepulcre@ua.es Received 15 November 2010; Accepted 7 March 2011 Academic Editor: Stephen Clark

Copyright © 2011 G. Mora and J. M. Sepulcre. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We give a partition of the critical strip, associated with each partial sum 1 + 2z + ••• + nz 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 by

which admits analytic solutions of the form za on the open set ^ = C \ (-to, 0] if and only if a is a zero of

For each integer n > 2, each function Gn(z) represents the nth partial sum of the Riemann zeta function Z(z) on the half-plane Re z < -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

f (z)+ f (2z) + ••• + f(nz) = 0, n > 2,

Gn(z) = 1 + 2z + ••• + nz.

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 n/2

Hn(z) = Gn(iz) = 1 + 2iz + ••• + ;

whose spectra, for each integer n > 2, are the finite sets

{ilnk : k = 1,...,n}. (1.4)

Furthermore, the functions Hn(z) have the property consisting on the existence of some value of x = Re z, say x0, such that either Re Hn(x0, y) = 0or Im Hn(x0, y) = 0 for all y e R. Indeed, x0 = 0 satisfies such property. Therefore, the Hn(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 by

2.T lim d-5)

x2-xi x2 - x1

where N(x1,x2, y1,y2) denotes the number of zeros of f (z) in the rectangle

xi < Re z < x2, y1 < Im z < y2, (1.6)

and y(y) is the mean function associated with ln |f (x + iy)| defined as

1 (T+a

<p(y) = Tlim rr ln\f(x + ^\dx- (L7)

TJ-T+a

The formula (1.5) is of asymptotic type, and it is based on the assumption of the existence of derivative of y(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|,

N(xi,x2,yi,y2) = ^(x2 - xi) + O(l), (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 Hn(z), inside the rectangle defined by the intersection of its critical strip with

the strip x1 < Re z < x2. Indeed, since the spectrum of Hn(z) is contained in the line segment [0, iln n] of the imaginary axis, a simple application of (1.8) leads to the formula

N(xi,x2,yi,y2) = —(x2 - xi) +0(1). (1.9)

Nevertheless, it is well known that the term 0(1) is an "obscure" function which we only know to represent a bounded quantity. In general, the term 0(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 0(1) when the function belongs to that special class of almost-periodic functions which contains, in particular, to our functions Hn(z). In fact, on this subject, we find in [3] the following result.

"There exist infinitely many rectangles x1 < Re z < x2, y1 < Im z <y2 in the critical strip of the function Hn(z) for which the number of zeros of Hn(z) is given by the formula

N(xi,x2,yi,y2) = (x2 - xi) + Q„, (1-10)

where Q.n is a real number with |Qn| < 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 + iy

y1 < y2 < ••• < yl, (1.11)

and Pv (z) is for v = 1,2,...,l a polynomial of degree < mv - 1 with

m1 + m2 + ••• + ml = n,

(1.12)

Pi(z)Pi(z)/ 0,

then the number N (gl, a,a + b) of the zeros (according to multiplicity) of the function

gi (z) = £ Pv (z)e^z (1.13)

in the infinite vertical strip

a < x < a + b (1.14)

satisfies the inequality

N(gj, a,a + b) - — (/// - /d)

<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 ¡1 < ¡2 < ■■■ < ¡l linearly independent and m1 = m2 = ■■■ = ml = 1, there exist infinitely many values for a, b such that, independently of i, the inequality

N(gi, a,a + b) - — (/// - /d)

(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,...,ak} of real numbers is said to be linearly independent if and only if any linear combination

with integers nj, implies that nj = 0 for all j = 1,..., k. For example, the set

{ln pi, ln p2,...,ln pk},

where p1,p2,.. .,pk are different prime numbers, is linearly independent. Nevertheless, for a given set of real numbers (x1,x2,...xl}, we can always suppose the existence of a basis {a1,a2,...,ak}. That is, on the one hand, (a1,a2,...,ak} is linearly independent and, on the other hand, for each m = 1,...,l, we can write

Xm = ^"'j nmjaj, (2.3)

where the nmj 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,...,ak} be a linearly independent set of nonnull real numbers. For arbitrary numbers b1,b2,...,bk and T, e > 0, there exists a real number t > T and integers n1, n2,...,nk such that

\taj - nj - bj \ <e, Vj = l,...,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 < Re z < x+T, y1 < Im z < y2, for some T > 0 and sufficiently large values of |yi| 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 (i.10) assuming that they adopt the normalized form

P (z) = 1 + 2 Wje^z, (2.5)

where the coefficients w; are nonnull complex numbers and the frequencies yj are positive real numbers so that

yi <f2 < ••• < fn. (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 e > 0 there exists a length l = l(e) such that every interval b < y <b +1 of length l on the imaginary axis contains at least one translation number t associated with e satisfying the inequality

\f (z + iT) - f (z)\< e, Vz eC. (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 e > 0. Then, any interval of length l, l = l(e,f), will be called an e-interval of almost periodicity of f.

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

Lemma 2.5. Let

P(z) = 1 + wje^2, wj e C \ {0} j=i

be an exponential polynomial with increasing positive frequencies < ••• < y„. Then, there exist two real numbers xi, x2 such that all the zeros of P(z) are in the strip

SP(z) = {z : x1 < Re z < x2}.

Furthermore, for „ = 1,2, there exists a value for y, say y0, such that either

{z : Re P(z) = 0} n {z : Im z = yo} = 0

(2.10)

{z : Im P(z) = 0} n {z : Im z = yj} = 0.

(2.11)

Proof. Since

lim P(x + iy) = 1,

P(x + iy) lim —-IL = i,

x ^ Wn&^n (x+l'y)

(2.12)

for any value of y, there exist x1 < 0 < x2 such that

|P(z) - 1| < 1, Vz with Re z < x1,

wne^nz

< 1, Vz with Re z > x2.

(2.13)

Hence, P(z) has no zero neither in the half-plane Re z < x1 nor in the half-plane Re z > x2. Consequently, all the zeros of P(z) are situated in the strip

xi < Re z < x2.

(2.14)

To prove the second part of the lemma, we will only consider the case Im P(z) = 0 (the

case Re P(z) = 0 is completely similar). In this case, for any positive integer n, the equation Im P(z) = 0 can be explicitly written as

^^(a, sm(^y) + fr cos(^y)) = (2.15) j=i

where a, = Re w, and ¡3, = Im w,. By defining

A,(y) = a, sin (f,-y) + fr, cos(f,y), for each ] = 1,...,n, (2.16)

equation (2.15) becomes

Ze^Ajiy) = 0. (2.17)

On the other hand, it is plain that the set of the zeros of each function A, (y), denoted by B,, is given by

Bj = J j- fjrkj - arctan ^ V k} e I (2.18)

where arctan(fr/a,) is taken as n/2 when a, = 0. Since ef1x > 0 for all real x, the case n = 1 easily follows by taking y = y0, for arbitrary y0 / Bi.

Now, assume that n = 2. If the sets B1 and B2 are distinct, suppose that there exists some y0 e B1 such that y0 / B2. Then, the right-line of equation y = y0 does not meet Im P(z) = 0. Indeed, if for some real x the point (x,y0) satisfies the equation Im P(z) = 0, then, from (2.17) and taking into account that y0 e B1, it necessarily would have A2(y0) = 0 and, therefore, y0 e 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

1 / \ 1 / \

— ( jrki - arctan — ) = — ( jrk2 - arctan — ), (2.19)

f 1 \ a1 / H2\ a2 /

and reciprocally. By defining the numbers

= t!l '' ~ /'2'

arctan(p1/a^j - f arctan(/a2)

equality (2.19) can be written as

ki - yk2 = b, (2.21)

which represents an equation with infinitely many solutions for integers k1 and k2. Let k1r k2 and k'i, k'2 be integers verifying (2.21). Then, by subtracting in (2.21), one has

(ki - k1) - y(k2 - k2) = 0, (2.22)

which implies that y must be necessarily a rational number (observe that it means, in particular, that the frequencies y1, y2 are linearly dependent) and, because of 0 < y1 < y2, the number y is a positive rational less than 1. On the other hand, since

arctan

arctan

(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

1 ft 1 &

y i =--arctan — =--arctan —. (2.25)

y1 ai y2 a2

Now, assume that, for any real number y, there exists a value of x such that

{z = x + iy : Im P{x,y) = 0} n {z : Im z = y} / 0. (2.26)

Thus, in particular, given y1, there exists a1 such that Im P(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 e (v,y1) and, therefore, Aj (y) / 0 for; = 1,2. Then, by assuming (2.26), if we divide (2.17) by ey1XA1(y) one has the following property.

For each y e (v, y1), there exists x such that the relation

1 ^Mg(/<2-/<i)x = o (2.27)

holds.

Now, by taking the limit in (2.27), when y ^ y\, it follows that the point (ai,yi) satisfies

1 + lim = o.

y ^ yi Ai( y)

(2.28)

However, since

lim ^44 = ^ y ^ yi Ai( y) ^îjwij

(2.29)

is positive, by substituting in (2.28), we are led to a contradiction. Consequently, the case B\ = 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) = i + £w^, wj e C \ {0} j=i

(2.30)

an exponential polynomial with increasing positive frequencies ¡i < ■■■ < ¡in forming a linearly independent set. Then, there exists a value for y, say y0, such that either

{z : Re P(z) = 0} n {z : Im z = yo} = 0

(2.31)

{z : Im P(z) = 0} n {z : Im z = yo) = 0.

(2.32)

Proof. For the sake of brevity, we will prove the lemma in the case Im P(z) = 0 (the case Re P(z) = 0 is completely similar). Consider the coefficients Wj 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)

Ji = j e J : a > > 0}, J2 = {j e J : a > 0,pj < 0},

J3 = [j e J : aj < 0,fr < ^, J4 = {j e J : aj < 0,fr > 0}.

Now, define the numbers

; 2 jt

8' 3 8' 5 8' 7 8'

£ V7e/,

if j e Ji, if j e J2, if j e J3, if j e J4.

(2.34)

Let us pick an arbitrary real number T and a positive e such that e < 1/4n. Then, by applying Theorem 2.2, there exists t > T and integers nj such that

taj - nj - bj | < e, Vj e J.

(2.35)

Hence, by substituting the values of aj and bj in the preceding inequality and multiplying by 2n, one has

n 4 + nj + 2nnj, if j e J1,

3 jt T + nj + 2nnj, if j e Ji,

5JT T + nj + 2nnj, if j e J3,

7JT T + nj + 2nnj, if j e J4.

(2.36)

where the nj's are real numbers such that |nj| < 1/2. Then, according to the definition of the Jk's, it is clear that

aj sin(tj + ßj cos(tj > 0, Vj e J.

(2.37)

Consequently,

Im P(x, t) = ^e^'x(aj sin (y;t) + ßj cos(fit)) > 0, Vx e j=1

(2.38)

and then the lemma follows by taking y0 = t. Corollary 2.7. Let

P(z) = 1 + w-jey,z, wj eC \ {0} j=1

be an exponential polynomial with increasing positive frequencies ¡i < ■■■ < ¡„ forming a linearly independent set. Then, there exist infinitely many rectangles {Rk} in the critical strip of P(z) such that either Re P (z) or Im P (z) is always positive at every point of the sides of each Rk that are parallel to the real axis.

Proof. By applying Lemma 2.5, determine the right lines of equations x = xi, x = x2 that define the strip xi < Re z < x2 where all the zeros of P(z) are comprised. Let m be an arbitrary integer, by taking Ti = m and by applying Theorem 2.2, just as we have done in Lemma 2.6, we determine a value tm > Ti such that Im P(x, tm) > 0 for all x e R. Now, again from Theorem 2.2, for T2 = tm there exists a value tm+i > T2 such that Im P(x,tm+i) > 0. Then, the four right lines of equations x = xi, y = tm, x = x2, y = tm+i define a rectangle, say Rm, such that Im P(z) is positive when z lies on any of the sides of Rm that are parallel to the real axis. By reiterating this process, we will obtain the infinitely many rectangles {Rk : k > m} desired. A completely analogous result we would have obtained if we had considered Re P(z) instead of Im P(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 = Im z, say y0, such that either Re f (x, y0)= 0 or Im f (x,y0)/ 0 for all x e R .By denoting this class by AS, it follows that AS is nonvoid. Indeed, from lemmas 2.5 and 2.6, AS contains all exponential polynomials of degree „ = i, 2 with increasing positive frequencies and all exponential polynomials of arbitrary degree with linearly independent positive frequencies. Then, in particular, G2(z) = i + 2z belongs to AS, and, since the frequencies log 2 < log 3 are linearly independent, the function G3(z) = i+2z+3z is also in the class As. However, although for any „ > 4, the frequencies log 2 < log 3 < ■■■ < log „ are always linearly dependent, all the approximants

G„(z) = i + 2z + ■■■ + „z (3.i)

of the Riemann zeta function Z(z), in the half-plane Re z < -i, belong to the class AS. Likewise, all the derivatives of G„(z) are in the class AS. To see that it is enough to check that for the value y0 = 0, Re G„k (x, 0) > 0, for all x e R and every k = 0, i, 2,.... Then, we are going to obtain our formula to count the zeros of the functions G„ (z) as a consequence of a general result on the functions of the class AS.

Theorem 3.1. Let f (z) be a function of the class AS. Then, there exist infinitely many rectangles {Rk} in the critical strip of f (z) such that the number of zeros inside each rectangle Rk, N(f (z); Rk), satisfies

< 2, (3.2)

where ¡d denotes the difference between the upper and the lower bounds of the spectrum of f (z) and hk is the height of Rk.

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 < Re z < x2}. (3.3)

On the other hand, since f (z) is an almost-periodic function, let ^TO_1 AjeHz be the Dirichlet series [7, p. 77] associated with f (z), denoted by

f (z) ~ £A^z. (3.4)

Then, the set of the Fourier exponents of the above series, also called Dirichlet exponents,

{¡j : j en}, (3.5)

forms the spectrum of f (z), say Sf (z). Now, because the lower and the upper bounds of the spectrum of f (z) enter in the spectrum, let us define

¡1 = min{ ¡j : j e N},

¡to = max{¡j : j eN}, (3.6)

i = ¡to- ¡1,

with ¡¡1, ¡to e Sf. Then, by considering the two almost-periodic functions

/(=} -1, J£L -1, (3.7)

from (3.4), it follows that these functions have associated Dirichlet series

TO A ' 1 TO A ' 1

j_1 A1 j_1 Ato

respectively. Now, as the Dirichlet exponents of the above series, ¡j+1 - ¡1 and ¡j+1 - ¡TO are strictly positive and negative, respectively, because of [7, Theorem 3.21], the functions

A\e>llZ

tend to zero as Re z ^ -to and Re z ^ +to uniformly with respect to y, respectively. Therefore, there exist two reals a, b with a < x1 < x2 < b such that

< 1, for Re z < a,

(3.10)

< 1, for Re z > b.

On the other hand, as f is of class AS, let y0 be a value of y = Im z such that, for instance,

Im f (x + iy0) > 0, Vx eR. (3.11)

Then, from continuity, given the interval K = [a, b] there exists 6 > 0 such that

Im f (x + iy0) > 6, Vx e K. (3.12)

Now, from Definition 2.4, by taking e1 = 6/2, let (0,Z1) be the e1-interval of almost periodicity of f (z), with l1 > 0. Then, Definition 2.3 involves the existence of a translation number t1 e (0, Z1) such that

If (z + iT1) - f (z) | < e1, Vz = x + iy0, with x e K. (3.13)

According to (3.12), inequality (3.13) implies

Im f (x + i(y0 + T1)) > ei, Vx e K. (3.14)

Then, the four right lines of equations x = a, y = y0, x = b, y = y0 + t1 define a rectangle S1 such that, from (3.12) and (3.14), Im f (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 e2 = 6/22, in the e2-interval of almost periodicity of f, (0, l2), with l2 > 0, there exists a translation number t2 e (0,l2) such that

|f (z + iT2) - f (z)| < e2, Vz = x + i(y0 + t^, with x e K. (3.15)

Then, inequalities (3.14) and (3.15) imply

Im f(x + i(y0 + t1 + t2)) > e2, Vx e K. (3.16)

Therefore, the four right lines of equations x = x1, y = y0 + t1, x = x2, y = y0 + t1 + t2 define a rectangle S2 such that Im f (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

{Sk : k eN},

(3.17)

with the property consisting of Im f (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 Sk. Now, the intersection of each Sk with the critical strip of f, defined by (3.3), is a new rectangle, say Rk. Then, we claim that the set

{Rk : k e N} (3.18)

is the desired family of rectangles. Indeed, firstly, we observe that the number of zeros of f (z) inside each rectangle Rk is equal to the number of zeros inside each rectangle Sk. Secondly, Rk and Sk have the same height hk. Then, the variation of the argument of f (z) on each rectangle Sk is held to the following considerations:

(1) since Im f (z) > 0on the sides of Sk that are parallel to the real axis, the variation of the argument of f (z) on these sides is less than n;

(2) from (3.10), the variation of the argument of f (z) on each side of Sk defined by the lines x = a and x = b is

H1 hk + 0a, with |0a| < n,

(3.19)

hk + 0b, with |0b| < n,

respectively. Therefore, noticing the previous considerations, the total variation of the argument of f(z) on each rectangle Sk is

(¡to - ¡1)hk + Of(Z),k, with |Of(Z),k| < 4n. (3.20)

Consequently, the number of zeros of f (z) inside each AS satisfies the inequality

< 2. (3.21)

Therefore, the Rk'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 Re z < -1, Gn (z) = 1 + 2z + ■■■ + nz, n > 2, can be partitioned in infinitely many rectangles {Rk : k e Z} such that the number of its zeros inside each rectangle Rk, N (Gn(z); Rk), satisfies

i\Ttn / \ r> \ hk ln n N(Gn(z);Rk) -

< 2, (3.22)

where hk is the height of Rk.

Proof. Firstly, from Lemma 2.5, for each integer n > 2, there exist two real numbers an, bn such that the critical strip of the zeros of Gn(z) is defined by

Scn(z) = {z : an< Re z < bn}. (3.23)

Now, starting from y0 = 0 and taking into account that G„(z) e , determine the family of rectangles {Rk : 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 e Sg.„(z) : 0 < Im z}. (3.24)

Now, defining the rectangle R-k as the conjugate of Rk-1 for k = 1,2,..., the desired partition of SG„(z) is formed by the rectangles of the family {Rk : k e Z}. Finally, noticing G„(z) = 0 if and only if G„ (z) = 0, one has

N(G„(z); R-k) = N(G„(z); Rk-i), k = 1,2..........(3.25)

and, since the set {ln k : 1 < k < „} is the spectrum of each G„(z), inequality (3.22) follows.

The corollary is then proved. □

References

[1] G. Mora, "A note on the functional equation F(z) + F(2z) + ... + F(nz) = 0," Journal of Mathematical Analysis and Applications, vol. 340, no. 1, pp. 466-475, 2008.

[2] B. J. Levin, Distribution of Zeros of Entire Functions, vol. 5 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, Revised edition, 1980.

[3] G. Mora and J. M. Sepulcre, "On the distribution of zeros of a sequence of entire functions approaching the Riemann zeta function," Journal of Mathematical Analysis and Applications, vol. 350, no. 1, pp. 409415, 2009.

[4] P. Turan, "On the distribution of zeros of general exponential polynomials," Publicationes Mathematicae Debrecen, vol. 7, pp. 130-136, 1960.

[5] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Clarendon Press, Oxford, UK, 1954.

[6] H. Bohr, Almost Periodic Functions, Chelsea Publishing Company, New York, NY, USA, 1947.

[7] C. Corduneanu, Almost Periodic Functions, John Wiley & Sons, New York, NY, USA, 1968.

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