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On the Modulus of the Selberg Zeta-Functions in the Critical Strip.

1 Introduction

Let s = [sigma] + it denote a complex variable. We start with the definition and some properties of the Riemann zeta-function. For [sigma] > 1, the Riemann zeta-function is given by the series

[zeta] (s) = [[infinity].summation over (n=1)] 1/[n.sup.s],

and can be analytically continued to the whole complex plane, except for a simple pole at s = 1 with residue 1. Trivial zeros of [zeta](s) are located at the negative even integers. The remaining, the so-called non-trivial zeros, lie on the critical strip 0 < [sigma] < 1. The Riemann zeta-function satisfies the functional equation

[zeta](s) = [2.sup.s] [[pi].sup.s-1] [zeta](1 - s) [GAMMA] (1 - s) sin [pi]s/2,

or [xi](s) = [xi](1 - s), where [xi](s) = 1/2 s(s - 1) [[pi].sup.-s/2 [GAMMA](s/2) [zeta](s), and [GAMMA](s) denotes the Euler gamma-function. The function [xi](s) is an entire function whose zeros are the non-trivial zeros of [zeta](s), see [19, [section]II].

In the paper [11], it was proved the following relation between functions [zeta](s) and [xi](s).

Theorem 1. The functions [zeta](s) and [xi](s) satisfy, for [absolute value of t] [greater than or equal to] 8 and [sigma] < 1/2, the inequality

Re [zeta]'(s)/[zeta](s) < Re [xi]'(s)/[xi](s).

Sondow and Dumitrescu proved in [17] the following theorem for the function [xi](s).

Theorem 2. The function [xi](s) is increasing in modulus along every horizontal half-line lying in any open right half-plane that contains no its zeros. Similarly, the modulus decreases on each horizontal half-line in any zero-free, open left half-plane.

In the same paper, the following reformulation for the Riemann hypothesis that all non-trivial zeros of [zeta](s) lie on the line [sigma] = 1/2 was given.

Theorem 3. The following statements are equivalent:

I. If t is any fixed read number, then [absolute value of [xi]([sigma] + it)] is increasing for 1/2 < [sigma] < [infinity].

II. If t is any fixed real number, then [absolute value of [xi](a + it)] is decreasing for -[infinity] < [sigma] < 1/2.

III. The Riemann hypothesis is true.

Later, Theorem 3 was reproved in [11] in a slightly different way.

Related properties of the functions [zeta](s) and [xi](s) in the critical strip were also investigated in [15].

In this paper, we ask whether Selberg zeta-functions have similar properties as the Riemann-zeta function has in Theorems 1 - 3. Note that, for Selberg zeta-functions, the analogue of the Riemann hypothesis is usually valid. We consider Selberg zeta-functions for cocompact and modular subgroups.

Let H be the upper half-plane, and [GAMMA] be a subgroup of PSL(2, R). Let [GAMMA]\H be a hyperbolic Riemann surface of finite area. The Selberg zeta-function [zeta](s) is defined [5], for [sigma] > 1, by

[mathematical expression not reproducible],

where {P} runs trough all primitive hyperbolic conjugacy classes of [GAMMA], and N(P) = [[alpha].sup.2] if the eigenvalues of P are [alpha] and [[alpha].sup.-1], [absolute value of [alpha]] > 1. The Selberg zeta-function has a meromorphic continuation to the whole complex plane [5].

If [GAMMA]\H is a compact Riemann surface of genus g [greater than or equal to] 2, we use the notation [zeta](s) = [Z.sub.C](s). If [GAMMA] = PSL(2,Z), then we denote [zeta](s) = [Z.sub.PSL(2,Z)] (s). Similarly, as the Riemann zeta-function, the Selberg zeta-function [Z.sub.PSL(2,Z)](s) has a meromorfic continuation to the whole complex plane, and satisfies the symmetric functional equation [8]

[XI] (s) = [XI] (1 - s),

where

[XI] (s) = [Z.sub.PSL(2,Z)] (s) [Z.sub.id] (s) [Z.sub.ell] (s) [Z.sub.par] (s),

and

[mathematical expression not reproducible]. (1.1)

The function [[GAMMA].sub.2](s) is called the double Barnes gamma-function, and is defined by the canonic product

[mathematical expression not reproducible],

where [[gamma].sub.0] denotes the Euler constant. The function [[GAMMA].sup.2](s) satisfies the relations

[mathematical expression not reproducible],

see, for example [1], [16] or [20].

The function [XI](s) is an entire function of order 2, and has zeros at the points [mathematical expression not reproducible] are the eigenvalues of the Laplace operator [3], [7]. The function [Z.sub.PSL(2,Z)](s) has poles and zeros at the following points [6]:

Poles of [Z.sub.PSL(2,Z)](s):

(1) s = 0; order 1,

(2) s = 1/2 - k, k [greater than or equal to] 0; order 1.

Zeros of [Z.sub.PSL(2,Z)](s):

(1) s = 1; order 1,

(2) s = -6k - j, k [greater than or equal to] 0,j = 1, 2, 3, 4, 6; order 2k + 1,

(3) s = -6k - 5, k [greater than or equal to] 0; order 2k + 3,

(4) s = [rho]/2, where [rho] are non-trivial zeros of [zeta](s)),

(5) s = 1/2 [+ or -] i[r.sub.n], n [greater than or equal to] 0.

We prove the following theorem.

Theorem 4. There exists a positive number C such that, for t > C and 0 < [sigma] < 1/2,

Re [XI]'(s)/[XI](s) < 0.

Furthermore, if we assume the Riemann hypothesis for [zeta](s), then there exists a positive number [C.sub.1] such that

Re [Z'.sub.PSL(2,Z)] (s)/[Z.sub.PSL(2,Z)] (s) < Re [XI]'(s)/[XI](s)

for t > [C.sub.1] and 0 < [sigma] < 1/4. Conversely, if

Re [Z'.sub.PSL(2,Z)] (s)/[Z.sub.PSL(2,Z)] (s) < 0

for t > [C.sub.1] and 0 < [sigma] < 1/4, then the function [zeta](s), for t > [C.sub.1], has zeros only for [sigma] = 1/2.

Theorem 4 is proved in the next section. Below, we formulate a couple of corollaries of Theorem 4. We also want to mention that a part of assertions of Theorem 4 can be obtained following the proof of Theorem 6.1 in [12].

Corollary 1. For a fixed sufficiently large t, the function [absolute value of [XI]([sigma] + it)] is decreasing for 0 < [sigma] < 1/2, and is increasing for 1/2 < [sigma] < 1 with respect to [sigma].

Corollary 2. If the Riemann hypotesis is true for [zeta](s), then, for a sufficiently large fixed t, the function [absolute value of [Z.sub.PSL(2,Z)]([sigma] + it)] is decreasing for 0 < [sigma] < 1/4 with respect to [sigma].

Proofs of Corollaries 1 and 2 follow from Lemma 1, functional equation [XI](s) = [XI] (1 - s) and equality [XI] ([bar.s]) = [bar.[XI](s)].

We return to Selberg zeta-functions attached to compact Riemann surfaces. The function [Z.sub.C](s) is an entire function of order 2 [4, [section]2.4, Theorem 2.4.25] and satisfies the functional equation [4, [section]2.4, Theorem 2.4.12]

[Z.sub.C] (s) = f (s)[Z.sub.C] (1 - s),

where

f(s) = exp(4[pi](g - 1) [[integral].sup.s-1/2.sub.0] v tan([pi]v) dv),

and g [greater than or equal to] 2 is the genus of a Riemann surface. The above functional equation is equivalent to M(s) = M(1 - s), where

[mathematical expression not reproducible].

The Selberg zeta-function [Z.sub.C](s) has trivial zeros at s = 1,0, -1, -2, ..., non-trivial zeros on the critical line [sigma] = 1/2 and also, possibly, on the interval (0,1) of the real axis, see [4, [section]2.4, Theorem 2.4.11] and [13]. In this sense, the analogue of the Riemann hypothesis holds for [Z.sub.C](s). Moreover, the following statement is true.

Theorem 5. There exists a positive number B such that the functions [Z.sub.C](s) and M(s), for t > B, 0 < [sigma] < 1/2, satisfy the inequality

RE [Z'.sub.C](s)/[Z.sub.C] (s) Re M'(s)/M(s) < 0.

Note that a part of Theorem 5 is proved in [9], namely,

Re [Z'.sub.C] (s)/[Z.sub.C] (s) < 0

for - c [less than or equal to] [sigma] [less than or equal to] 1/2 and t [greater than or equal to] [t.sub.0] > 0, where c > 0 is an arbitrary constant, and [t.sub.0] is a constant depending on c.

A couple of corollaries follow from Theorem 5 for functions [Z.sub.C] (s) and M(s).

Corollary 3. For a fixed and sufficiently large t, the function [absolute value of M([sigma] + it)] is decreasing for 0 < [sigma] < 1/2, and is increasing for 1/2 < [sigma] < 1.

Corollary 4. For a fixed and sufficiently large t, the function [absolute value of [Z.sub.C](a + it)] is decreasing for 0 < [sigma] < 1/2.

Proofs of Corollaries 3 and 4 are the same as proofs of Corollaries 1 and 2, and Theorem 5 is proved in Section 3.

2 Proof of the Theorem 4

Before the proof of Theorem 4, we state one lemma.

Lemma 1. (a) Let f be a holomorphic function in an open domain D and not identically zero. Let us also suppose Re f'(s)/f(s) < 0 for all s [member of] D such that f (s) [not equal to] 0. Then |f (s)| is strictly decreasing with respect to [sigma] in D, i.e., for each [s.sub.0] [member of] D, there exists [delta] > 0 such that [absolute value of f (s)] is strictly monotonically decreasing with respect to a on the horizontal interval from [s.sub.0] -[delta] to [s.sub.0] + [delta].

(b) Conversely, if [absolute value of f (s)] is decreasing with respect to [sigma] in D, then Re f'(s)/f9s) [less than or equal to] 0 for all s [member of] D such that f (s) [not equal to] 0.

The proof of the lemma is given in [11].

Remark 1. Of course, the analogous results hold for monotonically increasing [absolute value of f(s)] and f'(s)/f(s) > 0.

Now we prove Theorem 4.

Proof of Theorem 4. First we prove that

[mathematical expression not reproducible].

From the equality [XI](s) = [Z.sub.PSL(2,Z)](s)[Z.sub.id](s)Zell(s)Zva [GAMMA](s), we find that

[mathematical expression not reproducible].

Hence, to complete the proof it is sufficient to show that

ReU(s) > 0, t> [C.sub.1] > 0, 0 < [sigma] < 1/4.

By (1.1), we obtain

[mathematical expression not reproducible], (2.1)

where [mathematical expression not reproducible].

To prove the inequality ReU(s) > 0, we need to investigate the behavior of the functions [PSI](s), [[PSI].sub.2](s) and [zeta]'(2s)/[zeta](2s) in the region 0 < [sigma] < 1/4 and t > [C.sub.1] > 0. For the function [PSI] (s), the estimate [10]

[mathematical expression not reproducible],

holds. From this, we deduce that

Re[PSI](s) = log t + O (1/t), t [right arrow] [infinity], [absolute value of arg s] < [pi]. (2.2)

It is known [21] that, for -s [not member of] N

[mathematical expression not reproducible].

This and (2.2) show that

[mathematical expression not reproducible] (2.3)

for 0 < [sigma] < 1/4 and t > [C.sub.1] > 0.

From the formula [2]

[mathematical expression not reproducible],

we obtain that

[xi]'/[xi] (s) = [summation over ([rho]) 1/s - [rho],

where the summation runs over all non-trivial zeros of the Riemann zetafunction taken in conjugate pairs and in order of increasing imaginary parts. If [rho] = [beta] + i[gamma], then

[mathematical expression not reproducible].

If we assume the Riemann hypothesis, i.e., [beta] = 1/2, then Re[xi](s)'/[xi](s) > 0 for [sigma] > 1/2, and Re[xi](s)'/[xi](s) < 0 for [sigma] < 1/2.

On the other hand, from the equation

[xi](s) = (s - 1) [[pi].sup.-s/2] [GAMMA] (s/2 + 1) [zeta](s)

we get

[mathematical expression not reproducible].

This yields that, for [sigma] > 1/2,

Re [zeta]'(s)/[zeta](s) > 1/2 log t - 1/2 log 2[pi] + O (1/t), (2.4)

and, for [sigma] < 1/2,

-Re [zeta]'(s)/[zeta](s) > 1/2 log t - 1/2 log 2[pi] + O (1/t), (2.5)

In view of (2.1), (2.2), (2.3) and (2.5), we find that for t [right arrow] [infinity],

[mathematical expression not reproducible], (2.6)

where [a.sub.0] = 1/6 log 2[pi] + log [pi]/2 and c([sigma]) = log 1/2 + [sigma]/3 - 1/2. This shows that there exists a constant [C.sub.1] > 0 such that ReU(s) is positive for t > [C.sub.1] and 0 < [sigma] < 1/4. Hence, for t > C and 0 < [sigma] < 1/4,

Re [Z'.sub.PSL(2,Z)](s)/[Z.sub.PSL(2,Z)] (s) < Re [XI]'(s)/[XI](s).

We note that the restriction of [sigma] < 1/4 is due to the zeros of the function [zeta](2s).

Now we prove that

Re [XI]'(s)/[XI](s) < 0

for t > [C.sub.1] and 0 < [sigma] < 1/2. The function [XI](s) is an entire function of order two. It has a canonical product expansion [14], [18]

[mathematical expression not reproducible], (2.7)

where [??] runs over the nonzero roots of [XI](s), and a, b, c, and n are constants. This implies

[mathematical expression not reproducible].

If [??] = 1/2 + i[r.sub.n], n [greater than or equal to] 0, then the latter sum splits into two parts: for those [??] for which the numbers 1/2 + i[r.sub.n] are real, and for those [??] for which the numbers 1/2 + i[r.sub.n] are complex. There are only a finite number of real numbers 1/2 + i[r.sub.n]. Then

[mathematical expression not reproducible]. (2.8)

We see that the sum

[mathematical expression not reproducible]

is positive and unbounded as t [right arrow] [infinity]. Then, from equation (2.8), it follows that there exists a number C > 0 such that

Re [XI]'(s)/[XI](s) > 0

for t > C and 1/2 < [sigma] < 1. By a note after Lemma 1, for fixed t > C, the function [absolute value of [XI]([sigma] + it)] is monotonically increasing as a function of [sigma], 0 < [sigma] < 1/2. In view of the functional equation [XI](s) = [XI] (1 - s) and [XI] ([bar.s]) = [bar.[XI](s)], the function [absolute value of [XI]([sigma] + it)] is monotonically decreasing for t > C as a function of a, 1/2 < [sigma] < 1. So, the real part of its logarithmic derivative is negative, and the second assertion of the theorem holds.

The statement that if

Re [Z'.sub.PSL(2,Z)] (s)/ [Z.sub.PSL(2,Z)](s) < 0

for t > [C.sub.1] and 0 < [sigma] < 1/4, then the Riemann hypothesis is true, follows straightforward from Lemma 1 and the fact that the function [Z.sub.PSL(2,Z)](s) has zeros s = [rho]/2, where p are non-trivial zeros of [zeta](s). Recall that

[mathematical expression not reproducible],

where [a.sub.0] = 1/6 log 2[pi] + log [pi]/2 = 0.757 ....

Corollary 5. If 0 < [sigma] < 1/4, then

[mathematical expression not reproducible]

holds. If 1/2 < [sigma] < 1, then

[mathematical expression not reproducible]

holds, where c([sigma]) = log 1/2 +[sigma]3 - 1/2 and t [right arrow] [infinity].

Proof. The first part of the corollary follows from the fact Re ([XI]'/[XI] (s)) < 0, 0 < [sigma] < 1/2, and inequality (2.6). The second part is obtained analogically.

3 Proof of Theorem 5

Proof of Theorem 5. Recall that the Selberg zeta-function attached to compact Riemann surfaces satisfies the functional equation M(s) = M(1 - s), where

[mathematical expression not reproducible].

The function M(s) is an entire function of order two, and it has the same form of canonical product expansion (2.7) as the function [XI](s). So, for t > [t.sub.0] > 0, the function |M(s)| is monotonically decreasing with respect to 0 < [sigma] < 1/2.

Let

l(s) = exp ([[integral].sup..1/2-s.sub.0] v tan [pi]v dv).

To complete the proof, we need to show that

Re (l'(s)/l(s)) > 0

for 0 < [sigma] < 1/2, and t > [[??].sub.0]. By elementary calculation, we obtain

[mathematical expression not reproducible],

as t [right arrow] [infinity]. Taking B = max([t.sub.0], [[??].sub.0]) completes the proof.

In the same way the following corollary follows.

Corollary 6. If 0 < [sigma] < 1/2, then

-Re [Z'.sub.C](s)/[Z.sub.C] (s) > 2[pi](g - 1) x t x (1 + O ([e.sup.-2[pi]t])), t [right arrow] [infinity],

holds. If 1/2 < [sigma] < 1, then

-Re [Z'.sub.C](s)/[Z.sub.C] (s) > 2[pi](g - 1) x t x (1 + O ([e.sup.-2[pi]t])), t [right arrow] [infinity],

holds.

Proof. Proof is the same as that for Corollary 5.

4 Some remarks on the Riemann zeta-function

In this section, we present some remarks on the Riemann zeta-function [zeta](s), which could have been obtained proving Theorems 4 and 5.

Let, as above, [rho] = [beta] + i[gamma] be non-trivial zeros of [zeta](s). Recall that

[xi]'/[xi] (s) = [summation over ([rho]) 1/s - [rho],

where the summation is over all non-trivial zeros of the Riemann zeta-function taken in conjugate pairs in order of increasing imaginary parts. Also,

[xi]'/[xi] (s) = [xi]'/[xi] (s) + 1/2 [PSI] (s/2 + 1) -1/2 log [pi] + 1/s-1.

Comparing the latter equalities with

[zeta]'/[zeta] (s) = b - 1/s - 1 - 1/2 [PSI] (s/2 + 1) + [summation over ([rho])](1/s - [rho] + 1/[rho]).

where b = log 2[pi] - 1 - [[gamma].sub.0]/2, we have [19] that

[summation over ([rho])] 1/[rho] = 1 + [[gamma].sub.0]/2 - 1/2 log 4[pi]

The inequalities (2.4) and (2.5) give the bounds for the real part of the logarithmic derivative of the Riemann zeta-function in the half-planes [sigma] < 1/2 and [sigma] > 1/2, respectively. Assuming the Riemann hypothesis, allows to construct more precise bounds. For this we need some lemmas.

Lemma 2. Let N(T) be the number of zeros of [zeta](s) in the rectangle 0 < [sigma] < 1, 0 < t < T. Then, as T [right arrow] [infinity],

N (T) = T/2[pi] log T/2[pi] - T/2[pi] + R(T),

where -R(T) = O(log T). If the Riemann hypothesis is true, then R(T) = O (log T/log log T).

The proof of the lemma can be found, for example, in [19].

Lemma 3. For t > 1, the inequality

arctan t < [pi]/2 - 1/2t

holds.

Proof. We have that

[mathematical expression not reproducible].

Lemma 4. Let [[rho].sujb.1] = 1/2 + i[[gamma].sub.1], [[gamma].sub.1] = 14.134725 ..., be the first non-trivial zero of [zeta](s). Then

[summation over ([gamma]>0)] 1/1/4 + [([gamma] - t).sup.2] > 1/2 t/[[gamma].sub.1] + O(1/t)

and

[summation over ([gamma]>0)] 1/1/4 + [([gamma] - t).sup.2] > 1/8[pi]t log t/2[pi] + O(1/t)

Proof. By Lemma 2, summing by parts, we get

[mathematical expression not reproducible],

where the last inequality was obtained using that -[pi]/2 [less than or equalto] arctan v [less than or equal to] [pi]/2. This proves the first part of the lemma.

Similar arguments and Lemma 3 show that

[mathematical expression not reproducible].

It is well known that

[mathematical expression not reproducible].

Assume the Riemann hypothesis. Then, in view of Lemma 4, we obtain

[mathematical expression not reproducible].

Using this and (2.2), we find that

[mathematical expression not reproducible],

for 0 < [sigma] < 1/2, and

[mathematical expression not reproducible]

for 1/2 < [sigma] < 1 as t [right arrow] [infinity].

http://dx.doi.org/10.3846/13926292.2015.1119213

Acknowledgements

The first author is partially supported by grant (No. MIP-066/2012) from the

Research Council of Lithuania.

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Andrius Grigutis (a) and Darius Siauciunas (b)

(a) Faculty of Mathematics and Informatics, Vilnius University Naugarduko str. 24, LT-03225 Vilnius, Lithuania

(b) Institute of Informatics, Mathematics and E-Studies, Siauliai University P. Visinskio str. 19, LT-77156 Siauliai, Lithuania

E-mail(corresp.): siauciunas@fm.su.lt

E-mail: andrius.grigutis@mif.vu.lt

Received June 11, 2015; revised November 2, 2015; published online November 15, 2015
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