Zeros of the Selberg zeta function for non-compact surfaces

Transcription

1 Zeros of the Selberg zeta function for non-compact surfaces Abstract In the present paper we provide a rigorous mathematical foundation for describing the zeros of the Selberg zeta functions Z X for certain compact surfaces X with boundary, corresponding to convex cocompact Fuchsian groups Γ without parabolic points. For definiteness, we consider the case of symmetric pairs of pants where we show how Z X can be approximated by complex trigonometric polynomials on suitably large domain in Theorem.5. As our main application, we explain the striking empirical results of Borthwick [5] Theorem.4. Keywords Selberg zeta function non-compact surface configuration of zeros PACS M6 7C Introduction The Selberg zeta function Z X associated to a compact Riemann surface X with negative Euler characteristic and without boundary is a well known and much studied complex function. It is a function of a single complex variable defined in terms of the lengths lγ of the closed geodesics γ on the surface by analogy with the Riemann zeta function in number theory. Definition.. We can formally define the Selberg zeta function by Z X s = n γ e s+nlγ,. where the product is taken over all closed geodesics γ on X. It was shown by Selberg in 956 that for such surfaces the zeta function Z X has an analytic extension to the entire complex plane and that the non-trivial zeros can be described in terms of the spectrum of the Laplace Beltrami operator [6] and the Selberg trace formula.

2 INTRODUCTION Is Rs a b Figure : a A pair of pants X; b The zeros of the associated zeta function Z X in a critical strip; observe the difference in scales on the axes. The individual zeros are so close in the plot that they almost appear to be six continuous curves. Theorem. Selberg. Let X be a compact Riemann surface with negative Euler characteristic and without boundary. Then the function Z X has a simple zero at s = and for any zero s in the critical strip < Rs < we have that either s [,] is real, or Rs =. In the case of infinite area surfaces the situation is somewhat different, since the original trace formulae approach of Selberg no longer applies. However, it follows from the dynamical method of Ruelle that providing the surface X appears as a quotient space of H by a convex cocompact Fuchsian group then the analogous zeta function Z X still has an analytic extension to the entire complex plane. Unfortunately, this construction provides little effective information on the location of the zeros, other than that the first zero at s = δ,, the Hausdorff dimension of the limit set of the Fuchsian group. In interesting experimental work, Borthwick has studied the location of the zeros in classical examples of infinite area surfaces, such as pairs of pants [5]. In particular, for the case of symmetric pairs of pants he has observed a number of very interesting properties, some of which we will explain mathematically in this note. The starting point for our analysis has its roots in quite general results of Grothendieck and Ruelle. The study of the Selberg zeta function via the action of the Fuchsian group on the boundary H dates back to work of the first author approximately twenty five years ago [4], which provided the formulae used by Borthwick. However, it is only with the advent of superior computational resources, and the ingenuity of those that employ them, that the striking features seen in Figure have been revealed. The plot in Figure is fairly typical for the distribution of zeros in the critical strip for a sur-

3 INTRODUCTION face X with three boundary components each of which is sufficiently long. More precisely, a contemporary personal computer allows one to study surfaces with the length of boundary geodesics lγ 8. We denote by S X the zero set of the function Z X, i.e. S X : = {s C Z X s = }. Based on empirical evidence, one can make the following observations. Qualitative Observations. Let X be a pair of pants with boundary geodesics of the length b. Then the elements of S X follow a specific pattern, moreover, O : The vertical spacing of zeros is approximately π b. O : The pattern of zeros appears to lie on four distinct curves, which seem to have a common intersection point at δ + i π eb. O : The vertical apparent periodicity of the pattern of zeros is approximately πe b. However, we should stress that these qualitative observations are not completely rigorous for any fixed b >. For example the apparent periodicity in O is not a true periodicity as we see from the behaviour of the zeros near the vertical line Rs = δ in Figure. In fact, since the geodesic flow restricted to the non-wandering set is mixing, we know that there is only one zero with Rs = δ. Naud [] showed an even stronger result: for any ε > there is only finite number of zeros satisfying Rs > δ +ε. Nevertheless, partial numerical evidence illustrating the second observation O, yet showing how it can be reconciled with the rigorous results, is given in Table. Namely, we analyze values of zeros closest to the right boundary Rz = δ of the critical strip: { E := s S X for all s S X such that s s < π } eb we have Rs < Rs. We see that for all s E satisfying Is < there exist an s E such that s s + πe b. We will show that O O hold asymptotically, in a suitable sense, for b. The key to understanding the apparent asymptotic behaviour as b is the approximation of Z X by simpler functions with these precise properties, on suitable domains. Using estimates on the remainder of an absolutely convergent series formulation for Z X, obtained in Theorem.5 we get an approximation to Z X by determinants of certain explicit finite matrices, whose zero sets are easy to calculate analytically. We will be studying the zeta function restricted to a compact subset of the critical strip, Rs δ. To formulate our results precisely we need the following notation. This normalization makes formulae in subsequent calculations shorter. We have, in fact, verified this for larger values of Is, but we omit the numerics here.

4 INTRODUCTION Endpoints for strings of zeros with Rz δ b = π, z = δ =.46949, π expb = k z k z k z k i i i i i i5.8 b = 8, z = δ =.7887, π expb = k z k z k z k i i i i i i i i i i i i Table : Empirical estimates on zeros near the right boundary Rz = δ. Notation.. We will be using the following. The compact part of the critical strip of the height T we denote by RT = {s C Rs δ and Is T }. The compact part of the normalized critical strip of the height T we denote by RT = {s C Rs ln and Is T }. Let ŜX : = {s = σ +it Z σb X +ite b = } be the zeros of Z X mapped by σ +it bσ +ie b t. The surface X has an explicit dependence on b. We will be using a set of regularly spaced points on the lines Rs = and Rs = ln given by L = ln + iπz iπz. 4

5 INTRODUCTION Is Rs δ Rs Figure : A more careful look at the apparent periodicity near Rs = δ, the case lγ = 8. Let C = 4 j= C j, where { } C = ln cost t R ; { } C = ln + cost t R ; { C = ln eit } eit 4 e it t R ; { C 4 = ln eit + } eit 4 e it t R. The following result provides a rigorous explanation of the Observations O and O for large values of b. Theorem.4. The zero sets S X and ŜX satisfy the following.. There exist a subsequence s n S X and κ > such that sup dbs n,l = O b, s n S X Re κb. The affinely transformed zeros ŜX are close to C, more precisely, for any κ we have sup ds,c = O b, s ŜX Re κb 5

6 INTRODUCTION Observation O corresponds to the following remark. Remark.5. The four curves C,,C 4 have common points, more specifically 4 C j = j= { ln + iπ + k }, k Z. The set C is invariant with respect to vertical translation by σ + it σ + it + π..5 Is Rs a b Figure : a The dashed lines correspond to the curves C i and the circles correspond to the points of Ŝ X for b = 5; b A zoomed version in a neighbourhood of ln, π 4. A version of Theorem.4 part for a fixed domain Rk was also observed by Weich [7]. An element in the approach to this program touches on a more obvious question of how accurately we can approximate Z X using finitely many closed geodesics. In particular, we need to estimate the approximation error in terms of the lengths of the boundary geodesic. A similar analysis can be carried out for a punctured torus and for less symmetric surfaces. However, for simplicity of exposition we will concentrate on this example and leave the analysis in these other cases to another occasion. This project has had a long gestation period, having begun after the first author heard the original empirical results of David Borthwick presented at a conference on Quantum Chaos in Roscoff in June of. We are grateful to him for sharing his original Matlab code with us. We are grateful to F. Bykov for writing a new program. A preliminary announcement of these results was made by the first author at the conference Spectral problems for hyperbolic dynamical systems held in Bordeaux in May of 4. 6

7 SELBERG ZETA FUNCTION The zeta function and closed geodesics In this section we make some preliminary estimates on the lengths of geodesics used in its definition. Let D = {z = x + iy: z < } denote the Poincaré disc equipped with the usual Poincaré metric ds = dx +dy. This is a x y manifold with constant negative curvature. We can fix a value < α π and consider a Fuchsian group Γ α = R,R,R generated by reflections R,R,R with respect to three disjoint equidistant geodesics β,β,β, with end points e π j±αi D, j =,,, respectively. We can β b b β b β Figure 4: Three circles of reflection in Poincaré disk with pairwise distance b. write X = D /Γ, and assume that the surface X possess a metric of constant negative curvature, carried over from the Poincaré disk. In particular, X is a pair of pants i.e. topologically a sphere minus three disjoint disks. The length of the boundary geodesics of X we denote by b. In the notation introduced above, let us associate a closed geodesic γ to each conjugacy class in Γ. We shall denote by lγ its length in the hyperbolic metric of the surface, and by ωγ N its word length, i.e. the period of the cutting sequence, or the number of times it crosses the boundary of the fundamental domain of Γ. Our starting point is the following important result of D. Ruelle from 976. Theorem. after Ruelle. Let δ > be the largest real zero for Z X. In the notation introduced above, the infinite product. converges to a non-zero analytic function for Rs > δ and extends as an analytic function to C. Proof. The first part follows from more general results on Axiom A flows which we can apply to the geodesic flow restricted to the recurrent part []. The latter part follows from applying ideas from the work of Ruelle [5], see also [] for more details. Later we will establish a connection between the angle α and the distance b between geodesics of reflection. 7

8 . Approximating the lengths of closed geodesics SELBERG ZETA FUNCTION Our strategy for understanding the behaviour of the zeta function is based on the idea of quantitative approximation by functions of the following type. Definition.. We call a complex function f a trigonometric polynomial if it may be written in the general form where c,,c n R and λ,,λ n >. f s: = c e sλ + + c n e sλ n,. Trigonometric polynomials were famously studied by Lagrange in his work on planetary motions cf. [9] and in the work of Bohr on almost periodic functions [4]. We are going to approximate the zeta function Z X by trigonometric polynomials Z n, which appear as determinants of simple matrices C X. The curves C j, j =,...,4 appear afterwards as submanifolds containing the zero set of the determinants. To get an insight into how the matrices are constructed, we consider a related complex function /ζ defined by ζ s = Z Xs Z X s + = e slγ,. γ which shares the same zeros as Z X s in the strip < Res < δ, since Z X is real analytic for Rs >. The function ζ is the exact form of the zeta function studied by Ruelle. Intuitively, an approximation to the function /ζ could be obtained by replacing the lengths lγ of closed geodesics with some approximate values. In the following paragraph we present two possible approximations to lγ.. Approximating the lengths of closed geodesics In the present context the coefficients λ j, j =,...,n in. can be expressed in terms of closed geodesics, which in turn can be written in terms of the boundary length b. In particular, we have the following result. Proposition.. Let the length of a boundary geodesic on X be b. Then. A coefficient λ n is given by finite sums of lengths lγ of closed geodesics γ of word length ωγ k for k n;. The hyperbolic length lγ of a closed geodesic γ and the word length ωγ are related by cosh coshlγ = sinh ωγ b bp γ sinh. b for some polynomial P γ Z[x] of degree ωγ. 8

9 . Approximating the lengths of closed geodesics SELBERG ZETA FUNCTION Proof. The values for λ n in part will be a consequence of the expansion of the presentation for Z X as a series in.. Part follows from a straightforward expansion see Lemma 4.4 in 4.. The large symmetry group of the symmetric pair of pants means that many of the polynomials P γ are the same. This explains why in numerical computation there appears to be a lot of multiplicity in the coefficients λ j... First order approximation We may first consider the simplest approximation l γ = nb to the length lγ of closed geodesics of the word length ωγ = n. This has a simple geometric interpretation. To every closed geodesic we associate a periodic cutting sequence, which in turn uniquely corresponds to a periodic orbit of the subshift of finite type σ : Σ A Σ A on the words of alphabet of three symbols corresponding to three reflections. The transtition matrix for the subshift is given by the matrix A =, i.e. Σ A = {x = x n n= n= {,,}: Ax n,x n+ =, for n } and σx n = x n+ for n. Closed geodesics correspond to periodic points for σ, and n tran counts the number of non-prime closed geodesics corresponding to n-reflections. Using the standard formal matrix identity 4 we can define ζ by 5 deti za = exp n= z n n tran ζ s : = e sl γ = deti e sb A = 6e bs + 9e 4bs 4e 6bs, γ where the product is taken over prime closed geodesics []. Later we will see that the function /ζ is a good approximation to the function Z X in a small neighbourhood of the real line. We may exploit this idea, for instance, to deduce that the function 4 The equality holds true when the power series on the right hand side converges. In this case, the resulting analytic function has an analytic extension to C. For the values of z corresponding to the divergent formal series, the equality between the analytic extension and the determinant is understood. 5 As above, the infinite product converges in the right half plane Rs > and has an analytic extension to the left half-plane. 9

10 . Approximating the lengths of closed geodesics SELBERG ZETA FUNCTION Z X has a double zero at s =. Namely, we may rewrite the complex function ζ s = deti e bs A = 4e bs e bs and observe that s = is a double zero of ζ s... Second order approximation Now we would like to consider the next level of approximation to the lengths of closed geodesics, which turns out to be of the form l γ = nb + cγe b, where the coefficient cγ N is defined using the cutting sequence. We observe the following simple fact: Lemma.4. Consider a regular hyperbolic hexagon whose even sides are of length b > and whose odd sides are of length ε b then ε b = e b/ + e b + Oe b/ as b +. Proof. We recall that cf. [], Theorem 7.9. coshε b sinhb = coshb + coshb. The result follows by expanding the both sides in e b as b + and comparing the expansions. Now we are ready to describe the coefficient cγ. Lemma.5. Assume that γ corresponds to a cutting sequence of period n specified by j j j j 4 j n j n, where j k j k+ for k n and j n j. Then cγ = #{ k n: j k j k+ mod n } Proof. We will consider the geodesic segment γ jk j k+ j k+ γ corresponding to the subsequence of the cutting sequence j k j k+ j k+ and estimate its length. We will consider two cases separately: j k = j k+ and j k j k+. Let us first consider the case when j k = j k+, see geodesic segment γ in Figure 5 for example. We can approximate this geodesic segment by the geodesic segment consisting of the sides of the two hexagons and thus necessarily of length b up to an error Oe b independently of j k and j k+.

11 . Approximating the lengths of closed geodesics SELBERG ZETA FUNCTION R R γ R γ R R Figure 5: A geodesic segment γ, running along the side of two hexagons when R jk+ = R jk ; and a geodesic segment γ, traversing two hexagons when R jk+ R jk. Let us now consider the case j k j k+, see the geodesic segment γ in Figure 5 for example. We may approximate the length of this segment by double the length of the hypothenuse of the hyperbolic right-angled triangle one leg of which is equal to the edge of the fundamental domain of the length b and another leg equal to a half of the edge of fundamental domain of the length ε b. We can use the hyperbolic cosine law to write cosh lγ jk j k+ j k+ = cosh ε b coshb. Therefore we can write lγ jk j k+ j k+ = ln coshbcosh ε b + cosh bcosh ε b = ln coshbcosh ε b + e b + Oe b = = lncoshb + ln cosh ε b + ln We have asymptotic expansions = + e b + Oe b.4 lncoshb = b + ln + e b = b + e b + Oe 4b.5 and ln cosh ε b e e b/ +Oe b + e e b/ +Oe b = ln = ln + e b + Oe b..6

12 . Approximating the lengths of closed geodesics SELBERG ZETA FUNCTION Substituting.5 and.6 into.4, we conclude lγ jk j k+ j k+ = b + e b + Oe b, and the Lemma follows. Again, a routine calculation shows that the choice of j k and j k+ will change lγ jk j k+ j k+ by not more than Oe b. We may associate to a closed geodesic a periodic orbit of a subshift of finite type on the space of sequences of six symbols R j R j, where j, j and j j. The transition matrix is given by B j j 4 j j = if j = j, otherwise. As one would expect, we have that n tr B n is equal to the number of closed geodesics of word length ωγ = n. In order to keep the additional information of the second order approximation to the length of closed geodesics, consider the matrix B j j 4 j j z = if j = j ; if j = j and j = j 4 ; z if j j and j j 4 ; z otherwise. With a suitable labeling, we obtain z z z z z z z z z Bz = z z z z z z z z z.7 Lemma.6. The coefficients of the polynomial n trbn z = d n z n + d n z n + + d z + d are given by d k = #{γ : ωγ = n and cγ = k}. Proof. Follows immediately from the definition.

13 APPROXIMATION RESULTS Now we return to the complex function ζ to the length of the closed geodesics: given by. and consider the first approximation Now we define l γ = nb + cγe b. ζ s : = γ e sl γ = deti e sb Bexpse b In the next section we will show that there exists a trigonometric polynomial Z n which gives a good approximation to both the function Z X and the determinant deti e sb Bexpse b on a large rectangle < Rs < δ and < Is < expb. Combining this information with some technical estimates, we will deduce location of zeros of Z X from the zeros of the determinant. Approximation results The Selberg zeta function, initially defined in terms of lengths of all closed geodesics, is not an object which can be easily computed numerically: one has to be satisfied with a finite although large set of geodesics. Lemma.. There are exactly p n = 4 n + closed geodesics of the word length ωγ = n. Proof. Follows by induction in n from the recurrent relation p n = 4 n + p n, p = 6. Naturally, a fundamental question arises: to what extend does an approximating function Z n, which uses geodesics of word length at most ωγ n actually reflect the features of Z X? In this section we construct trigonometric polynomials Z n and quantify the error carefully.. The expansion We can begin by considering the function in two complex variables Z X s,z = n γ z ωγ e s+nlγ,. which converges for z sufficiently small and Rs sufficiently large. We can follow Ruelle [5] in re-writing the infinite product as a series Z X s,z = + n= a n sz n. by taking the Taylor expansion in z about. It is then easy to see that a n s is defined in terms of finitely many closed geodesics with word lengths at most n. In fact this series converges to a bianalytic function for both z,s C:

14 . The expansion APPROXIMATION RESULTS Theorem. Ruelle [5]. Using the notation introduced above, there exists C = Cs > and < θ < such that a n Cθ n and thus the series. converges. In particular, we can deduce that Z X z,s is analytic in both variables. We next make an easy observation. Lemma.. The odd coefficients vanish i.e., a = a = =. Proof. The fundamental domain for X consists of two hexagons glued across three of the alternating six edges. Since any closed geodesic γ crosses the fundamental domain an even number of times it follows that ωγ N in the infinite product. and thus only the even terms a,a 4, can be non-zero in.. We can formally rewrite the zeta function. as z Z X s = exp m m m= ωγ=m e slγ e lγ = exp m= b m sz m. where e m slγ if m is even; e b m s = ωγ=m lγ if m is odd; and then expand the exponential as a power series and obtain coefficients a n comparing. with., since these series converge provided Rs >. In particular, we can easily check that the first three non-zero terms are: and in general, a s = b s; a 4 s = b 4 s + b s ; a 6 s = b 6 s + b sb 4 s b s ; a n s = n n a j sb n j s. j= Combining the latter with.4, we deduce that each of the coefficients a n s is a complex trigonometric polynomial..4 4

15 . Main theorems APPROXIMATION RESULTS. Tail estimates We begin by defining a sequence of functions Z n, approximating Z X. Definition.4. We define the complex trigonometric polynomial Z n s = n a k s correspond to those in. Our main approximation result is the following. k= a k s, where the terms Theorem.5. Let X be a symmetric pair of pants with boundary geodesics of the length b. Then we may approximate Z X on the domain RT by the complex trigonometric polynomial Z n so that sup RT Z X Z n ηb,n,t where T = T b = e κb for some constant κ > independent of b and n, such that. for any n 4 we have ηb,n,t b = O b as b. for any b we have ηb,n,t b = O e bk n as n.i for some k > which is independent on b and n. For a fixed b this theorem estimates the number of terms a n needed to uniformly approximate Z X to any given error, on an exponentially growing domain. On the other hand for a given n this Theorem estimates the difference between Z X and Z n as b on an exponentially growing domain. Remark.6. The constant k in Theorem.5 should satisfy inequality < k < κ 6, although this bound is not sharp. A sharp bound can be obtained using the same argument, but the formulae will be more complicated. Example.7. We can fix a surface by choosing the length of boundary geodesics b and plot the zeros for the approximating trigonometric polynomials Z n = + a + + a n for n =,,,6. For instance, in Figure 6 zeros of polynomials approximating Z X with b = 5 are shown. In particular, this more refined result shows that numerical results for Z n can be used to justify Observation O in a domain Re k b. Since in practice n is bounded above by computational considerations, we may assume that it is fixed. Even with a modern computer, one will not be able to consider n > 6 in a reasonable time 6. Moreover, in practical applications b cannot be chosen too large either due to computer restrictions because of accumulation of errors while dealing with 6 The most time-consuming part is the Newton method used to locate a zero starting from a point of a lattice on RT. The time taken by this calculation grows exponentially with n. The total number of the searches is proportional to the area of RT, which is proportional to T. 5

16 ℑs ℑs NUCLEAR OPERATORS AND ANALYTIC FUNCTIONS ℑs ℜs. ℜs ℑs ℜs.5 d Z8 s.5. ℜs c Z6 s ℑs b Z4 s ℑs a Z s.5 ℜs e Z s.5. ℜs f Z s Figure 6: Plots of the zeros Zn s for n =,,..., 6 when b = 5. small numbers. The coefficients bm defined by.4 involve a sum of 4m + terms of the order exp smb with < ℜs <.5, say. The bound 4m + is equal to number of closed geodesics of the word length m, see Lemma.. 4 Nuclear operators and analytic functions The proof of Theorem.5 is based on the original approach in [5] and the interpretation in [4]. We begin by recalling some abstract results, essentially due to Grothendieck, on nuclear operators. We then complete the section by relating the length of the boundary geodesics to the contraction on the boundary corresponding to reflections, generating the group Γα. 6

17 4. Nuclear Operators 4 NUCLEAR OPERATORS AND ANALYTIC FUNCTIONS 4. Nuclear Operators The convergence of the series. in Theorem. will follow from estimates of Ruelle [5], after Grothendieck [6]. We summarize below the general theory. Let B be a Banach space. Definition 4.. We say that a linear operator T : B B is nuclear with exponentially decreasing coefficients if there exist for each n. w n B, with w n B = ;. ν n B, with ν n B = ;. λ n R, with < λ <, C > satisfying λ n Cλ n such that T f = λ n w n ν n f. 4. n= Lemma 4. after Grothendieck. A nuclear operator on a Banach space is trace class, and we can write where We may also expand a n = deti zt = exp deti zt = + n= n= z n n trt n. a n z n, det [ν jk w jl ] n k,l= λ j...λ jn. 4. j < < j n Applying estimates of Grothendieck and Ruelle, we obtain an explicit bound. a n C n n n/ λ nn+/, 4. where n n/ bounds the supremum norm of the matrix 7 [ν jk w jl ] n k,l=. 7 We follow Ruelle in including the term n n/ although this can be improved upon by looking at Hilbert spaces of analytic functions. For instance, O. Bandtlow and O. Jenkinson [] have shown that we can supress n n/ by working with Hardy spaces, but then λ would be different, too. 7

18 4. Constructing the Banach space 4 NUCLEAR OPERATORS AND ANALYTIC FUNCTIONS 4. Constructing the Banach space Given four points z < w < w < z on the boundary H, we define the cross ratio by [z,w,w,z ] = z w w z z w w z. We recall the following classical formula cf. [] 7.. Lemma 4.. Let L,L be two disjoint geodesics in H with end points z,z and w,w. The distance dl,l between L and L satisfies [z,w,w,z ] = tanh dl,l /. By assumption, the group Γ = R,R,R is generated by reflections with respect to three disjoint geodesics, which we denote by β, β, and β, respectively. Without loss of generality, we may assume that the geodesic β j has end points e π j More precisely, by straightforward calculation using Lemma 4. we get ±θ i, for j =,, and a small real number θ. Lemma 4.4. Let β and β be two disjoint geodesics in H with end points e that the distance dβ,β = b. Then sinθ = coshb. ± π ±θ i and assume Remark 4.5. In notations and under the hypothesis of the last lemma, we have an asymptotic relation θ = e b + e b + oe b as b. To define the Banach space, we fix a small ϕ < θ and introduce three additional geodesics υ j with end points e π j ±ϕ i, j =,,. We may consider the disk D as a subset of C and formally extend the geodesics υ j to circles υ j C. Furthermore, let {U j } j= be three compact disks in C such that U j = υ j cf. Figure 7. The Banach space of bounded analytic functions f : j= U j C on the union i= U i we denote by B. We supply it with the supremum norm f B : = f. 4. Transfer operators We can now define transfer operators L s, acting on the Banach space B of bounded analytic functions on j= U j. Definition 4.6. For each s C we can define L s f z = where χ Uk is the indicator function of U k. k= χ Uk z R jz s f R j z, 4.4 j k 8

19 5 PROOF OF THE APPROXIMATION RESULT We can apply the general theory of nuclear operators to the transfer operators by virtue of the following Lemma 4.7. The operator L s : B B is nuclear. Proof. We observe that the operators f f R j are nuclear and R j zs are analytic [5]. Thus L s are nuclear, too. In particular, applying Lemma 4. to T = L s allows us to recover Theorem., see [5] for details. 5 Proof of the approximation result In this section we give a proof of Theorem.5. We will need the following simple technical estimate. Lemma 5.. Let x n be a sequence of real numbers satisfying x n exp pn qn for some constants p,q >. Then for any n > we have that π p x k k=n q exp exp q n p 4q q Proof. The result follows by straightforward calculation using the classical bound fot the error function n exp t d t π exp n. We now turn to the proof of Theorem.5. This follows the same lines as [8]. The central new idea is that the disks U, U and U used to define B are allowed to depend on b. Proof. Without loss of generality we may assume that the geodesic β j has end points e π j±θi D for j =,,. We choose three additional geodesics υ j with end points e π j±ϕi D for some < ϕ < θ, that we will specify later, see Figure 7 for details. We may consider D as a subset of C with usual Euclidean metric, and then complete υ j to full Euclidean circles υ j C. We define U j C to be compact disks with U j = υ j. It turns out that the calculations are much easier in the upper half model of the hyperbolic space H. We choose the map Sz = i +z z to change the coordinates. Then the geodesic β j has end points sin π j ±θ +cos π j ±θ H, and its Euclidean radius and centre are given by, respectively ε j = sin π j + θ π j sin + cos π j θ + θ + cos π j θ 5. c j = sin π j + θ π j sin + cos π j + θ + θ + cos π j θ. 5. 9

20 5 PROOF OF THE APPROXIMATION RESULT U υ β b β b θ b ϕ β υ U υ U Figure 7: Three geodesics β j in D giving rise to the reflections R j ; and three additional geodesics υ j which are used to define domain of the analytic functions in B. sin π j ±ϕ +cos π j The end points of υ j are ±ϕ H and the euclidean radius is r j = sin π j + ϕ π j sin + cos π j ϕ + ϕ + cos π j ϕ We can consider the Banach space B to be the space of bounded analytic functions on j= U j C with the supremum norm. 5. We see that the reflection R j with respect to the geodesic β j in H is given by R j z = ε j z c j +c j. We deduce that for any distinct j,k,l the image R j U k U l U j, provided ε j z c j < r j for any z U k U l. We know that for all z U k U l we have z c j >, thus it is sufficient to chose θ and ε j such that ε j < r j. Using 5. by straightforward calculation we may estimate θ ε j θ + Oθ and ϕ r j ϕ + Oϕ for small values of θ and ϕ. Hence it is sufficient to choose θ and ϕ such that 4θ < ϕ. Using Lemma 4.4, we see θ = e b + e b + oe b. In particular, it is sufficient to choose ϕ = e bκ for some < κ < and b sufficiently large. Then e bκ r j e bκ + Oe bκ, 5.4 e b ε j e b + Oe b. 5.5 Using the Cauchy integral formula for z U k and R j z U j we can write f R j z = f ξ πi U j ξ R j z dξ

21 5 PROOF OF THE APPROXIMATION RESULT and then R jz s f R j z = R j zs πi U j f ξ ξ R j z dξ. Since L s is a nuclear operator, it satisfies 4.. More precisely, we may write L s f z = = = j= j= R jz s f R j z k=,k j χ Uk z R jz s R j z c j n n= πi λ n w n zν n f, n= U j f ξ ξ c j n+ dξ χ Uk z k=,k j where, w n B, ν n B, and λ n R + satisfy conditions of Definition 4.. We may choose for any j {,,} w n+ j R jz s R jz c j n πi χ Uk z 5.6 k=,k j f ξ ν n+ j dξ 5.7 U j ξ c j n+ with normalization w n = ν n =. Then for any j {,,} λ n+ j = R j s Uk U l R j c j n Uk U l f ξ πi U j ξ c j where k j and l j. We may observe that ξ c j = r j and conclude λ n+ j R jz s Uk U l εn j r n+ j or more precisely, using formulae 5. and 5., we obtain an upper bound λ n max R j jz s Uk U l max max j R jz s Uk U l { ε n/ ε n/, r ε r /, r n/ ε 4/ r / max j n/ ε j r n/ j. r n/ ε 4/ r / n+ dξ, 5.8 ε n/ } r n/ Comparing this with the definition of the nuclear operator 4., we get explicit bounds for parameters λ and Cs. λ n max k sup max z U k j k ε s j z c j s ε 4/ r / max j n/ ε j r n/ Csλ n, j

22 5 PROOF OF THE APPROXIMATION RESULT with the choices λ = Cs = max k ε j max j r j sup max z U k j k ε s j z c j s Using the bounds 5.4 and 5.5 for ε j and r j, we conclude λ = 5.9 ε 4/ r /. 5. max ε j e κ b. 5. r j Furthermore, we see that for any j k for all z U k we have argz c j arcsin r j c k c j ϕ. Therefore for s = σ + it, z c j s = exp ln z c j +iargz c j σ +it = z c j σ exp argz c j t exp4ϕt, since by construction inf z Uk z c j >. Using 5.4 and 5.5, we deduce Cs = max k sup max z U k j k ε s j z c j s ε 4/ r / max k sup max z U k j k ε σ j z c j s 4ebκ+4/ 4ε σ j e 4ϕt+bκ+4/ e ln4 bσ+4ϕt+bκ+4/. 5. Substituting bounds 5. and 5. into Ruelle s inequality 4. and taking into account t < T for s = σ + it RT, we obtain an upper bound a n s C n sλ nn+/ n n/ bκ + 4 nn + b κ exp ln4 bσ + + 4ϕtn bκ + 4 nn + b κ exp ln ϕT n ln since exp bnσ. In order to estimate the tail of the series constant k < such that which is equivalent bκ + 4 n ln ln b ln + nlnn + nlnn, 5. a n s using Lemma 5., it is sufficient to find a n=4 b κ n + n + ln + nlnn κ k κ ln b < n < bn κ k lnn bk κ ln. 5.5

23 6 ASYMPTOTIC RESULTS FOR LARGE B It is clear that the inequality doesn t hold for any n but it does hold for all b and n 4 with the choices κ =.5 and.95 k <. Therefore we obtain an upper bound a n s exp 4ϕT n b κ k n, for all n We recall that ϕ = e κb and applying Lemma 5. with the choices p = 4e κb T and q = b κ k 6, we get an estimate Z X s Z n s = a n σ + it ηb,n,t b, where T b = k e κb for some k > and all b, n 4: k=n ηb,n,t b = 6π = b κ k exp 4k b κ k exp Therefore we have the desired asymptotic estimates: b κ k 6. for any n 4 we have ηb,n,t b = O b as b ;. for any b we have ηb,n,t b = O e bk n ; as n. k n b κ k hold with the choices, for example, < k κ k 6, where < κ < and k are chosen so that 5.4 holds Asymptotic results for large b We can now use the approximation result in Theorem.5 for fixed n and large b to prove the results in Theorem.4. Theorem 6.. Given a κ > as in Theorem.5, the complex analytic function Z X sb converges uniformly to Z 6 sb on the domain Re κb, more precisely, sup s Re κb s s b Z X Z 6 = O as b +. b b Proof. In Appendix we explain how to compute the lengths of closed geodesics. By a straightforward manipulation with some help from a computer we can show that a n s = exp nbs b+ Oexp b s for n = 8,, and therefore a n s/b = exp ns b + Oexp b s/b as b. The result follows from Theorem.5 which gives the bound for Z X Z 4.

24 6 ASYMPTOTIC RESULTS FOR LARGE B Remark 6.. Using the estimates for the hyperbolic length of closed geodesics of the word length ωγ 6, presented in the Appendix, we may compute the first few non-zero coefficients a s = 6e bs e b a 4 s = 5e 4bs 6e 4bs + e b + e b s + Oe b a 6 s = e 6bs e 6bs + e b + e b s 6e 6bs + e b + e b s + + 4e 6bs + e b + e b s + Oe b. We see that s s s s b Z 6 = + a + a 4 + a 6 6e s + 9e 4s 4e 6s = deti e s A. b b b b Now we are ready to prove the first part of Theorem.4 Proof. We shall show that on the domain Re κb we have that the function Z X s vanishes at s n b such that lim b s n b b = ln + iπn. The function deti e s A vanishes at {iπn,ln+iπn}, for n Z. For any sufficiently small η > we have that the closed balls Uln + iπn,η = {s C: s ln + iπn η} contains no more zeros. Let us denote ε = inf s U deti e s A >. Using Theorem 6., we can now choose b sufficiently large so that we have s inf Z X deti e s A < ε s U b. It then follows by Rouché s Theorem [] that for any n N the function Z X s has exactly one zero s n b, satisfying s n b b ln + iπn < η. Remark 6.. Weich [7] has observed this result for a case of the domain of the fixed size, i.e. Rc, where c is independent of b. This is consistent with an old result [], which states that the largest real zero of Z X satisfies δ ln b as b. This implies asymptotic spacing of individual zeros suggested in O. We now turn to the problem of describing the distribution of the zeros O and O. In Section.. we introduced a matrix B, closely connected to the zeta function. In the following lemma we show its determinant appoximates the zeta function and quantify the error. 4

25 6 ASYMPTOTIC RESULTS FOR LARGE B Lemma 6.4. Let us recall.7 z z z z z z z z z Bz = z z z z z z z z z The real analytic function Z σb +ite b converges uniformly to deti exp σ itbe b Be it, and more precisely, σ Z b + iteb det I exp σ itbe b Be it = Oexp b as b +. Proof. This follows by straightforward calculation of the first coefficients and the determinant. Let us introduce dummy variables x: = exp σ itbe b and y: = e it with x < and y =. Then det I exp σ itbe b Be it 6 = deti xby = x k P k y, 6. k= where P k Z[ ] are some polynomials with integer coefficients. More precisely, we can compute: At the same time P y, P y 6, P y = 5 6y, P y = 4y + 6y 4 + y 6, P 4 y = 5 6y + 7y 4 6y 6, P 5 y = 6y 4, P 6 y = y 6. Z σ b + iteb = σ a j j= b + iteb With some help of a computer, one can deduce by straightforward manipulation that a n σb + ite b = x n P n y + Oe b. We shall illustrate this using formulae for the coefficients from Remark 6.. a σ b + iteb = 6exp σ itbe b e b = 6x e b = 6x + Oe b = xp + Oe b. 5

26 6 ASYMPTOTIC RESULTS FOR LARGE B Similarly for a 4 : σ a 4 b + iteb = 5e 4σ 4itbeb 6e 4σ 4itbeb + e b + e b σ/b+ite b + Oe b = = 5x 6x + e b + e b σ/b + e b + e b ite b + Oe b = = 5x 6x e it + Oe b + Oe b = = x 5 6y + Oe b = x P y + Oe b, where we have been using the fact that + e b + e b e b = e + O e b. Now we can prove the second part of Theorem.4. Proof. We find that the matrix Be it has exactly four different eigenvalues β k t, k =,...,4: β t = e it β t = e it + β t = eit 4 e it eit β 4 t = eit 4 e it eit Therefore we deduce that the zero set of the determinant det I exp σ itbe b Be it belongs to the subset { σ,t R k : expσ + itbe b = β k } The four equations expσ = β k t give us four curves { } C = ln cost t R ; { } C = ln + cost t R ; { C = ln eit } eit 4 e it t R ; { C 4 = ln eit + } eit 4 e it t R. which correspond to the curves suggested in O. Since the curves C j do not have horizontal tangencies σ const, without loss of generality we may define the neighbourhoods as follows: BC j,ε: = { σ,t σ ln β j t > ε } 6

27 7 L-FUNCTIONS AND COVERING SURFACES To complete the argument we shall show that for all ε > and T > there exists b > such that for any b > b the zeros of the function Z σ b + ite b with σ and t e κb belong to a neighbourhood B k C k,ε of the union of the curves k C k. Indeed, given ε > and a point z = σ +it outside of ε-neighbourhood of 4 j= C j we see that the determinant deti exp σ it be b Bexpit > exp 6εexpε 6 > is bounded away from zero and the bound is independent of b. Therefore for b sufficiently large all zeros of the function Z X σ/b + ite b belong to the ε-neighbourhood of 4 j= C j. We have concentrated on the particular case of the symmetric pair of pants whose boundary curves have the same lengths. However, the same method works in the case that the boundary curves have different length as well as in the case of symmetric punctured torus, and allows to explain the nature of the patterns of zeros described in the sections 5. and 5. of [5]. 7 L-functions and Covering surfaces Our results have concentrated on a special class of surfaces, but can be easily adapted to cover a large class of geometrically finite surfaces of infinite area. There is a fairly simple method for constructing quite complicated surfaces using the basic pair of pants V. We can write V = H /Γ. We then write a cover V = H /Γ for V in terms of a normal subgroup Γ < Γ. Let us denote by G = Γ/Γ the finite quotient group. Let γ be a closed geodesic on V and then this is covered by the union of closed geodesics γ,,γ n on V. Let R χ be an irreducible representation for G of degree d χ with character χ = trr χ. The regular representation of G can be written R = χ d χ R χ. where G = dχ χ Definition 7.. Given s C we define Lz,s, χ = γ det I z g e s+nλg RgΓ where gγ is a coset in G. Lemma 7.. For characters χ and χ we can write Lz,s, χ + χ = Lz,s, χ Lz,s, χ. If H < G is a subgroup and χ is a character of H the we can write G = m i= Hα i and define the induced character χ of G by for g G. χ g = α i gα i H 7 χα i gα i

28 7 L-FUNCTIONS AND COVERING SURFACES Lemma 7. Brauer Frobenius. Each non-trivial character χ is a rational combination of characters χ i of G induced from non-trivial characters χ i of cyclic subgroups H i. There exist integers n,,n k with and thus Ls,z, χ n = nχ = k i= α i gα i n i χ i H χα i gα i Since it is easier to deal with cyclic covering groups. We need the following. Lemma 7.4. Let χ be a character of the subgroup H < G and let Ls, χ be the L-function with respect to the covering Ṽ of Ṽ /H. Then Ls,z, χ = Ls,z, χ. The proof is analogous to that of the proof of Proposition in []. This leads to the following. Lemma 7.5. If χ is an irreducible non-trivial character of G then Ls, χ n is a product of integer powers of L-functions defined with respect to non-trivial characters of cyclic subgroups of G. Finally this means that we can write the zeta function ZṼ s,z in terms of the L-functions L V s,z, χ for V. Lemma 7.6. We can write ZṼ s,z = L V s,z, χ d χ. χ irreducible where the product is over all irreducible representations of G. In particular, the zeros for ZṼ s,z will be the union of the zeros for the L-functions L V s,z, χ d χ. Example 7.7. We can take a double cover Ṽ for a pair of pants V, which corresponds to a sphere with four holes. The corresponding covering group is simply Z and the zeta function ZṼ s is then the product of:. the zeta function Z V s for the original pair of pants;. the L-function L V s,z, χ corresponding to the representation χ : π V Z where χg = ng where ng counts the number of times the generator a, say, occurs in g. In particular the zeros for ZṼ s are a union of the figures for these two functions. 8

29 ℑs ℑs EXAMPLES OF THE COEFFICIENTS ℑs A ℜs a zeros for ZV s.5. ℜs b zeros for LV s, z, χ.5. ℜs c superposition Figure 8: Zeros of the zeta function and zeros of the L-function in the case `γ,, = 9. A Examples of the coefficients In this Appendix we present the asymptotic formulae for the hyperbolic length of the first closed geodesics, which then lead to the asymptotic expressions for the first few non-zero coefficients a, a4, a6. Using the identity trr j R j... R jn, `γ j, j,... jn = Arcosh relating the length of the closed geodesic corresponding to the cutting sequence of period n to the matrices defining the reflections, we compute the lengths of the closed geodesics for n =,,, 4. The case n =. It has been established in Lemmas.5, and., there are exactly 6 geodesics of the length `γ j j = b. The case n = 4. There are 6 geodesics of the length 4b and geodesics of the length `γ j j j j = Arcosh coshb + cosh b. The case n = 6. There are 4 + = 66 homotopy classes of closed geodesics; among which there are 6 geodesics of the length 6b and of the length `γ j j j j j j = Arcosh 4 cosh b + 6 cosh b = 6b + 6e b + Oe b. There are 8 geodesics of the length b `γ j j j j j j = Arcosh 8 cosh cosh b = 6b + 4e b + Oe b. 9

30 REFERENCES REFERENCES Finally, there are 6 geodesics of the length lγ j j j j j j = Arcosh 4cosh b + cosh b coshb = 6b + e b + Oe b. The case n = 8. There are = 58 homotopy classes of closed geodesics; among which there are 6 geodesics of the length 8b. Moreover, there are 4 geodesics of the length lγ j j j j j j j j = Arcosh 4cosh b + 4cosh b + 8cosh 4 b + = 8b + e b + Oe b ; and another 48 geodesics of the length lγ j j j j j j j j = Arcosh coshb 4cosh b + 4cosh b + 8cosh 4 b = 8b + e b + Oe b. In addition, we have geodesics of the length lγ j j j j j j j j = Arcoshcosh b + 8cosh b + 8cosh 4 b = 8b + 4e b + Oe b ; and 48 geodesics of the length lγ j j j j j j j j = Arcoshcoshb 4coshb + coshb + 4coshb + = 8b + 4e b + Oe b ; and another 48 geodesics of the length lγ j j j j j j j j = Arcosh coshb + 8cosh b + 8cosh 4 b = 8b + 4e b + Oe b. Finally, there are 4 geodesics of the length lγ j j j j j j j j = Arcoshcoshb 6coshb + coshb + 8coshb + = 8b + 6e b + Oe b ; and further more 48 geodesics of the length lγ j j j j j j j j = Arcosh coshb + cosh b + 8cosh 4 b = 8b + 6e b + Oe b. References [] Ahlfors, L. V. Complex analysis. An introduction to the theory of analytic functions of one complex variable. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York, 978. xi+pp.

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