IC/91/94
INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
ARITHMETIC, GEOMETRY AND DYNAMICS IN THE UNIT TANGENT BUNDLE OF THE MODULAR ORBIFOLD
Alberto Verjovsky
INTERNATIONAL ATOMIC ENERGY AGENCY
UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION 1991 MIRAMARE - TRIESTE
IC/90/94
INTRODUCTION International Atomic Energy Agency The modular group PSL(2, Z) and its action on the upper half-plane H together with and with its quotient, the modular orbifold, are fascinating mathematical objects. The study United Nations Educational Scientific and Cultural Organization of modular forms has been one of the classical and fruitful objects of study. If one considers INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS the group PSL(2,R) and we take the quotient M = PSL(2,R)/PSL(2,Z) one obtains a three dimensional manifold that carries an enormous amount of arithmetic information. Of course this is not surprising as the elements of PSL(2,Z) are Mobius transformations given by matrices with columns consisting of lattice points with relatively prime integer coefficients and therefore M contains information about the prime numbers. On the other hand from the dynamical systems viewpoint, M is also very interesting—it has a flowwhic h is an Anosov flow. The flowa s well as its stable and unstable foliations correspond to three ARITHMETIC, GEOMETRY AND DYNAMICS IN THE UNIT one-parameter subgroups: the geodesic flow, and the stable and unstable horocyclic flows TANGENT BUNDLE OF THE MODULAR ORBIFOLD* (see the next section). All these flows preserve normalized Haar measure m, and are ergodic with respect to this measure. By a theorem of Dani [Da] the horocyclic flows have a curve •"(v)i (v > 0), of ergodic probability measures. These ergodic measures are supported in ALBERTO VERJOVSKYJ closed orbits of period y of the corresponding horocyclic flow. If we denote by my the measure corresponding to y > 0, then m, converges to m as y -» 0 ([Da] [Za]). We will International Centre for Theoretical Physics, Trieste, Italy. also give a proof of this. However what is most interesting for me as a dynamical systemist is the remarkable connection found by Don Zagier between the rate of approach of m, to m and the Riemann Hypothesis. Zagier found that the Riemann Hypothesis holds if and only if one can find a nonzero smooth function / with compact support on M such that 4 1/2 mv(f) = m(f) + oiy*' -') for all e > 0. He also proved that m,(/) = m(jF) + o(y ). In the Appendix we will review Zagier's approach as well as the extension given by Sarnak ([Sa]). The purpose of this paper is to analyze the dynamics and geometry of the horocyclic now to show that the exponent 1/2 is optimal for certain characteristic functions of sets ABSTRACT called "boxes". Of course this is very far from disproving the Riemann Hypothesis since such a characteristic function is not even continuous. By geometric means we reduce the The interplay between the geometry, dynamics and arithmetic of the modular orbifold analysis of the convergence of m, to ro to a lattice point counting. Thus the fact that are studied. The Riemann hypothesis looms upon this structure. the exponent cannot be made better than 1/2 is similar to the circle problem in which we count the number of lattice points with relatively prime coordinates inside a circle. If instead we take a smooth function / with compact support and take the sum, T.{y), of the values of the function over all lattice points (yn,ym) with y > 0 and (n,m) G Z then j/2£(y) will converge to the integral of / over the plane as y -* 0 and the error term will be oiy"), as y -t 0, for all a > 0. This follows by the Poisson summation formula using MIRAMARE - TRIESTE the fact that the Fourier transform of / decays very rapidly at infinity May 1991 We will use many properties of SL(2, R), its quotients by discrete subgroups and the two locally free actions of the proper real affine group on thse quotients. Good references for each subject are [A], [G], [L], [Ma] and [Ra].
0. PRELIMINARIES
*To be submitted for publication Let G := SL(2,R) denote the Lie group of 2 x 2 matrices of determinant one, with real fPermanent address: CINVESTAV Del IPN, Apdo. Postal 14-740, Mexico D.F., Mexico. coefficients. The Lie algebra of G, 0 := st(2,R) , consists of real 2x2 matrices of trace
Typeset by zero. This Lie algebra has the standard basis: By euclidean translation to the origin and clockwise rotation of 90 degrees we have a trivialization $ : TjH -* H x S1 given by 0 0 A = B = 0 - c = 1 o
To A, B and C correspond the left-invariant vector fields X, V, and Z respectively in where v is a hyperbolic unit tangent vector anchored at z 6 ^H. For example, using (0.2), SL(2,R).These vector fields induce, respectively, the nonsingular flows: -j, ,_» y(^ x) gives an explicit identification and it will be the one we will use here. If -v = f" M £ G, we will let 7(2) = -j denote the corresponding element in G. g,;G->G lc «J cz -(- a Using trivialization (0.2), we have Explicitly: (0.3) y(«,0) = (7{2),0-2arg(cz + d)) a b c d In this notation 8 is to be taken modulo 2ir, where the angle of a unit vector is measured from the vertical counter-clockwise. a b i *i (0.1) The three basic vector fields X, Y and Z descend toGs XjH and the flows induced by c d *([ ])- 0 lj' them correspond to the geodesic flow, unstable horocyclic flow and stable horocyclic flow, a b respectively. c d D- Geometrically these flows can be described as follows. Let z e HI and let vz £ XjH be a unit vector based at 2. This vector determines a unique oriented geodesic 7, as well + + To simplify notation, let us write: g := {gt}t€E, h := {hf}t&t and h •= {ht }tem- as two oriented horocycles C and C~ which pass through z are orthogonal to 7 and Consider the upper half-plane, H = {z = (x,y) := x + iy \ y > 0} C C equipped with the tangent to the real axis. Then v' := gt(vt) is the unit vector tangent to 7, following the 2 2 1 2 + metric ds = (l/y )(dx' + dy ). With this metric H is the hyperbolic plane with constant same orientation as 7 at the point at distance t from z. The vectors w = h+(vt) and + negative curvature minus one. w~ = hv(yt) are obtained by taking unit vectors tangent to C and C~, respectively, at G acts by isometries on H as follows: distances u and v respectively, and according to their orientations (see Figure 1).
a b c d where z = x + iy, y > 0. The action is not effective and the kernel is the subgroup of order two, {/, —/}, consisting of the identity and its negative. Let G = 5/{1,-1} := PSL(2,R). Note that G is the group of Mobius transformations that preserve H and is in fact its full group of orientation- preserving isometries. The action of G on HI can be extended via the differential to the unit tangent bundle which we shall denote henceforth by T\R. If 7 £ G and y' denotes its differential acting on unit vectors, we have: Figure 1
*It is because of this geometric interpretation that the flows g, h+ and h~, defined originally in G are called geodesic and horocyclic flows respectively. Formulae (0.1) tell us H HI that the orbits of the respective flows are obtained by left-translations of the one-parameter subgroups where Px is the canonical projection. Naturally, "unit vector" refers to the hyperbolic metric. 0 e-t*l "Ho 1 • "" U Every one-parameter subgroup is conjugate in G to one of the above, and h+ is conjugate When F is co-compact both h+ and h are minimal flows on M(T)- In particular, the toft". horocyctic flows do not contain periodic orbits. This was proved by Hedlund [He]. It is Henceforth we will equip G with the left-invariant riemannian metric such that {X, Y, Z} also a consequence of a result of Plante JP1]. Suppose for instance that one orbit of h+ is is an oriented orthonormal framing. We will call this metric the standard metric. not dense. Then the closure of this orbit contains a non-trivial minimal set £ and Plante By the standard riemannian measure or Baar Measure, we will mean the measure, m, showed that E must be both a 2-torus and a global cross section for the geodesic flow, induced by the volume form £1 which takes the constant value one in the oriented framing implying that M(F) would be a torus bundle over S1. But this is impossible since under {X, Y,Z}. Since G is unimodular, the measure m is bi-invariant. the hypothesis, F cannot be solvable. Let J4.2(R) denote the proper a/pne group: It was shown by Furstenberg [Fu] that for F co-compact both h+ and h~ are strictly ergodic flows with m as their unique invariant measure (this also implies minimality of A2(R) = {T : R -» R | T(r) = ar + 6; a, 6 6 R, a > 0}. k+ and h~). Since the geodesic flow is transitive, it contains a set of the second Baire category of dense orbits and it also contains a countable number of periodic orbits whose Let us parametrize A2(R) by pairs (a, 6) with a,b 6 R, a > 0. union is dense in M(T). The number of such periodic orbits as a function of their periods There are two monomorphiams, A2(R) «—» G, given by: grows exponentially. Margulis has a formula relating the topological entropy of g and the rate of growth of periodic orbits. This formula was also latter obtained by Bowen. These (0.4) facts are interesting because the horocyclic flow is the limit of conjugates of the geodesic flow as can easily be seen by considering the vector fields Xt = tX + Y as t tends to zero. We see from these inclusions that the pairs {X, Y] and {X,Z} generate Lie algebras When F is a nonuniform lattice (i.e. when F is discrete, M{T) is not compact, and + iaomorphic to the Lie algebra of the affine group: m(M(T)) < co) then g, h and h~ are still m-ergodic. However, for nonuniform lattices both h* and h~ are not minimal. IX,Y] = [Y,Z] = If F C G is any discrete subgroup, then F\H := S(T) is a complete hyperbolic orbifold. If F is co-compact this means that S(T) is a compact surface provided with a special metric As a consequence, G contains two real analytic foliations by planes whose leaves are and a finite number of distinguished points or conical points labelled by rational integers. the orbits of the free actions of the affine group in G (or simply, the leaves are obtained The complement of the conical points is isometric to a surface of curvature minus one by left-translations of the two copies of the affine group). These two foliations, denoted (in general incomplete), and of finite area. Each distinguished point has a neighborhood respectively by T+ and T~, intersect transversely along the orbits of the geodesic flow. which is isometric to the metric space obtained by identifying isometrically, the two equal The geodesic flow is an Anosov flow and T* and T~ are its itnstable and stable foliations. sides of an isosceles hyperbolic triangle. These two equal sides have to meet at an angle Each leaf of T* and T~ inherits from the standard metric, a metric of constant negative 2"7>/9) where p/q is the rational number attached to the distinguished point. Of course, curvature equal to minus one. The geodesic flow permutes the orbits of k+ as well as the distinguished points correspond to the equivalence classes of fixed points of elliptic the orbits of h~. It dilates uniformly and exponentially the orbits of h.+ and contracts elements of F. S(T) is obtained by identifying sides of a fundamental domain which is a uniformly and exponentially the orbits of ft". finite hyperbolic polygon in H by elements of F. A classical theorem of Selberg (which The flows ft"1"an d h~ in the unit tangent bundle of H are conjugate by the so-called also holds for PSL(2, C)) asserts that F contains a subgroup F of finite index and without "flip map" which sends a unit tangent vector to its negative. Therefore, any dynamical or elliptic elements. Then 5(f) is a branched covering of S(F) and it is a complete hyperbolic ergodic property that holds for h+ also holds for h~. surface without conical points and finite area. In the unit tangent bundle Ti(S(F)) = G/T For any discrete subgroup T C G, the basic vector fields X, Y and Z descend to the we have the geodesic and horocyclic flows and there is a finite covering map from Ti(S(T}) quotient onto G/T = M(T] whose deck transformations commute with the three flows. In this way, M(r):=PSL(2,R)/r, for any discrete F, we can now speak of the horocyclic and geodesic flows on an orbifold. M(F) plays the role of the unit tangent bundle of 5(F) when F has elliptic elements. and they induce flows which we will still denote by g, h+ and k~. The standard metric, invariant form ft and Haar measure m descend to A/(F). All If F is a nonuniform lattice then S(F) is a noncompact, complete, hyperbolic orbifold of three flows g, h+ and h~ defined in M(F) preserve m. Therefore g is a volume-preserving finite area. The orbifold is obtained by identifying isometrically, pairs of sides of an ideal Anosov flow. hyperbolic polygon of finite area. It then follows that the fundamental polygon has a finite When r is a discrete, co-compact subgroup, then g, Ji+ and h~ are all m-ergodic and number of sides and there must be at least one vertex at infinity. Each vertex at infinity g is a topologically transitive Anosov flow. In this case M(F) is a Seifert bundle over a is the common end point of two asymptotic geodesies which are identified by a parabolic compact two dimensional orbifold. The exceptional fibres are due to the elliptic elements element of F (see Figure 2). of F. Both foliations !F+ and F~ are transverse to the fibres and every leaf is dense. In this way we obtain a "cusp" for each equivalence class under F of the set of vertices
6 conical point
cusp oup Figure 3
is one of the reasons why the modular orbifoli carries >o much arithmetical information. The horocycles corresponding to the cusp are regularly immersed closed curves in S(F). "Near" the cusp all such horocycles are embedded. A necessary condition for such a closed conical horocycle to be embedded is that its hyperbolic length is sufficiently small. However when point the hyperbolic lengths of these closed horocycles are big, they are no longer embedded and Figure 2 they self-intersect. The number of self-intersections grows as their lengths grow and they tend to "fill up" all of S{T). (see Figure 4). at infinity. Also we obtain a one-parameter family of closed horocycles in S(T). We can assume by conjugation in G, that the cusp is the point at infinity in the upper The orbifold S(T) has a finite number of cusps and conical points. If we compactify half-plane so that the family of asymptotic geodesies associated to the cusp is the family of S(T) by adding one point at infinity for each cusp we obtain a compact surface S(V). It parallel vertical rays oriented in the upward direction. By conjugation we may also assume is natural to label these new points by the symbol oo (or by ooc if we want to specify the that the parabolic subgroup associated to the cusp is the following subgroup: cusp c) even though the "angle" at a cusp is zero. The genus, area, number of cusps and conical points are related by a Gauss-Bonnet type of formula. In this paper the most important orbifold will be the modular orbifold which corresponds rao = to F = PSL(2,Z). Its area is, by Gauss-Bonnet formula, equal to ff/3. An important number for us will be ir2/3 which is the volume of PSL(2,R)/PSL(2,Z). or equivalently, the group of translations: Tn(z) = z+n; n € Z. When the above happens we say that the point at infinity is the standard cusp. The modular group PSL(2,Z) has Notation. M := M(SL(2,Z)) = PSL(2,K)/PSL(2,Z). the standard cusp at infinity. The horocycles corresponding to the standard cusp at infinity are the horizontal lines in The modular orbifold is obtained from the standard modular fundamental domain by the upper half-plane. By formula (0.3) we see that the points ((z, y), it) and ((x + 1, y), TT) identifying sides as shown in Figure 3. are identified by the differential of elements of Too. Therefore, associated to the standard The modular orbifold has just one cusp and two conical points with labels | and j. M cusp there exists a one-parameter family of periodic orbits of h+ in the unit tangent bundle is a Seifert bundle over the modulax orbifold with two exceptional orbits corresponding to of S(T): the conical points. Both foliations T* and T~ have dense leaves. 7, = {((*,¥).*) € Ti(5(r)) | 0 < x < 1} (y > 0). Let F C G be any nonuniform lattice. Assume for simplicity that T does not have elliptic elements. Consider all asymptotic geodesies corresponding to a given cusp. The The periodic orbit yy has period (or length) 1/y. union of all these geodesies covers the orbifold S(T). In fact, given any point of S(T) In general since any cusp can be taken to the standard cusp at infinity, we see that asso- there exists a countable dense set of unit tangent vectors at this point so that if we take ciated to each cusp, there exists a one-parameter family of closed orbits for the horocyclic a semi-geodesic starting at the point and tangent to one of these vectors, then it will flow where the natural parameter is the minimum positive period. Conversely, given any + converge to the cusp. In the case of the modular group the angle between any two of these periodic orbit 7 of h , then 7 is included in such one-parameter family. This is so since vectors at any point where two such semi-geodesies meet is a rational multiple of 2K. This the diameter of g~t({y}) tends to zero as t tends to infinity and thus it tends towards the
S where 1 € 7 is any point in 7 and T is the period of 7. It is evident that m(-y) is ft+-invariant and it is ergodic for fe+. A theorem of Dani [Da] asserts that the normalized standard riemannian measure and measures of type m(i) are the only ergodic probability invariant measures for h+. It also follows from Dani's work that for a nonuniform lattice, an orbit of h+ is either dense or else it is a periodic orbit. Suppose that F has only one cusp and that this cusp is the standard cusp at infinity. Let M(T) denote the one-point compactification of M(T). Let us extend k+ by keeping the point at infinity fixed. Let mv denote the ergodic measure concentrated in the unique periodic orbit of period 1/j/. Let 6^ denote the Dirac measure at the point at infinity. Another result of Dani is that m, converges weakly to the point-mass at infinity as y —» oo and my converges weakly to the normalized Haar measure m, as y —t 0. The geometric significance of this result is clear: as the period decreases, the periodic orbit becomes smaller and tends to the cusp. On the other hand, as the period increases, the horocyclic orbit gets longer and wraps around M(T): it is almost dense. The latter case means that as the period grows the horocyclic orbits tend to be uniformly distributed with respect to the normalized Haar measure. Let Ca(M(T)} denote the Banach space of continuous real-valued functions of M(T), with the sup norm. Let C* = [C°(M(T))]m denote its topological dual with the weak*- topology. Let D : R+ - C; D(y) = ms, (y > 0) where RJ" denotes the multiplicative group of positive real numbers.
Dani's Theorem. [Da]. The measure mv converges in the weak*-topology to the nor- + Figure 4 malized Haar measure m as y —* 0. The only ergodic measures ot h are D(y) for y > 0, and rn. point at infinity of some cusp. Exactly the same reasoning plus the fact the h+ is ergodic The weakly*-compact convex envelope of the image of the curve D is the set of all with respect to m implies that the only minimal subsets of ft+ are M(T) and horocyclic invariant probability measures of h+. periodic orbits. Using the "flip" map we see that everything we just said is also true for The rate of approach of mB to the Haar measure m (as y —» 0) is intimately related to the Riemann Hypothesis. Don Zagier [Za] found a remarkable connection between the Riemann As a consequence of the above remarks, we obtain that when T is a nonuniform lattice, hypothesis and the horocycles in the orbifold 5(PSL(2, Z)) = PSL(2, Z)\H (i.e., in the mod- there exist injectively and densely immersed cylinders C,, t = l,...,r, where r is the ular orbifold). Here we will use a particular case of Sarnak's result [Sa] (which generalizes number of cusps. These cylinders are mutually disjoint and their union comprises the the previously mentioned theorem of Zagier) for the case M = PSL(2,R)/PSL(2,Z): totality of periodic orbits of h+. Naturally, these cylinders are distinct leaves of the Theorem (Zagier). Let f be any C°° function defined on M and with compact support. unstable foliation of the geodesic flow. Everything that we have discussed so far follows Then: by analyzing the locally free actions of the affine group on SL(2, R) and on its quotients by discrete subgroups. (0.5) 0J. For nonuniform latices T, the horocyclic flows h+ and ft" are not uniquely ergodic in A/(r) since both contain periodic orbits. For each periodic orbit 7 of h+ let 01(7) denote FUrtiiermore, the above error term can be made to be ofy3'*"') for every e > 0 if and only the Borel probability measure which is supported in 7 and which is uniformly distributed if the Riemann Hypothesis is true. with respect to its arclength i.e., if / : M(T) -» R is a continuous function, then In particular it follows from the above theorem that mv converges vaguely (namely as a distribution) to m, as y —* 0. If U C M is any open set then (0.5) implies that ), /) := »»,(/) = I j /(*?(*)) dt, {as y - 0)
10 Figure 5 Figure 6 where xu >s the characteristic function of U. P. Sarnak [Sa] developed Zagier's result to other discrete subgroups of PSL(2, B), namely The base and the common length can be chosen small enough so as to have an embedded nonuniform lattices. It will be clear that many of the ideas contained in the present cube. In order to compute the volume (with respect to the normalized Haar measure) of paper (which incidentally, are reasonably simple and geometric) could be applied to these a standard box it is better to take a lift of the box in the covering SL(2, R), compute the more general cases, at least when the nonuniform lattices are arithmetic (for example, for volume there, and then normalize. Let C C SL(2,R) be a box, let £ be the common length of the unstable segments of the box (to be called the "height" of the box). Denote by A congruence subgroups). the hyperbolic area of the base B. Then, as we will show latter: In the present paper we will show that the exponent 1/2 in the error term of (0.5) is optimal for certain characteristic functions X'J- Namely, we will prove the following (see m(C) = At = (area of base) x "height". Theorem 3.12): Theorem. There exists an open set U C M and a positive constant K > 0 depending A formula reminiscent of our elementary school days! onJy on U such that: Therefore, if C is a box in M we obtain by normalization: (0.6) for 0 < y < 1/2. m(C) = ^ {At). Furthermore, if a > 1/2, then a The set of all boxes in M is a basis of the topology of M and generates the
11 12 Figure 6 suggests Ampere's Law of electromagnetism. Let us imagine a steady unit Therefore, we see from (1.2) that: current flowing around the horocyclic orbit 71 then the normalized integral of the magnetic 3_ field induced in the boundary of the square is n(y, C). 7T2 There is another way in which we can compute mv(C). Let Ct = 9-t(C), (t > 0), be the image of a standard box by the geodesic flow reversing time. Then, since the geodesic Using (1.1) we obtain: flow preserves m, we have: f gt(Ctn-n) = cnyt (1.3) JL »-0 7T2 \m(Ct) = m(C); * € R- Formula (1.3) implies that m converges vaguely to m as y —» 0. Let B(t) be the base of C< and let A(t) be its area. Let l(t) be the height of C . Then: t t However, what is important for us is that the exponent 1/2 in the error term in (1.2) is optimal. This will follow from the fact that n(y,C) grows as a classical arithmetic function: A(t) = e'A(O) = e'A, the summatory of Euler's if function: where A is the area of the base of C. We also have: ifr£R,r>l. £(t) = e-'e(0) = e-'£, 1=1 n
"(y.C)-*^-1/2), wiiere ~ denotes asymptotic equivaJence as y —» 0. Therefore, Mertens' theorem will show the validity of (1.2). Incidentally, the number ~h ~ 5[C(2)]-I which appears in Mertens' formula is equal to the volume of M. Hence, weak convergence mv —> m (as y -+ 0) is equivalent to Mertent' theorem. This theorem appears in any standard text book in number theory, for instance, Apostol [Ap], Hardy and Wright [HWj and Chandrasekharan [Ch]. See also [Da]. Figure 7 2. SOME LEMMAS Definition 2.1. Let SI be the volume form in G such that: Let J(y) — {Ji(y), Jj(y),..., Jn(j,c)(y)} be this collection of open intervals of equal length. The distribution of J(y) in the basic horocyclic orbit looks seemingly random. This is not exactly the case: the midpoints of the intervals are like a circular Farey sequence of order y'1. The connection between the growth and distribution of circular Farey sequences where Xp,Yp,Zp € TpG are tangent vectors at p. Let {•, •) be the riemannian metric such and the Riemann Hypothesis are well-known theorems of Franel [fr] and Landau [Lan] (see that {X, V, 2} is an oriented orthonormal framing. Let || - || be the norm induced by this also Edwards [E, p. 263]). The problem of understanding the distribution of J(y) can be metric. viewed as a problem of equidistribution in the sense of Hermann Weyl. We shall prove later that J{y) has the pattern just described. In fact, we will show: Throughout this paper we will use this metric. The measure m determined by Q is the standard riemannian measure or Saar measure.
1 If F C SL(2,R) is any discrete subgroup then X, Y, Z, fi, m and (•,•} descend to (1.2) ~y- (aay ~> 0). M(T) := SL(2,R)/r, and they also descend to any quotient PSL(2,R)/I\
13 14 Definition 2.2. Let M = PSL(2,R)/ PSL(2,Z). We will let m denote the normalized where COS0 = d(c2 + d2)^, sinS = c{ 0 we have Using the monotnorphisms of the affine group into G given by formulae (0.4), we obtain a b] _ fc-1 0] [l col [0 -1] the commutation rule: c oj *" [ 0 c\ [0 1 j [1 0 J ' a 1 6 1 aH a (231 \ ° 1 f 1 -f ] \ ° 1 This gives the explicit AMK. decomposition. We also have, if d ^ 0, The modular function of the affine group is: ^)-i 0 1 [ cosfl sinfll [ [ 2 1 [c dj"[o 0 (c'+rf )*] [-sin« cosflj
1 ! if rf = o, As a manifold G is the product S xB . The group G can be decomposed as a product [a 6] _ fl ac~l] fc-1 Ol [0 -ll in two ways using Iwasawa'e decompositions: G = MAK, and G = AMK, where JV is the [c Oj ~ [0 1 J [ 0 cj [l 0 J ' nilpotent group of matrices: Let A = * £ G, and let us write the unique decomposition
, = [l i|[v 0 1 [ cos0 sintfl. Q A is the diagonal group: [0 1J [0 j/-1 J [-sine cosej ' y
Then the map rp : SL(2,R) -» T(H, which identifies SL(2,R) as a double covering of -{[: .!•]!•>•}• PSL(2,R):=TiHisgivenby and tC is the compact circle group:
n JC = (r(«):\ = -sinf°°f*f ™cosff;jl 9e Therefore the measure induced in TjH by this identification is given by the volume form Therefore any , £ G can be written in a unique way as follows: 1 dxdydS
a b _\l x]\y 0 1 [ cocoss 0 sin 61 Hence, if (7 C TjH is any open set then by Fubini's theorem we have: c d 1° ij L° r'J [-«si•n S cos & j where i, 8 £ R and JI > 0. (2.4) m{U) = [ dV = I f " Ah(U(6))d<>, Also it can be written in a unique way (using (2.3)) as follows: Ju 2 Jo 0 I [l y-2il [ cosfl sin0] where U{6) = {z € H | (z,9) € U] is the "slice" of V corresponding to 6 and c d y-1] [o 1 J [-sinfl cos^J " dxdy These two parametrizations of G give different expressions for the Haar measure written = jj 75 in terms of dx, dy and d8. is its hyperbolic area. From Formula (2.4) we obtain that if U C T]H is S1-saturated (i.e. it is a union of circle 1 a b 0 l u cos* sine fibres), U =D x S , then c d d2)1/2 0 l ~ain9 cosfl m(u) =
15 16 In particular if we take the fundamental domain of the action of PSL(2, Z) in TiH consisting Proposition 2.7. of all unit tangent vectors based in the modular fundamental doniain in H we have: 2 ! 2 = Ah(V{6)) =\T f sec 9 = («i - uo)(e-" - e" ') sec 6 m(M)=3/7r2. UtQ Jut y 1 Therefore: Now let A = A(u,t,v) e G be as follows: 1 ul [«
2 2 1 2 then a(A) = e'/ (i> + I)" / , so we have: e-'»-
A- 1,
Then (2.6) is the N"A>C decomposition of A and we have:
t<> cos* = (u3 + I)"1/2, sinC = — v(v2 + I)"1'*, andu = -tan0. {1 81 (U\ — "f^w — u \(v — v \(e~ — f~'M 2 Let U C SL(2,R) be the closed set: Another way to compute m(U) is the following. Let / C R3 be the cube:
U = {A(u,t,v) | u0 < u < ui, v0 < v < vi, t0 < t < ti] 3 / = {(u, *, v) e R | u0 < u < ui v0 < v < vt t0 < t < U }• where A(u, t, v) is as defined in (2.5). Let if : I ~* U be the pararoetrization of U given by f{u, t, v) := A(u, t, u), where A(u, t, v) Let U C PSL(2,R) be the projection of U. Then: is given by (2.5). Let 9U = •=-, dt = •=-, 9^ = •=- be the standard vector fields in /. ou a at ia a ov88 3 for tlle m(U) = i Then, (v.(3u)(Ui4iV), ^,(9,)(„,(,„),v*( o)(i.,*,«)) b ' tangent space TA(Uitil,)G. Comparing this basis with the basis (X^,itiV),Y^Uitt^,ZjnK>tiVy) given by the basic vector fields at A(u, t, v), we see that the change of bases is given by the matrix: where Oo,8i € (-7r/2, TT/2) are the unique numbers such that u0 = — tan0o and - tan Si, and where k(6) = Ah(U(&)). 1 Using formula (2.6) we obtain for S\
2 U(0) = l(-(l/2)e' sin(2fl) + u) + (e' cos 6)i | u0 < u < u,, ta < t < d } c H. Since the determinant of this matrix is e ' we have: We have that {/(#) has the same hyperbolic area as: 1 1 1 /"' /*' f i 2 ~m(U) = ^ / / e~'dudtdv V(0) = {usec 0 + e't 11 < i <
Let us consider now the geodesic flow gt : G —> G. We have for t € R and p € G: We thus obtain: g't(Yr) = e'K,,,,) g*t{Zf) = e"^,^,,
17 18 where g* is the differential. Then: (ii) Any left-translation of a box is also a box: aC(x;a,b,c) = C(ax;a,b,c); a € G. (iii) Since the center and the parameters of the box can be chosen arbitrarily, it follows Hence, {31} is an Anosov flow leaving invariant the splitting: that the interiors of the boxes (i.e. the open boxes) generate the topology of G and the tr-algebra of its Borel subsets. + TG = E © E~ © E Let us compute the Haar measure of the box C := C(x;a, b,c). Using a left translation by I"1 and the fact that G is unimodular, it is enough to compute the volume with where in this Whitney sura E+, E~ and E are the line bundles spanned by Y, Z and X parameters (e; a, b, c) where e is the identity element. But in this case using formula (2.9) respectively. The fact that g is Anosov implies that it is structurally stable and its periodic we have: orbits are dense. This accounts for its very rich dynamics. 2 The differential of the geodesic flow acts on the canonical framing as follows; Lemma 2.11. m(C(x;a,&,c)) = a^e^ - e'^ ) = 2ac[sinh(£)]. Let us return now to T* and T~• If x € G, let us denote by I+(i) and L~{x) the leaves of T* and T~ which contain the point x. Explicitly: Therefore, the Jacobian of gt is identically equal to one and the geodesic flow preserves ft. !+(*) = {#(*(*)) I M e *} + A similar calculation shows that ft is also preserved by h and h~. I-(x)={Ji-(j,(*))K*€R}. The circle group K, is the one-parameter subgroup corresponding to _j I ^ 3l(2,R). 0 Definition. Let C = C(x; a, b, c) be a closed box. Then the base of C is: Therefore the vector field W = Y — Z induces a free action of the cirlce on G. The foliations :F+ and T~ tangent to E+ © E and E~ ffi E are respectively the unstable and stable foliations of the geodesic flow and are also obtained by left translations of the two copies of the affine group in G. Both foliations are transverse to W and every leaf of these Clearly, we have 0(C) C £-(x) and Ah(${C)) = 2csinh(£). foliations intersects each circular orbit of W in exactly one point. This is the geometric From now on we will work in G = PSL(2,R) and in M = PSL(2,R)/ PSL(2,Z). In interpretation of the two Iwa.iawa decompositions. Given any measure (or more generally, these two manifolds we have the unstable and stable foliations f+ and T~'• any Schwartz distribution) one can disintegrate the given measure with respect to each We define standard boxes in M exactly the same way. In fact, they are the projections of the foliations. If T C G is any discrete subgroup, then the vector field W descends of the boxes in G. However, we will only consider embedded boxes in M. to GjT = Af (r) and induces a periodic flow which gives M(T) the structure of a Seifert If C = C(x;a, b,c) C G is a box, then: fibration over a hyperbolic orbifold. The foliations JF+ and T~ descend to M{T) and their leaves are transverse to the fibres of the Seifert fibration. If m(M(T)) < 00 then every m{C) = ac (sin leaf is dense. With respect to the induced metric, each leaf in any of the two foliations is isometric to H. Formula (2.8) can be obtained directly by disintegration. If C — C(T; a, b, c) Q M, then its Haar measure is given by m = -^ac (sinh(6/2)). Definition. Let JSC. Let a, b and c be positive reals. Then a standard box or simply it* a closed box, denoted by C(x; a, 6, c), or simply by C if the parameters are understood, is Let us recall the identification 0 : PSL(2,R) -* XiH, which assigns to each Mobius the subset of G defined as follows: transformation We call x the center of the box. Thcaction of PSL(2,R) on = {(z,8) | z € H,0(mod 2>r)} is given by: Remark 2.10. a{z,9) = *),8 - 2arg(cz + d)), c{z) = ^±±. (i) The image under the geodesic flow of a closed box is another closed box: (Recall that the angles are measured counter-clockwise from the vertical.) + gt(C(x; a, b, c)) = C(S»(x); e~'a, b, e'c); t € R. Let I (e) and L~(e) be the unstable and stable leaves through the identity e g G. 19 20 + L~ are the basic unstable and basic liable decays very rapidly as y —> 0 or y —' oo) then the corresponding probability invariant Definition. := £ and + + leaves, respectively: measure for k is concentrated in P{L ). Let C denote the set of all boxes in M which are adapted, i.e., C € C •» j3(C) C P{L~). Then C is a basis for the topology of M. Hence, to see how the measures mv, approach the normalized Haar measure m, it is enough to estimate with precision mt(C) for all C £ C. Definition. An adapted box C C G is a standard box such that /3(C) C £~. Fbr each * 6 R, let A< be the horizontal line which is parametrized by: If a € G and C(o; a, 6,c) is an adapted box centered at a, then a = (xo + i}ft>, 0) and \,{s) = 3 e R. Let A, : R-» XiH be defined by A((a) = (Aj(a), IT). Thus At parametrizes the horocycle with equation y = e~* and Po Aj parametrizes with arclength as parameter the unstable horocyclic orbit of period y = e~', (See Figure 8). i.e. it parametrizes: Ti = 9I(TO)- Let A = SL(2,Z), c / 0. Let A(z) = denote the corresponding cz + d modular Mobius transformation and let A' : XiH —> TiH be the induced map in the unit tangent bundle. The image of A, under A is the horocycle which is the circle tangent to the real axis at the point * and whose highest point is: (2.12) z = -+e'c-ii; (c^O). Figure 8 This fact follows immediately since the point with biggest ordinate in A(A() corresponds to the unique real number $Q for which 4-{A o A )| is real. Hence SQ = — *. We also We see that bases of adapted boxes can be identified with rectangles in H whose sides ( obtain that A'(~/t) intersects L~ only at the point: are parallel to the coordinate axis. We will not distinguish the base and the corresponding rectangle in H. Let P : G -* M be the covering projection and let P : M -* £(PSL(2, Z)) be the Seifert J fibration onto the modular orbifold. Then P(L^) and P{L~) are called the basic unstable When c = 0, then A is a horizontal translation by an integer and A( and yt are kept and stable leaves of the corresponding geodesic flow. P(L+) and P(L~) are the cylinders invariant by A and A' respectively. + mentioned before which contain all periodic orbits of fc an d h~ respectively. Definition. For t — 0 we have the basic horocycle: Both P(L+) and P(L~) are dense in M. If * € 5(PSL(2,Z)) and S1^) = P'1^) denotes the circular fibre over z, then Hi ={(z,l)€H|ar eR} =A0. The basic koToball is the boundary of H\ in the extended hyperbolic plane (the closure L+(Q):=S1(z)nP(L+) of H in the Riemann sphere): L-(Q):=Sl(z)nP(L-) have the property that if a,0 6 X+(Q) (or £~(Q)) then there exists 6Q € Q such that The images by elements of PSL(2, Z) of the basic horocycle are called the Ford circles and the images of the basic horoball are called the Ford discs. Two Ford discs either coincide or else they are tangent at a point in H and have disjoint interiors. The hyperbolic area of a Ford disc is one. where r* : M —> M, 8 g R is the periodic flow induced by W = Y — Z. It follows from formula (2.12) that the Ford discs are tangent to the real axis at rational This simple fact happens to be very important for number theory. The reason is clear points (except for the basic horoball which is tangent at the point at infinity), and every (apart from the fact that the rationals are involved): any invariant measure for h+ corre- rational point is a point of tangency. All these facts are important for number theory sponds to a Choquet measure on the image of the curve D : [0, co] -+ C*. If this measure (read the very last paragraph in Raderaacher's classic book in complex functions). (See has compact support (or even if the density of a Choquet measure at J?(0) and D(*x>) Figure 9). 21 22 The subgroup of integer translations of the modular group identifies (x,y) £ H with ( n,y) e H, and ((i,y),0) with ((i + n, ji),0); n 6 2. Since we want adapted boxes in M which are embedded, it is enough to consider standard boxes in M which are projections of standard boxes in TtH whose bases lie in the half-open strip 0 < x < 1 (see Figure 10). Figure 9 Let F be the set of all Ford discs that intersect the strip 0 < x < 1, y > 0. For r > 0 1 let Fr denote the subset of Ford discs in F that intersect the half-plane y > r" . Then the circles in Fr are tangent to the real axis at exactly the rational points in [0,1] which 1 2 belong to the Farey sequence of order r / . Therefore, its cardinality \Fr\ is given by where ip denotes Euler's totitnt function. Everything follows just by looking at formula Figure 10 (2.12). Let Q be a rectangle which corresponds to the base of an adapted box C. For each Let C C M be a box such that ff(C) is the rectangle Q := Q(ai,a2; ft,#z), where i € R let T, : H -. H be defined by 0 < ai < c*j < 1, 0 < ft < /32, denned by: t Q = {(*,V) € H | <*i < x < at, 0i < y < fa}. (2.13) Tt{x,y) = {x,e~ y) We have that ,%_,(C)) = T,(/3(C)) where Tt is given by formula (2.13). Therefore: and let Qt = Tt(Q). Then Qt is the base of the adapted box g-t(C). As t —t oo , Qt starts intersecting more and more Ford discs. Let n(t) denote the number n(0 := #{ff.(7o) n 0(C)} = #{70 n gt{C))} of Ford discs whose highest point is contained in Qt, then n(t) has the same growth type i 6 PSL(2,Z) I A'(7,)n {((*,»),0) € T,H | (X,y) € Q} ? 0} as that of FWey sequences contained in a fixed interval. At this point it is clear the connection between Farey sequences, the ergodic measures = a e SL(2,Z), cjtO I A'( ,)n{((x,!/),0)€ TiH | (x,y) € <3} ? k [c Theorem (Franel-Landau). Let FN = {/i,/2,..- ,/•(«)} be the Farey sequence of order JV consisting of reduced fractions j £ (0,1], arranged in order of magnitude. Let We thus have the following: &n = fn — Q(fj)> in = 1,... ,4(JV)) be the amount of discrepancy between /„ and the corresponding fraction obtained by equidividing the intervaJ [0,1] into *(JV) equaJ parts. Proposition. The number of points in which the horocyclic orbit gt(lo) intersects the base of the box /3(C) as a function of t is given by Then, a necessary and sufficient condition for the Riemann hypothesis is that; (2.14) Let R^. = {(",«) eR1 I v > 0} denote the open upper half-plane with the Euclidean metric. An alternative necessary and sufficient condition is that: Let * : R^. -» H, be the function: • (JV) (2.15) *(«, v) = - + v-2i, (v> 0). \6i\ = 0{N^') for all t > 0, (Landau). This simple function has the following remarkable properties: 23 24 Proposition. * is an orientation-reversing diffeomorphism from the upper euclidean half- Let A(() = ^,(A(0)), for t £ R, where A(0) = A is the trapezium of the previous plane onto the hyperbolic plane. It sends rays emanating from the origin onto the family paragraph. Let Tt be the transformation defined by formula (2.13), then we have: of geodesies which are verticaJ h'nes and the famiiy of horizontal lines onto the famiiy of horizontal horocydes. The absolute value of the Jacobian of $, with respect to the (2.18) *o^t = T, otf. Euclidean and hyperbolic metrics is 2. Therefore, At{U) = \Ak{i{U)) where U C R+ is any open set and ACJ A/, denote the Euclidean and hyperbolic areas, respectiveiy. 'il lenils Hence, by formula (2.14) we have: the integer lattice points with relatively prime coordinates in the euclidean upper half-plane onto the pointi of tangency of the Ford circles in the hyperbolic plane (2.19) n(t) = #{(«, b) € Z+ x Z+ I {a, 6} = 1, (a, b) € A(t)}. The usefullness of this proposition is that it reduce* the problem of counting the number of intervals in which a closed korocycle intersects a box to a euclidean lattice point counting 3. MAIN RESULTS Let A be any trapezium in R+ whose boundary consists of two horizontal lines and two Now, all that it is left is to estimate n(t). We will need the following: segments which are collinear to the origin. Then *(A) is a rectangle in H whose sides are parallel to the coordinate axis (see Figure 11). Theorem (3.1). (Mertens) Let 0< <*i < a2 < 1. For each I > 0, let T(1) be the triangle in R\ defined by r(l) = {(«,«;) e R\ I v < t,ai < ~ < 0.2}. b, -,— Let N(£) be the number of lattice points with relatively prime integral coordinates • contained in r(l). Then where \t}(t)\ is bounded by (2 + 2\/§) (1 + I/log 2) < 24 for all t > 2. Figure 11 Proof. This result is classical and the method of proof starts with Gauss. However, I will If A is bounded by the lines u = a^v, u = ajv and the lines v = 6], v = 62, then give a proof here since I will need the method for the following lemmas. I will adapt the 2 2 proof given in Chandraaekharan [Ch, p. 59]. *(A) = {(x,y) 6 H I a, < x < at, b~ < y < frj }. Then, 2 Let A(i) = A(l)£ = (a2 -Ql)^/2, and p(t) = p(lX be the Euclidean area and perimeter of T{£), respectively. Let K(t) = {(a, 6) e Z+ x Z+ I (a, ft) € r(«)}, and and let N(t) ^= \K{t)\ be its cardinality. Then A{t) ~ V2p(t) < 7T(<) < A{t) + y/2p(e), for ail * > 0. Therefore Hence (2.16) A (A) = - C (3.2) 7f(t) = A(l) + a(i)p(£) where \a(t)\ < \fo for all I > 0. Of course we knew (2.16) since the Jacobian of * is —2, but we wanted this fact explicit For £>1: (in fact the formula (2.16) for all trapeziums implies that the Jacobian is ±2). For each (eR, let ftt: R^. -* R^. be the homothetic transformation; (n,m)eK(f> /2 /a (2.17) fit(u v) = (e' u, e* u). (an empty sum will be by definition equal to zero). 25 26 •.i, m 1* ip Si Therefore and Then, since {n,m} = d •» = 1, it follows that there exists a bijective (where [•] is the greatest-integer function). spondence between the sets Finally, since A{t) = A(l)P < \P, and p(l) < 2 + V^ we have that Bi(d) = {(n,m) € T(() \ {r>,m} = d) and B2(d) = {(n',m') | (nW) € T(e/d),{n',m'} = 1} where (1 < d < t). where By definition, the number of elements of Bjfd) is equal to N(t/d). Hence, <24. O Corollary 3.3. The {allowing estimates hold: N(t) Applying the Mobius inversion formula, we obtain N(l)= £ and where /i is the Mobius function. By (3.2), we have Remark. Let f (t) := {{u, v) 6 T(£) | V < f.} denote r(t) minus its base, and let iV(f) denote the number of lattice points with relatively prime coordinates contained in T(£). Then = N{t) + tf(*)«, where \K{£)\ < 1. Since £*(•) is bounded for all t > 0, and |^(d)| < 1, we have Therefore + i)( 0, 0 < QI < a2 < 1, 0 < 0i < fa. Let A(£) be the trapezium in R+ defined as = {(u,u)eR^. |QI Therefore, by Theorem (3.1) and (3.4) we have 27 28 Corollary 3.5. The following equality holds are also parallel to the horizontal axis). Therefore it must contain at least -^—^^ _2 lattice points and at most t"i~ttt?" +" + 3 such points. Hence, if [£ + l\ is a prime, then where (3.10) ) wbere |,,W|< 3. < 40[ft - A + A log ft - log ft +1] for aJi I > max{2,2/Sf1, e">}. Prom (3.8) and (3.9) we obtain Corollary 3.6. Tie following formula, holds: Hence The following lemma depends only on two properties: the infinitude of primes and the fact that tp(p) = p - 1 if p is a prime. (3.11) Lemma 3.7. Keeping the notation of Theorem (3.1) we have: *+• ti-\ = Now consider formula (3.11) for all I > 0 such that [£ + 1] is a prime. Then, using (3.10) we have for every e such that -oo < e < |. Proof. Suppose the lemma is not true for some e € (—oo, 1/2). Then there exists a bounded function Bt(£), bounded for all £ > 0, such that (recalling that J4(1) = (aj — ai )/2). But now we arrive to a contradiction since under the hypothesis, R(t) is bounded for all £ > 0 whereas L{£) tends to infinity when t tends to (3.8) _6^ 2 infinity by a sequency {£„} such that [£ + 1] is prime. • w n flemori S./S. The fact that the set of all t > 0 for which [£ + 1] is a prime, is infinite, is Define H(t) by all that we used. Hence, it does not depend on the triangle. Remark S.1S. When e above is very close to 1/2, then 6(£) := [£$ - ^] I3/2— oscillates 3 then and becomes unbounded extremely slowly. Also it follows that 0s,e(£) := [^$ - e] £*+ /2 is 0(1) as f —* oo if and only if c = ^j and S < —1. When aj and QJ are rational and we let t go to infinity through natural numbers then N(£)/A(£) are (not very good) rational approximations of ^. In the context of Corollary 3.5, let &(£) denote the trapezium A(£) minus its open base: where u(£) is the number of lattice points with relatively prime coordinates contained in (< + l)(*)i £(<) = {(u, v) € R% | 0 < «! < «/v < a2 < 1, A< < *> < W}- = {(a, 6) e Z+ y = 1, („, b) € 1), 6 = [i + 1]}. Then if Jv(f) denotes the number of lattice points with relatively prime coordinates contained in 2(£), we have If [£ + 1] is a prime, then «;<*) = #{a € N j Ol < a/[< + 1] < «2, o < [< + 1]}. Now, r(t + l)-r(£) is a trapezium of unit height, with its bottom side missing and whose (3.H) = N(£) + Z{£)P l, where -1 < {(<) < 0, for aU f > 0. non parallel sides have positive slope greater than or equal to one (and whose parallel sides x 29 30 n 1 d Corollary 3.15. For every e > 0 such that -oo < e < 1/2 we have where m = min{n € N | £ < *~ /32~ } an where JV(.), r(-) are exactly as in Theorem 3.1. Therefore, we have lim (3.16) /. Suppose the corollary is false for some e,t £ (-oo,l/2). Then, just as in Lemma (3.7), there exists a function Be(£) bounded for all I > 0 such that But (3.16) implies that, 1/2 , Then by (3.14), we have where i?((f) is bounded for all I > 0. But this contradicts Lemma (3.7), therefore, Corol- lary 3.6 must be true. • N(t) 6 3 6 6 < f / j-< Let us recall the functions X,, * and fit given by formulas (2.13), (2.15) and (2.17) and A.(A(f)) ^(A(0 ) ^2 Ae(A(f)) connected by formula (2.18): * o p, = Tt o *. Let C C M be any adapted box whose base fi(C) corresponds to the rectangle o have QCH, Q:={(x,y)\ai where ^ is a function bounded as follows: for all t > 2max{log2, log 2 + ^ log ft, & * }. n=0 FVom (3.17) we obtain 31 32 And hence we obtain: 4. APPENDIX The Rankin-Selberg Method and the Metlin transform of m,. In all that follows we will borrow from [Za] and [Sa]. Prom Corollary (3.15) we obtain Let / : M —• R be any C°° function with compact support. Consider for s € C with R(s) > 1 the Mellin-type transform for every a > 1/2. (E(s),f):= Changing the parameter: y = e~* and denoting as usual by m, the horocyclic measure concentrated in the unstable horocyclic periodic orbit as period \jy (y > 0) we obtain We may think of / as a function / : 2iH -+ R which is F-invariant (V = PSL(2,Z)). using formula (1.1) Then Theorem 3.18. Let C C M be any adapted box. Then Jo where x, y and 9 are the parameters of SL(2, R) described before. Let K = sup{3( ) | (z,9) e supp(/)}. Then if »{J) > 1: where \Kc(y)\ is bounded by some postivie constant for all 0 < y < 1/2. furthermore, f i this positive constant can be chosen to be the same for every adapted box contained in C. (Compare with theorem 1.4) (4-1) * = «(*). Corollary 3.19. The following holds: Therefore, we see that (i) The integral denning 1 + < for all t > 0. Therefore G/( ) is holomorphic in R(.s) > 1. We have proved everything in Theorem (3.18). The fact that there exists a constant k (ii) Because of inequality (4.1) it follows that such that \K (y)\ < k for all 0 < y < 1/2 and every box C CC follows from (3.17). c E{.) : {»{,) > 1} -, [Crr = ?>{M) Theorem 3.20. Let C be any adapted box. Then defines a weakly holomorphic function with values in the distribution space, V(M), (3.21) U^[|mJ,(C)-mir(C)|y«] = +co of M. In fact, (4.1) implies that for all so> with £(«o) > 1, ^(^o) is complex valued infinite measure concentrated in P(L+) and invariant under hf. Where kf acts for alia > 1/2. on distirbutions S as follows: Theorem 3.22. Let f : M —tTR.be any continuous function with compact support. Then, there exists a positive constant K(f), depending only off, such that (h+SJ) = (S,fohtt),f € C?(At),t € R. 1/2 (3.23) \my{f)-Mf)\ 33 34 where * denotes convolution and Hence M(f,s)= f{z)E{z,s)dz. Js(D With this formula we see that M(f, s) enjoys the same properties of E(z, s): and "H(s,.) acts on 2ir-periodic vector functions by convolution in the 8 variable. (v) Let o = sup{3i(p); C(p) = 0}- Then, by Mellin inversion formula, we have for every (i) M(f,s) has a meromorphic continuation to all of C. It has a simple pole at » = 1. / € C?(M)x (ii) Res,=1 M(f,s) = i /S(r) /(u) du. (iii) Af *(/, s) := *-'V(s)({2s)M{fts) is regular in C - {0,1} and it satisfies the func- i H tional equation (/) ) (the validity of this inversion is shown in [Sa]). E(z, s) is an Eisenstein Series: Changing variables and classical growth estimates of C(J)>r(s) (Titchmarsh formulae (14.25) (14.2.6) p.283 [Ti]) we have the validity of the following inversion: *(.,.)= £ 9(7.)* = I E drs* (4.2) mv[t} = Remark. The author has found an interesting connection with Hurwitz zeta function by The right-hand side of (4.2) is a superposition of functions which are O (l — f — e) considering horocyclic measures concentrated on equally spaced closed horocycles ap- for all f. > 0 (y —• 0) hence, ms(f) has the same order for all / € Cf(M). This is proaching the cusp. Also one could give a geometric proof of the theorem of Franel-Landau the connection with the Riemann Hypothesis found by Zagier. To prove the Riemann Hypothesis it is enough to find a C2 function,/, with compact support in M such that the 3 4 £ ACKNOWLEDGEMENTS error term \my(f) - mv(/)| can be made o(y ' ~ ) for all e > 0 I would like to thank S. G. Dani, E. Ghys, S. Lopez de Medrano, C. McMullen, B. Ran- Eisenstein Series. Let C^(5(F)) denote the space of functions which decay very rapidly as the argument of the function approaches the cusp. Namely: the space of functions dol, D. Sullivan and especially Bonaventure Loo for his encouragement in writing this / : H -* C which are T-invariant and such that f(x + iy) = O(y~N) for all N, Then, since paper and also for his generous help while typing it. f(x + iy) is periodic of period one in the x-variable we may develop it into a Fourier series REFERENCES [A] D. V. Anosov, Pro. Steklov Inst. Math. 90 (1967), Geodesic flows on closed manifolds with negative curvature. Let C(f,y) = J f(x + iy)dx denote the constant term of its Fourier expansion: [Ap] T. Apostol, Introduction to the theory of number), Springer-Verlag, Berlin, Heidel- o berg, New York, 1979. [Ch] K. Chandrasekharan, Introduction to Analytic number theory, Springer-Verlarg, Berlin, Heidelberg, New York, 1968. Let M{f,s) be the Mellin transform of C(f,y): [Da] S. G. Dani, Invariant measures of horospherical flows on noncompact homogeneous spaces, Invent. Math. 47 (1978), 101-138. M(f,s)= H )y-X^-, (»(,)> 1). [E] H. M. 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