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

1 IntroductionLet 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.

References

[1] E.W. Barnes. The theory of the G-function. Quart. J. Math., 31:264-314, 1899.

[2] H.M. Edwards. Riemann's zeta-Function. Academic Press, New York, 1974. Reprinted by Dover Publications, Mineola, N.Y. 2001.

[3] J. Fischer. An approach to the Selberg trace formula via the Selberg zeta-function. Lecture Notes in Mathematics, 1253. Springer-Verlag, Berlin, 1987.

[4] D.A. Hejhal. The Selberg trace formula for PSL(2, R). Vol. 1. Lecture Notes in Mathematics 548, Springer-Verlag, Berlin-New York, 1976.

[5] D.A. Hejhal. The Selberg trace formula for PSL(2, R). Vol. 2. Lecture Notes in Mathematics 1001, Springer-Verlag, Berlin-New York, 1983.

[6] H. Iwaniec. Prime geodesic theorem. J. Reine Angew. Math., 349:136-159, 1984.

[7] S. Koyama. Determinant expression of Selberg zeta-functions. Transactions of the American Mathematical Society, 324:149-168, 1991. http://dx.doi.org/10.1090/S0002-9947-1991-1041049-7.

[8] N. Kurokawa. Parabolic components of zeta-functions. Proc. Japan Acad.. Ser. A Math. Sci., 64:21-24, 1988. http://dx.doi.org/10.3792/pjaa.64.21.

[9] W. Luo. On zeros of the derivative of the Selberg zeta-function. Amer. J. of Math., 127(5):1141-1151, 2005. http://dx.doi.org/10.1353/ajm.2005.0032.

[10] W. Magnus, F. Oberhettinger and R.P. Soni. Formulas and theorems for the special functions of mathematical physics, volume 52 of Die Grundlehren der mathematischen Wissenschaften. Springer Berlin Heidelberg, Berlin, Heidelberg, New York, Tokyo, 1966. http://dx.doi.org/10.1007/978-3-662-11761-3.

[11] Y. Matiyasevich, F. Saidak and P. Zvengrowski. Horizontal monotonicity of the modulus of the Riemann zeta-function and related functions. arXiv:1205.2773v1 [math.NT], 2012.

[12] M. Minamide. On zeros of the derivative of the modified Selberg zeta-function for the modular group. The Journal of the Indian Mathematical Society, 80(34):275-312, 2013.

[13] B. Randol. Small eigenvalues of the Laplace operator on compact Riemann surfaces. Bulletin of the American Mathematical Society, 80:996-1000, 1974. http://dx.doi.org/10.1090/S0002-9904-1974-13609-8.

[14] B. Randol. On the asymptotic distribution of closed geodesics on compact Riemann surfaces. Trans. Amer. Math. Soc., 233:241-247, 1977. http://dx.doi.org/10.1090/S0002-9947-1977-0482582-9.

[15] F. Saidak and P. Zvengrowski. On the modulus of the Riemann zeta-function in the critical strip. Mathematica Slovaca, 53(2):145-172, 2003.

[16] P. Sarnak. Determinants of Laplacians. Communications in Mathematical Physics, 110(1):113-120, 1987. http://dx.doi.org/10.1007/BF01209019.

[17] J. Sondow and C. Dumitrescu. A monotonicity property of Riemann's xi-function and reformulation of the Riemann hypothesis. Periodica Mathematica Hungarica, 60(1):37-40, 2010. http://dx.doi.org/10.1007/s10998-010-1037-3.

[18] E.C. Titchmarsh. The Theory of Functions. 2nd edition, Oxford University Press, 1939.

[19] E.C. Titchmarsh. The Theory of the Riemann zeta-function. Oxford Science Publications. 2nd edition, Oxford University Press, 1986.

[20] M.F. Vigneras. L'equation fonctionnelle de la fonction zeta de Selberg du groupe modulaire PSL(2,Z). Asterisque, 61:235-249, 1979.

[21] E.T. Whittaker and G.N. Watson. A Course of Modern Analysis. Fourth edition, Cambridge University Press, 1965.

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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