Prime Geodesic Theorem for Compact Riemann Surfaces Dzenanˇ Gusiˇ C´

Home , Prime geodesic

INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume 13, 2019 Prime Geodesic Theorem for Compact Riemann Surfaces Dzenanˇ Gusiˇ c´

Abstract—As it is known, there have been a number of attempts to obtain precise estimates for the number of primes not exceeding as x → ∞, for compact, d-dimensional locally symmetric x. A lot of them are related to the ones done by Chebyshev. Thus, spaces of real rank one, where π (x) is the correspond- a good deal is known about them and their limitations. The truth, Γ or otherwise, of the Riemann hypothesis, however, has still not been ing counting function, i.e., a yes function counting prime established. In this paper we derive a prime geodesic theorem for a geodesics of the length not larger than log x. compact Riemann surface regarded as a quotient of the upper half- The same prime geodesic theorem was derived by Gangolli- plane by a discontinuous group. We assume that the surface at case, Warner [9] when the underlying locally symmetric space is not considered as a compact Riemannian manifold, is equipped with necessarily compact but has a finite volume. classical Poincare metric. Our result follows from the standard theory of the zeta functions of Selberg and Ruelle. The closed geodesics The first refinement of such prime geodesic theorem (in the in this setting are in one-to-one correspondence with the conjugacy case of non-compact, real hyperbolic manifolds with cusps) classes of the corresponding group, so analysis conducted here is was given by [18] (see, [17] for a related work). Later, it was reminiscent of the relationship between the distribution of rational further improved by Avdispahic-Gu´ siˇ c´ [1]. primes and Riemann zeta function. By analogy with the classical In the case of compact hyperbolic Riemann surfaces (Γ is arithmetic case and the fact that the Riemann hypothesis is true in our setting, one would certainly expect to obtain an analogous a co-finite discrete subgroup of PSL (2, R)), prime geodesic error term in the prime geodesic theorem. Bearing in mind that the theorem with error terms is given by corresponding Selberg zeta funcion has much more zeros than the Riemann zeta, the latter is not satisfied however. X s 3 α π (x) = li (x n ) + O x 4 (log x) Keywords—Compact Riemann surfaces, prime geodesic theorem, Γ 3 upper half-plane, zeta functions, Laplace operator. 4

as x → ∞, where sn is a zero of the corresponding Selberg I.INTRODUCTION 1 zeta function, and α = − 2 [14] (α = −1 [19]). HE positive integers other than 1 may be divided into two Note that there have been many works: Iwaniec [15], Luo- T classes: prime numbers (which do not admit of resolution Sarnak [16], Cai [3], Soundararajan-Young [21], to achieve into smaller factors), and composite ones which do. better error terms for a speciﬁc arithmetic discrete subgroup We shall denote by π (x) the number of primes not exceed- Γ ⊆ PSL (2, R). ing x. In the case of compact, locally symmetric spaces of real The statement rank one, the best known error term in the prime geodesic theorem was given by Avdispahic-Gu´ siˇ c´ [2] (see also, [10], π (x) x → 1 [13], [12]). log x In this paper we pay our particular attention to the case of compact hyperbolic Riemann surfaces, and the corresponding as x → ∞, is known as the prime number theorem and prime geodesic theorem. represents the central theorem in the theory of the distribution of primes. It is known that the problem of deciding its truth or II.PRELIMINARIES falsehood captured particular attention of mathematicians for almost a hundred of years. Let Ω be a compact Riemann surface, regarded as a quotient Any generalization of the prime number theorem to the of the upper half-plane H+ by a discontinuous group Γ. more general situations is known as a prime geodesic theorem. We assume that H+ is equipped with the metric Almost all generalized papers treat the case of hyperbolic y−2 (dx)2 + (dy)2 . Riemann surfaces (see e.g., [14], [19]). Denote by A the volume of Ω. Gangolli [8] (see also, [7], [22]) and DeGeorge [4] inde- As it is known, an element γ ∈ Γ, γ 6= 1 can be put into pendently proved that normal form z → N (γ) z, with N (γ) > 1. π (x) The number N (γ) is the same within a conjugacy class, Γ → 1 xd−1 and is called the norm of the element. (d−1) log x Thus, l (γ) = log N (γ) is the length of the closed geodesic corresponding to the conjugacy class of γ. Dz.ˇ Gusiˇ c´ is with the Department of Mathematics, Faculty of Sciences and Mathematics, University of Sarajevo, Zmaja od Bosne 33-35, 71000 Sarajevo, Recall that a closed geodesic is called primitive if it is not Bosnia and Herzegovina e-mail: [email protected] a positive integral power of any geodesic other than itself.

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Consequently, we define Λ(γ) to be log N (γ0), where γ = n γ , and γ0 is primitive. −1 0 Y −sl(γ0) Let g be the genus of Ω. ZR (s) = 1 − e , We assume that g ≥ 2. γ0∈P Γh It is understood that Γ ⊆ PSL (2, ). R Re (s) > 1, and is called the Ruelle zeta function. Ω −1 The Gaussian curvature of is . Define N (t) to be the number of zeros of Z (s) on the Γ P Γ Γ Let h resp. h denote the set of the -conjugacy classes critical line on the interval 1 + i x, with 0 < x ≤ t. of hyperbolic resp. primitive hyperbolic elements in Γ. 2 The Selberg zeta function is defined by (see, e.g., [20, p. 8], III.PRELIMINARY RESULTS [14, p. 72]) The following results will be applied in the sequel. +∞ Theorem 1. [14, p. 81, Th. 4.25.] Z (s) is an entire function Y Y −(s+k)l(γ ) Z (s) = 1 − e 0 , of order 2. γ0∈P Γh k=0 Theorem 2. [6, p. 509, Prop. 7.] Suppose Z (s) is the ratio Re (s) > 1. of two nonzero entire functions of order at most n. Then, there By [14, p. 72, Th. 4.10.]: is a D > 0 such that for arbitrarily large choices of r Z (s) is an entire function, 0 Z (s) has trivial zeros s = −k, k ≥ 1 with multiplicities Z Z (s) (2g − 2) (2k + 1), |ds| ≤ Drn log r. Z (s) s = 0 is a zero of multiplicity 2g − 1, r s = 1 is a zero of multiplicity 1, Theorem 3. [14, p. 102, Th. 6.4.] Let s = σ + i T , where 1 the non-trivial zeros of Z (s) are located at ± i rn. 2 −1 ≤ σ ≤ 2, and T 6= rn for all rn. Then, More precisely, the zeros in the critical strip 0 < Re (s) <

1 are located at points which are solutions of the equations 0 Z (s) X 1 s (1 − s) = λn, where λn runs through the sequence of = O (T ) + 1 . eigenvalues (omitting λ = 0) for the problem ∆f + λf = 0 Z (s) s − − i rn 0 |rn−T |≤1 2 on Ω, where ∆ is the Laplace operator on Ω. Theorem 4. [14, p. 102, Prop. 6.6.] Suppose that 0 < ε < Thus, the numbers rn are normalized by the condition π 1, s = σ + i t, t ≥ 1000. Then, Arg (rn) ∈ 0, − 2 , so the non-trivial zeros are located precisely at the points 0 Z (s) = O ε−1t 1 Z (s) s = + i r , n 2 n 1 for σ ≥ 1 + ε. s˜ = − i r . 2 n 2 n AIN RESULT Hence, apart from a finite number of exceptional zeros s0, IV. M s˜0,..., sM , s˜M concentrated along [0, 1], the non-trivial zeros The following theorem is the main result of this paper. of Z (s) all lie on the line Re (s) = 1 (the Riemann hypothesis 2 Theorem 5. Let Ω be as above. Then, is true for Z (s)). Let πΓ (x) be the number of prime geodesics over Ω, whose M length is not larger then log x. 3 X sn −1 In [19], the author derived a prime goedesic theorem for πΓ (x) = li (x) + li (x ) + O x 4 (log x) compact Riemann surfaces. n=1 The main purpose of this paper is to give yet another proof as x → ∞, where sk ∈ [0, 1], k ∈ {1, 2, ..., M} is a zero of of the same theorem (with the same error terms) by using the corresponding Selberg zeta function Z (s). different means. Note that one may interpret l (γ) as the period of a periodic Proof: Suppose that k ≥ 2 is an integer. 1 orbit for the geodesic flow on Ω. Let x > 1, and c > 2 · 2 = 1. This suggests defining a zeta function by We define,

−1 X Y −sτ(γ) ψ0 (x) = Λ(γ) , ZR (s) = 1 − e , N(γ)≤x γ∈P Z x ψj (x) = ψj−1 (t) dt, where P is the set of periodic orbits, and τ (γ) is the period 0 of γ. In our setting, this zeta function reads as where j = 1, 2,... .

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As it is known, Thus,

1 C1 + i T˜ − α > (3) −1 X j 2 T˜ ψj (x) = (j!) Λ(γ)(x − N (γ)) . N(γ)≤x for α ∈ S , where S is the set of zeros of the Selberg Furthermore, by [19, p. 244], Sel Sel zeta function Z (s). We put c+i ∞ 0 1 Z Z (s) × 2π i Z (s) c−i ∞ R (T ) × s−1 (s + 1)−1 ... (s + k)−1 xsds 1 = s ∈ C : |s| ≤ T, Re (s) ≤ ∪ k 2 1 X N (γ) = Λ(γ) 1 − + 1 k! x s ∈ : ≤ Re (s) ≤ c, −T˜ ≤ Im (s) ≤ T˜ , N(γ)≤x C 2 c+i ∞ Z 0 1 Z (s + 1) q × ˜2 1 2π i Z (s + 1) where T = T + 4 . c−i ∞ Bearing in mind the choice of the value T˜, and the location −1 −1 −1 s × s (s + 1) ... (s + k) x ds. of the zeros of Z (s), we conclude that no pole of Hence, 0 ZR (s) −1 −1 −1 s+k c+i ∞ 0 − s (s + 1) ... (s + k) x 1 Z Z (s) ZR (s) × 2π i Z (s) c−i ∞ occurs on the square part of the boundary of R (T ). × s−1 (s + 1)−1 ... (s + k)−1 xs+kds− Furthermore, we may, without loss of generality, assume

c+i ∞ 0 that no pole of 1 Z Z (s + 1) × 2π i Z (s + 1) 0 c−i ∞ ZR (s) −1 −1 −1 s+k −1 −1 −1 s+k − s (s + 1) ... (s + k) x × s (s + 1) ... (s + k) x ds. ZR (s) −1 X k = (k!) Λ(γ)(x − N (γ)) N(γ)≤x occurs on the circular part of the boundary of R (T ) (see, e.g., [18, p. 98]). =ψk (x) . Now, we are in position to apply the Cauchy integral As already noted (see, [20, p. 9]), formula to the integrand of ψk (x) over R (T ), to obtain Z (s + 1) ZR (s) = . (1) 0 Z (s) Z Z (s) − R × Hence, ZR (s) R(T )+ 0 0 0 Z (s) Z (s + 1) Z (s) −1 −1 −1 s+k − = − R . (2) × s (s + 1) ... (s + k) x ds Z (s) Z (s + 1) ZR (s) 0 X ZR (s) Consequently, =2π i Ress=z − × ZR (s) z∈R(T ) ! c+i ∞ 0 1 Z Z (s) × s−1 (s + 1)−1 ... (s + k)−1 xs+k , − R × 2π i ZR (s) c−i ∞ × s−1 (s + 1)−1 ... (s + k)−1 xs+kds where R (T )+ denotes the boundary of R (T ) with the anti- clockwise orientation. =ψk (x) . Therefore, Let T1 0 be arbitrary number. Following [19, p. 243], we consider the segment of the ˜ c+i T 0 critical line 1 + i t, with T − 1 < t ≤ T + 1. 1 Z Z (s) 2 1 1 − R × Applying the Dirichlet principle, we obtain that there exists 2π i ZR (s) (4) 1 ˜ c−i T˜ a 2 + i T in the segment whose distance from any root of Z (s) is greater than C1 , for some ﬁxed C > 0. −1 −1 −1 s+k T˜ 1 × s (s + 1) ... (s + k) x ds

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c+i ∞ 0 0 1 Z Z (s) X ZR (s) − R × = Ress=z − × 2π i Z (s) ZR (s) R z∈R(T ) c+i T˜ ! −1 −1 × s−1 (s + 1) ... (s + k) xs+kds+ × s−1 (s + 1)−1 ... (s + k)−1 xs+k − c−i T˜ Z 0 0 1 ZR (s) 1 Z Z (s) − × − R × 2π i ZR (s) 2π i ZR (s) c−i ∞ CT × s−1 (s + 1)−1 ... (s + k)−1 xs+kds, × s−1 (s + 1)−1 ... (s + k)−1 xs+kds+ 1 +δ+i T˜ 2 0 1 Z Z (s) − R × we estimate 2π i ZR (s) 1 ˜ 2 +i T c+i ∞ 0 −1 −1 −1 s+k 1 Z Z (s) × s (s + 1) ... (s + k) x ds+ − R × 2π i Z (s) 1 −i T˜ R 2Z 0 c+i T˜ 1 ZR (s) − × × s−1 (s + 1)−1 ... (s + k)−1 xs+kds (6) 2π i ZR (s) 1 ˜   2 +δ−i T +∞ Z dt × s−1 (s + 1)−1 ... (s + k)−1 xs+kds+ =O xc+k  = O xc+kT˜−k .  tk+1  ˜ c+i T 0 ˜ 1 Z Z (s) T − R × 2π i ZR (s) Similarly, 1 ˜ 2 +δ+i T −1 −1 −1 s+k × s (s + 1) ... (s + k) x ds+ c−i T˜ Z 0 1 +δ−i T˜ 1 ZR (s) 2 Z 0 − × 1 ZR (s) 2π i ZR (s) − × c−i ∞ 2π i ZR (s) (7) −1 −1 c−i T˜ × s−1 (s + 1) ... (s + k) xs+kds −1 −1 −1 s+k × s (s + 1) ... (s + k) x ds, =O xc+kT˜−k .

Combining (4)-(7), we obtain + where CT is the circular part of R (T ) . By [19, p. 242, Lemma 1], c+k −k ψk (x) − O x T˜

0 X ZR (s) = Ress=z − × 0 0 ZR (s) Z (s) X −s Z (s + 1) z∈R(T ) = Λ(γ) N (γ) + ! Z (s) Z (s + 1) −1 −1 γ∈Γh × s−1 (s + 1) ... (s + k) xs+k −

0 1 Z Z (s) for Re (s) > 1. − R × 0 2π i Z (s) Z (s) R Therefore, R is bounded in any half-plane of the form CT ZR(s) (8) Re (s) > 1 + ε. × s−1 (s + 1)−1 ... (s + k)−1 xs+kds+ 1 +δ+i T˜ Since 2 0 1 Z Z (s) − R × 2π i ZR (s) 1 +i T˜ ψk (x) 2 −1 −1 ˜ −1 s+k c+i T 0 × s (s + 1) ... (s + k) x ds+ 1 Z Z (s) R 1 −i T˜ = − × 2 (5) Z 0 2π i ZR (s) 1 ZR (s) ˜ − × c−i T 2π i Z (s) −1 −1 R −1 s+k 1 ˜ × s (s + 1) ... (s + k) x ds+ 2 +δ−i T

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  −1 −1 × s−1 (s + 1) ... (s + k) xs+kds+ ˜ X 1 =O T + O   ˜  1 ˜ 1  c+i T 0 ˜ + i T − − i rn 1 Z Z (s) |T −rn|≤1 2 2 − R ×   2π i Z (s) R 1 +δ+i T˜ X 2 =O T˜ + O T˜ 1 −1 −1   −1 s+k ˜ × s (s + 1) ... (s + k) x ds+ |T −rn|≤1 1 +δ−i T˜ 2 0 ˜ ˜2 ˜2 2 1 Z Z (s) =O T + O T = O T = O T − R × 2π i ZR (s) c−i T˜ ˜ 1 1 for s = σ1 + i T , 2 ≤ σ1 ≤ 2 + δ. × s−1 (s + 1)−1 ... (s + k)−1 xs+kds. Thus,

1 +δ+i T˜ 2 0 Now, we estimate the integrals on the right hand side of (8). 1 Z Z (s) − R × By (1) and Theorem 1, a meromorphic extension over C 2π i ZR (s) 1 ˜ of the Ruelle zeta function ZR (s) is a quotient of two entire 2 +i T (10) functions of order 2 over C. × s−1 (s + 1)−1 ... (s + k)−1 xs+kds Thus, by Theorem 2, 1 +δ+k −k+1 =O x 2 T .

0 1 Z Z (s) Similarly, − R × 2π i ZR (s) 1 ˜ 2 −i T CT Z 0 1 ZR (s) × s−1 (s + 1)−1 ... (s + k)−1 xs+kds − × 2π i ZR (s)  0  1 +δ−i T˜ Z 2 (11) 1 +k −k−1 ZR (s) 2 −1 −1 =O x T |ds| × s−1 (s + 1) ... (s + k) xs+kds ZR (s) (9) CT 1 +δ+k −k+1   =O x 2 T . Z 0 1 +k −k−1 ZR (s) 1 =O x 2 T |ds| By Theorem 4 and (2), we obtain that for s = σ + i T˜,   1 2 ZR (s) |s|=T + δ ≤ σ1 ≤ c, 1 +k −k+1 =O x 2 T log T . 0 0 0 Z (s) Z (s) Z (s + 1) − R = − ZR (s) Z (s) Z (s + 1) ˜ 1 0 By Theorem 3 and (2), we obtain that for s = σ1 + i T , 2 Z (s + 1) ≤ σ ≤ 1 + δ, =O δ−1T˜ − . 1 2 Z (s + 1)

0 Z (s+1) s = σ + i T˜ 1 + δ ≤ σ 0 0 0 Since Z(s+1) is bounded for 1 , 2 1 Z (s) Z (s) Z (s + 1) − R = − ≤ c, it follows that ZR (s) Z (s) Z (s + 1) X 1 0 =O T˜ + − ZR (s) −1 ˜ −1 s − 1 − i r − = O δ T = O δ T ˜ 2 n ZR (s) |T −rn|≤1 0 Z (s + 1) ˜ 1 . for s = σ1 + i T , 2 + δ ≤ σ1 ≤ c. Z (s + 1) Hence,

0 ˜ c+i T 0 Since Z (s+1) is bounded for s = σ + i T˜, 1 ≤ σ ≤ 1 1 Z Z (s) Z(s+1) 1 2 1 2 − R × A 2 2π i Z (s) + δ, N (t) = 4π t + O (t), and (3) holds true, we estimate R 1 ˜ 2 +δ+i T × s−1 (s + 1)−1 ... (s + k)−1 xs+kds 0 (12) ZR (s)  c+i T˜  − 0 Z Z (s) ZR (s)  c+k −k−1 R  =O x T |ds| ˜ X 1 ZR (s) =O T + 1 + O (1) 1 +δ+i T˜ s − − i r 2 ˜ 2 n |T −rn|≤1 =O δ−1xc+kT −k .

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Similarly, or M 1 3 +δ−i T˜ X sn −1 2 0 π (x) = li (x) + li (x ) + O x 4 (log x) . 1 Z Z (s) Γ − R × n=1 2π i Z (s) R This completes the proof. c−i T˜ (13) × s−1 (s + 1)−1 ... (s + k)−1 xs+kds V. CONCLUSION =O δ−1xc+kT −k . The idea for this research comes from [18], where the Combining (8)-(13), taking into account (2), the fact that k author applied the contour integration over circular bound- ≥ 2, and letting T → ∞, we obtain that aries. Recently, in [10] and [13], the author derived general results, which are analogous to the present result, in the case of compact, even-dimensional and odd-dimensional locally ψk (x) symmetric Riemannian manifolds of strictly negative sectional 0 X Z (s) curvature, respectively. For a weighted form of the even- = Res × s=z Z (s) dimensional result, we refer to [11]. In the future work z∈A0,k the author plans to consider a weighted form as well as a ! logarithmic form of the corresponding prime geodesic theorem −1 −1 −1 s+k × s (s + 1) ... (s + k) x − in the case of compact symmetric spaces formed as quotients of the Lie group SL4 (R) (see, [5]). 0 X Z (s + 1) Res × s=z Z (s + 1) REFERENCES z∈A1,k ! [1] M. Avdispahic´ and Dz.ˇ Gusiˇ c,´ ”On the error term in the prime geodesic −1 −1 −1 s+k theorem,” Bull. Korean Math. Soc., vol. 49, pp. 367–372, 2012. × s (s + 1) ... (s + k) x [2] M. Avdispahic´ and Dz.ˇ Gusiˇ c,´ ”On the length spectrum for compact locally symmetric spaces of real rank one,” WSEAS Trans. on Math., X X vol. 16, pp. 303–321, 2017. = cz (0, k) − cz (1, k) , [3] Y. Cai, ”Prime geodesic theorem,” J. Theor. Nombres Bordeaux, vol. 14, z∈A0,k z∈A1,k pp. 59–72, 2002. [4] D. L. DeGeorge, ”Length spectrum for compact locally symmetric where A , i ∈ {0, 1} denotes the set of poles of spaces of strictly negative curvature,” Ann. Sci. Ec. Norm. Sup., vol. i,k 10, pp. 133–152, 1977. [5] A. Deitmar and M. Pavey, ”A prime geodesic theorem for SL4,” Ann. 0 Glob. Anal. Geom., vol. 33, pp. 161–205, 2008. Z (s + i) s−1 (s + 1)−1 ... (s + k)−1 xs+k, [6] D. Fried, ”The zeta functions of Ruelle and Selberg. I,” Ann. Sci. Ec. Z (s + i) Norm. Sup., vol. 19, pp. 491–517, 1986. [7] R. Gangolli, ”Zeta functions of Selberg’s type for compact space forms and of symmetric spaces of rank one,” Illinois J. Math., vol. 21, pp. 1–41, 1977. c (i, k) [8] R. Gangolli, ”The length spectra of some compact manifolds of negative z curvature,” J. Diff. Geom., vol. 12, pp. 403–424, 1977. 0 Z (s + i) [9] R. Gangolli and G. Warner, ”Zeta functions of Selberg’s type for some = Ress=z × noncompact quotients of symmetric spaces of rank one,” Nagoya Math., Z (s + i) vol. 78, pp. 1–44, 1980. ! [10] Dz.ˇ Gusiˇ c,´ ”Prime geodesic theorem for compact even-dimensional −1 −1 −1 s+k locally symmetric Riemannian manifolds of strictly negative sectional × s (s + 1) ... (s + k) x . curvature,” WSEAS Trans. on Math., vol. 17, pp. 188–196, 2018. [11] Dz.ˇ Gusiˇ c,´ ”A Weighted Generalized Prime Geodesic Theorem,” WSEAS Trans. on Math., vol. 17, pp. 237–251, 2018. Putting k = 2, we end up with [12] Dz.ˇ Gusiˇ c,´ ”On the logarithmic derivative of zeta functions for compact, odd-dimensional hyperbolic spaces,” WSEAS Trans. on Math., vol. 18, pp. 176–184, 2019. ψ2 (x) X X [13] Dz.ˇ Gusiˇ c,´ ”On the Length Spectrum for Compact, Odd-dimensional, = cz (0, 2) − cz (1, 2) . (14) Real Hyperbolic Spaces,” WSEAS Trans. on Math., vol. 18, pp. 211– 222, 2019. z∈A z∈A 0,2 1,2 [14] D. Hejhal, The Selberg trace formula for PSL (2, R). Vol. I. Lecture Note that the equality (14) represents the equality from Notes in Mathematics 548. Berlin–Heidelberg: Springer–Verlag, 1976. 0 [15] H. Iwaniec, ”Prime geodesic theorem,” J. Reine Angew. Math., vol. 349, Theorem 1 in [19, p. 245] for k = 2. pp. 136–159, 1984. [16] W. Z. Luo and P. Sarnak, ”Quantum ergodicity of eigenvalues on Now, proceeding in exactly the same way as in [19, pp. 245- 2 PSL2 (Z) \ H ,” Inst. Hautes Etudes Sci. Publ. Math., vol. 81, pp. 246], one obtains the equality from Theorem 2 in [19, p. 245]. 207–237, 1995. Taking into account our notation, the aforementioned equal- [17] M. Nakasuji, ”Prime geodesic theorem for higher-dimensional hyper- ity may be written in the following, equivalent form bolic manifold,” Trans. Amer. Math. Soc., vol. 358, pp. 3285–3303, 2006. [18] J. Park, ”Ruelle zeta function and prime geodesic theorem for hyperbolic manifolds with cusps,” in Casimir force, Casimir operators and Riemann M hypothesis, Berlin 2010, pp. 89–104. sn X x 3 [19] B. Randol, ”On the asymptotic distributon of closed geodesics on 4 ψ0 (x) = + O x , compact Riemann surfaces,” Trans. Amer. Math. Soc., vol. 233, pp. 241– sn n=0 247, 1977.

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[20] D. Ruelle, Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval. CRM Monograph Series Vol. 4. Providence: American Mathematical Society, 1993. [21] K. Soundararajan and M. P. Young, ”The prime geodesic theorem,” J. Reine Angew. Math., vol. 676, pp. 105-120, 2013. [22] M. Wakayama, ”Zeta functions of Selberg’s type associated with ho- mogeneous vector bundles,” Hiroshima. Math. J., vol. 15, pp. 235–295, 1985.

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