LET Y = Γ\G/K = Γ\X be a compact, n dimensional

Transcription

1 Prime geodeic theorem for compact even-dimenional locally ymmetric pace of real rank one Muharem Avdipahić and Dženan Gušić Abtract We improve the error term in DeGeorge prime geodeic theorem for compact even-dimenional locally ymmetric iemannian manifold of trictlegative ectional curvature Inde Term locally ymmetric pace prime geodeic theorem Selberg eta and uelle eta function I INTODUCTION LET Y Γ\G/K Γ\X be a compact n dimenional n even locally ymmetric iemannian manifold with trictlegative ectional curvature where G i a connected emi-imple Lie group of real rank one K i a maimal compact ubgroup of G and Γ i a dicrete co-compact torion-free ubgroup of G We aume that the iemannian metric over Y induced from the Killing form i normalied o that the ectional curvature of Y varie between 4 and A well known a prime geodeic C γ over Y correpond to a conjugacy cla of a primitive hyperbolic element γ Γ Let π Γ be the number of prime geodeic C γ of length l γ whoe norm N γ e lγ i not larger than ee Section 3 DeGeorge [7] derived the following form of the prime geodeic theorem with an error term π Γ log e αu u du η a + where η i a contant uch that 2n α η < α and α n + q with q depending on whether X i a real a comple or a quaternionic hyperbolic pace or the hyperbolic Cayley plane repectively ee ection I and V of [7] Integrating by part one eaily deduce a weaker form of the prime geodeic theorem π Γ α α log 2 M Avdipahić i with the Department of Mathematic Univerity of Sarajevo Zmaja od Bone Sarajevo Bonia and Heregovina mavdipa@pmfunaba Dž Gušić i alo with the Department of Mathematic Univerity of Sarajevo Zmaja od Bone Sarajevo Bonia and Heregovina phone: ; denang@pmfunaba a + where f g mean lim + f g Note that 2 wa alo proved by Gangolli [2] and by Gangolli-Warner [4] when Y ha a finite volume By adapting Hejhal technique [6] [7] Park [9] refined the correponding reult of Gangolli-Warner [4] for real hyperbolic manifold with cup Inpired by andol approach [20] we [] further improved Park reult [9] to the form π Γ 3 2 d0<jk 2d0 k li jk 3 2 d0 log 3 a + where d 0 2 n j k k 2d 0 k j k i a mall eigenvalue in [ ] d2 of k on π σk λ jk with j k d 0 + i λ j k or j k d 0 i λ j k in 3 2 d ] 0 2d 0 k i the Laplacian acting on the pace of k form over Y and π σk λ jk i the principal erie repreentation Note that the error term in 3 i in accordance with the bet known etimate in the cae of compact iemann urface ee eg [20] [4] The main purpoe of thi paper i to improve the error term in the prime geodeic theorem of DeGeorge [7] for compact even-dimenional locally ymmetric iemannian manifold of trictlegative ectional curvature o to correpond to 3 We hall ue the eta function of Selberg and uelle decribed by Bunke and Olbrich [6] In particular we utilie the fact that for even n thee function are meromorphic function of order not larger than n ee [2] [3] II PELIMINAIES In the equel we follow the notation of [6] Aume that G i a linear group Let g k p be the Cartan decompoition of the Lie algebra g of G a a maimal abelian ubpace of p and M the centralier of a in K with the Lie algebra m Let Φ g a be the root ytem and Φ + g a Φ g a a ytem of poitive root Let n α Φ + ga n α ISSN:

2 be the um of the root pace Then the Iwaawa decompoition G KAN correpond to the Iwaawa decompoition g k a n Define ρ 2 α Φ + ga dim n α α Let a + be the half line in a on which the poitive root take poitive value Put A + ep a + A Let σ ˆM By [6 p 27] there i an element γ K uch that i γ σ ee alo [6 p 23 Prop 2] Here i : K M i the retriction map induced by the embedding i : M K where K and M are the repreentation ring over Z of K and M repectively In [6 p 28] the author introduced the operator A d γ σ and A Yχ γ σ Thee operator correpond to pace X d and Y repectively Here χ i a finite-dimenional unitary repreentation of Γ and X d denote a compact dual pace of the ymmetric pace X Let E A be the family of pectral projection of a normal operator A Put and for C m χ γ σ Tr E AYχ γσ {} m d γ σ Tr E Ad γσ {} Definition [6 p 49 Def 7] Let σ ˆM Then γ K i called σ admiible if i γ σ and m d γ σ P σ for all 0 L σ Here P σ rep L σ denote the polynomial rep the lattice given by [6 Definition 3 p 47; ee alo p 40] In particular L σ T ɛ σ + Z where T and ɛ σ { 0 2} are given by the ame definition By [6 p 49 Lemma 8] there eit a σ admiible γ K for every σ ˆM III ZETA FUNCTIONS Since Γ G i co-compact and torion-free there are only two type of conjugacy clae: the cla of the identity e Γ and clae of hyperbolic element Let Γ h rep PΓ h denote the et of the Γ conjugacy clae of hyperbolic rep primitive hyperbolic element in Γ It i well known that every hyperbolic element g G i conjugated to ome element a g m g A + M ee eg [2] [4] Following [6 p 59] we put l g log a g For C e > ρ the Selberg eta function i defined by the infinite product ee [6 p 97] Z Sχ σ + γ0 PΓh k0 det σ m γ0 χ γ 0 S k Ad m γ0 a γ0 n e +ρlγ0 where σ and χ are finite-dimenional unitary repreentation of M and Γ repectively S k i the k th ymmetric power of an endomorphim n θn and θ i the Cartan involution of g For C e > the uelle eta function i defined by the infinite product ee [6 p 96] Z χ σ det σ m γ0 χ γ 0 e lγ0 γ 0 PΓ h n A known the uelle eta function can be epreed in term of Selberg eta function ee { eg [9] [] By [6 pp 99 00] there eit et I p τ λ τ ˆM } λ uch that Z χ σ n p0 Z Sχ + ρ λ τ σ p 4 τλ I p Let Λ rep Υ denote the et of all element λ rep τ that appear in 4 Note that [6 p 3 Theorem 35] give precie decription of the location and the order of the ingularitie of Z Sχ σ We have proved the following theorem Theorem A [2 p 528 Th 4] If γ i σ admiible then there eit entire function Z Z 2 of order at mot n uch that Z Sχ σ Z Z 2 where the ero of Z correpond to the ero of Z Sχ σ and the ero of Z 2 correpond to the pole of Z Sχ σ The order of the ero of Z rep Z 2 equal the order of the correponding ero rep pole of Z Sχ σ IV AUXILIAY ESULTS Lemma 2 If γ i σ admiible then where p n 2k n 2 P σ w p n 2k w n 2k k0 2T n 2 k! c n 2 k k 0 n 2 c n 2 n 2! 2T and the number c k are defined by the aymptotic epreion ISSN:

3 Tr e ta dγσ 2 t 0 k n 2 c k t k Proof By [6 pp 47 48] P σ 0 0 P σ w P σ w and P σ w w Q σ w where Q σ i an even polynomial Hence P σ i an odd polynomial Moreover P σ i a monic polynomial of degree n ee eg [5 pp 7 9] [22 pp ] Put n 2 P σ w p n 2k w n 2k p n k0 By [6 p 8] Q σ w 2T i! c i+ i 0 n 2 p n 2k q n 2k 2 n 2 k0 q n 2k 2 w n 2k 2 where q 2i In other word 2T n 2 k! c n 2 k k 0 n 2 Thi complete the proof Lemma 3 Let H be a half-plane of the form e < + ε ε > 0 minu the union of a et of congruent dik about the point T N ɛ τ σ + ρ λ λ Λ τ Υ Then there eit a contant C uch that for H Z χ σ Z χ σ C n Proof The identity 4 implie Z χ σ Z χ σ n p p0 τλ I p Z Sχ + ρ λ τ σ Z Sχ + ρ λ τ σ ecall [6 p 3 Theorem 35] Now it i enough to prove that if K i a half-plane of the form e < ρ + ε ε > 0 minu the union of a et of congruent dik about the point T N ɛ τ σ τ Υ then there eit a contant C S uch that Z Sχ τ σ Z Sχ τ σ C S n for K and all τ Υ The proof i independent of the choice of τ We implify our notation by omitting the latter By [6 p 8 Th 39] Z Sχ σ ha the repreentation 5 Z Sχ σ det A Yχ γ σ det A d γ σ + 2 dimχχy χx d ep dimχχy χxd n 2 m Hence ee [6 pp 20 22] c m 2m m! Z Sχ σ det Ad γ σ Z Sχ σ det A d γ σ + D + Z Sχ σ D Z Sχ σ ep π T Z Sχ σ e 0 2 dimχχy χx d m r r 2 2m r r 2 dimχχy χx d { tan πw P σ w T ɛσ 2 cot πw T ɛσ 0 tan πw K Pσw T ɛσ 2 0 cot πw T ɛσ 0 } 2 dimχχy χx d dw dw Conider the cae ɛ σ 2 The cae ɛ σ 0 i dicued imilarly The identity 6 implie Z Sχ σ Z Sχ σ Z Sχ σ Z Sχ σ + KP σ tan π T 6 Z Sχ σ i bounded on the complement of the union Since Z Sχ σ Z Sχ σ i bounded on every half-plane e > ρ+ε ε > 0 we conclude that Z Sχ σ i bounded on K Moreover tan π T of congruent dik about the point T k + 2 T k + ɛσ k Z Thi complete the proof Lemma 4 Let c d c < d If γ i σ admiible then there eit a equence {y j } y j + a j + uch that for c d Z χ + i y j σ Z χ + i y j σ O yj 2n Proof Conider the identity 5 It i enough to prove that there eit a equence {y j } y j + a j + uch that Z Sχ + i y j τ σ Z Sχ + i y j τ σ O yj 2n for a b and all τ Υ where a c ρ b d + ρ We conider the interval I given by i t t 0 < t t 0 + where t 0 > i fied ISSN:

4 It uffice to prove that there eit y t 0 t 0 + ] uch that Z Sχ + i y τ σ Z Sχ + i y τ σ O y 2n 7 for a b and all τ Υ Let S be the et of all ingularitie of all eta function Z Sχ τ σ τ Υ Let N t be the number of element in S on the interval i 0 < t Let N t be the number of ingularitie of Z Sχ σ on the ame interval By [6 Th 35] thee ingularitie are given in term of eigenvalue of A Yχ γ σ σ for ome σ admiible γ σ K Hence according to [8 p 89 Th 9] N t D t n t n log t for ome eplicitly known contant D However the O term doe not improve our reult For the ake of implicity we take N t O t n Conequently N t O t n It follow immediately that the number of ingularitie of Z Sχ σ on I i O t n 0 Similarly the number of element in S on I i O t n 0 ie it i at mot C t n 0 for ome contant C Denote by I 2 the interval i t t < t t Since I 2 I the number of element in S on I 2 i at mot C t n 0 Let u divide the interval I 2 into + C t n 0 equal interval By the Dirichlet principle one of them doe not contain any element from S Let i y be the midpoint of uch an interval We hall prove that y atifie 7 for a b and all τ Υ The proof doe not depend on the choice of τ Υ We implify our notation by omitting it ie we prove that Z Sχ + i y σ Z Sχ + i y σ O y 2n for a b By Theorem A Z and Z 2 are entire function of order at mot n Hence there are canonical product epreion for Z and Z 2 of the form ee eg [9 p 509] Z i ni e gi α i\{0} α ep α + 2 2α + + n 2 nα n i 2 where i i the et of ero of Z i n i i the order of the ero of Z i at 0 g i i a polynomial of degree at mot n Therefore Z Sχ σ Z Sχ σ n n 2 + g g 2 + i n α α We have i2 i y α 2 > 3 4 α i\{0} C t n C t n 0 + C y + 3 n C 2 4 for ome contant C 2 and all α i i 2 Now for a mall fied ε > 0 and the choice + i y a b we have Z Sχ σ Z Sχ σ n n 2 + g g k β A k β n β where β denote a ingularity of Z Sχ σ and Since get A {β β T N ɛ σ β > ρ + ε} A 2 {β 0 < β ρ + ε} A 3 {β β i t ρ + ε < t t 0 } A 4 {β β I } A 5 {β β i t t > t 0 + } A 6 {β β i t ρ + ε < t t 0 } A 7 {β β I } A 8 {β β i t t > t 0 + } β A β n β A converge and β y for β A we n β O β β A β n β Furthermore A 2 i a finite et Hence β A 2 n β O β β A 2 β n β O O ince β y ρ ε > C 3 y for ome contant C 3 and all β A 2 Similarly β y t 0 + > 4 and β > ρ + ε for β A 3 Hence β A 3 n β O β O 3 β A β n β O t 0 n O y 2n β A3 If β A 4 then β i y β > C2 7 4 > C 4y for ome contant C 4 Therefore β A 4 n β O β β A 4 β n β O and β > y β A 4 O t n 0 O y n O y 2n Similarly β t y > C 5 t for ome contant C 5 and β i t A 5 One ha ISSN:

5 n β β β A 5 O y + n β n O dn t β tn+ β A 5 t 0+ + O t 2 dt O t 0 + O t 0+ If β A 6 then β > y + ρ + ε > y and β > ρ + ε Hence n β β O y n β n β β A 6 O y n O y 2n β A 6 β A 6 Similarly β > y + t 0 > y and β > t 0 > y 7 4 > C 4y for β A 7 We have n β β O y n β n β β A 7 O y O β A 7 β A 7 If β A 8 then β y + t > t for β i t A 8 Therefore β A 8 n β O β O β A 8 β n β + t 0+ t n+ dn t O Finally n n 2 O y and g g 2 O We obtain Thi complete the proof Z Sχ σ Z Sχ σ O y 2n V PIME GEODESIC THEOEM Theorem 5 Let Y be a compact n dimenional n even locally ymmetric iemannian manifold with trictlegative ectional curvature Then π Γ n p0 +O p+ τλ I p n+ρ n+ log li pτλ n+ρ ] n+ pτλ a + where pτλ i a ingularity of the Selberg eta function Z S + ρ λ τ Proof We fi a χ ˆΓ A already mentioned there eit a σ admiible γ σ for every σ ˆM Fi σ ˆM and chooe ome σ admiible γ σ We implify our notation by omitting χ and σ in the equel For g Γ let n Γ g # Γ g / g where Γ g i the centralier of g in Γ and g i the group generated by g If γ Γ h then γ γ nγγ 0 for ome γ 0 PΓ h For γ Γ h we introduce Λ 0 γ Λ 0 γ nγγ 0 log N γ 0 By [6 pp ] We define where Z Z γ Γ h Λ 0 γ N γ e > 8 ψ j 0 ψ 0 ψ j t dt j 2 9 γ Γ hnγ Λ 0 γ Let k 2n be an integer and > c > By [8 p 3 Th B] and 8 c+i c i Λ 0 γ γ Γ h k! Z Z + + k d c+i c i Λ 0 γ γ Γ h Nγ γ Γ hnγ k! d N γ + + k Nγ Λ 0 γ N γ On the other hand by [8 p 8 Th A] Hence ψ k k! ψ k γ Γ hnγ c+i c i k k Λ 0 γ N γ k Z +k d Z + + k ISSN:

6 Aume that c i not a pole of the integrand of ψ k Z By Lemma 3 Z O n on the line e c Furthermore by Lemma 4 there eit a equence {y j } y j + a j + uch that Z t + i y j Z t + i y j O yj 2n [ ] for t c c Fi ome y j By contruction of {y j } we know that no pole of Z Z occur on the line Im y j Applying the Cauchy reidue theorem to the integrand of ψ k over the rectangle c y j given by vertice c i y j c + i y j c + i y j c i y j we obtain We have c+i y j c i y j c y j O O Z +k Z ++k e + c +i + c i c i c i yj c +i c i Similarly c +k c +i c +i y j c +i c +k c+i y j c +i y j c i c i y j c i y j c i y j d Z +k Z ++k c +i yj + c +i Z +k Z ++k c i c +i yj c +i d O c+i yj + c +i yj c +k Z +k Z ++k d k n+2 Hence by 0 and 5 O c +k Z +k Z ++k Z +k Z ++k Z +k Z ++k d d yj dv c i yj + c i yj O +k c 0 dv O +k c v k n+2 d O c+k y k+ 2n j d O c +k d O c+k y k+ 2n j where ie c+i y j c i y j n p+ p0 Z +k d Z + + k c +k τλ I p c y j c p τ λ k e c+k y k+ 2n j c p τ λ k Z S +ρ λτ +k Z S+ρ λτ ++k Letting j + c in we get c+i c i n p+ p0 n ψ k p+ p0 Z +k d Z + + k τλ I p τλ I p A pτλ k A pτλ k where A pτλ denote the et of pole of k Z S +ρ λτ +k Z S +ρ λτ ++k Take k 2n By 2 ψ 2n n p0 n p0 p+ p+ τλ I p τλ I p A pτλ 2n c p τ λ k c p τ λ k 2 c p τ λ 2n A pτλ c p τ λ 3 where for the ake of implicity we denote by A pτλ the et of pole of Z S +ρ λτ +2n Z S +ρ λτ ++2n and by c p τ λ the reidue at Z S +ρ λτ Z S +ρ λτ +2n ++2n correpond to ome τ λ I p for ome p {0 n } By [6 p 3 Theorem 35] the ingularitie of Z S + ρ λ τ are: at ± i ρ + λ of order m γ τ τ if 0 i an eigenvalue of A Y γ τ τ at ρ + λ of order 2m 0 γ τ τ if 0 i an eigenvalue of A Y γ τ τ at ρ + λ T N ɛ τ of order 2 n 2 voly volx d m d γ τ τ in thi cae > 0 i an eigenvalue of A d γ τ τ Here γ τ i ome τ admiible element in K Note that the ingularitie of Z S + ρ λ τ at ρ + λ T N ɛ τ are all le than ρ + λ Furthermore the ingularitie of Z S + ρ λ τ that correpond to A Y γ τ τ are contained in the union of the interval ISSN:

7 [ + λ λ] with the line ρ + λ + i An overlap between thee two kind of ingularitie may occur inide [ + λ ρ + λ ee [6 pp 4 5] The integer 0 2n are imple pole of +2n Thee integer may alo appear a imple pole ++2n Z S +ρ λτ of Z S +ρ λτ ie a ingularitie of Z S + ρ λ τ Denote by I pτλ the et of uch integer Put I pτλ to be the difference {0 2n} \I pτλ The et of the remaining ingularitie pτλ of Z S + ρ λ τ will be denoted by S pτλ eaoning a in [6 pp 88 89] we write Z S + ρ λ τ Z S + ρ λ τ opτλ + + i a pτλ i i where i a ingularity of Z S + ρ λ τ and o pτλ order of Now for pτλ S pτλ c pτλ p τ λ lim pτλ Z pτλ i S +ρ λτ +2n Z S+ρ λτ ++2n lim pτλ opτλ pτλ pτλ pτλ + + a pτλ i pτλ i pτλ pτλ +2n +2n ++2n o pτλ pτλ pτλ pτλ + pτλ + 2n Let j I pτλ We have c j p τ λ lim j Since and d d +2n Z S+ρ λτ ++2n + j 2 Z S +ρ λτ + j 2 Z S + ρ λ τ Z S + ρ λ τ + + 2n + o pτλ j + a pτλ i j + +2n ji i + l o pτλ j +2n l0 d d + l + o pτλ j a pτλ j + j 2 Z S +ρ λτ l0 +2n + j +2n l0 +2n Z S+ρ λτ ++2n i the l o pτλ j l0 + opτλ j l0 we obtain +2n log + l a pτλ +l c j p τ λ opτλ j l0 + l 2n l0 j +2n + o pτλ j a pτλ j + j d d opτλ j l0 j+l + opτλ j l0 j+l j+2n log 2n Finally let j I pτλ Now We denote: l0 c j p τ λ lim + j Z S +ρ λτ j Z S+ρ λτ Z S j + ρ λ τ Z S j + ρ λ τ j+l + apτλ j + l +2n +2n +l l0 j+2n +2n ++2n j+2n l0 j + l + I 2n {0 2n} { } n+ρ B pτλ j I 2n c j p τ λ O n+ B pτλ I 2n \B pτλ S pτλ S pτλ S pτλ Spτλ \S pτλ { } Cpτλ pτλ S pτλ pτλ 2n C 2 pτλ { pτλ S pτλ 2n < pτλ 2n + n+ρ n+ } n+ } 5 6 Cpτλ 3 { pτλ S pτλ 2n + n+ρ n+ < pτλ n+ρ { Cpτλ 4 pτλ S pτλ n + ρ } n + < pτλ Now we can write A pτλ c p τ λ B pτλ c p τ λ k Cpτλ k B pτλ c p τ λ + c p τ λ S pτλ Conider the um over Cpτλ in 7 c p τ λ 7 ISSN:

8 Since Cpτλ Spτλ S pτλ and 2n < + λ for Cpτλ it follow from 4 that C pτλ C pτλ c p τ λ o pτλ 2 n 2 voly volx d l0 +2n + + 2n k T 2n++ɛτ m d T k ɛ τ γ τ τ T k ɛτ +2n T k ɛ τ ρ + λ + l The fact that γ τ i τ admiible element yield m d γ τ τ P τ for all 0 L τ T ɛ τ + Z In particular m d T k ɛ τ γ τ τ P τ T k ɛ τ for k T 2n + ρ + λ + ɛ τ We obtain O O C pτλ c p τ λ k T 2n++ɛτ k T 2n++ɛτ Hence by Lemma 2 O C pτλ c p τ λ k T 2n++ɛτ P τ T k ɛ τ T k ɛ τ +ρ λ 2n 2n+ 2n++T ɛ τ 2n+ P τ T k ɛ τ T 2n+ k 2n+ k n+2 O 8 The um over B pτλ in 7 i a finite one Therefore by the definition of B pτλ n+ρ c p τ λ O n+ 9 B pτλ The um over Cpτλ 2 i a finite one a well Hence by 4 C 2 pτλ C 2 pτλ c p τ λ o pτλ +2n ++2n O n+ρ n+ Combining 3 and 7 20 we obtain ψ 2n n p+ p0 p+ n + p0 τλ I p B pτλ τλ I p Cpτλ 3 c p τ λ c p τ λ + 20 n p+ p0 p+ n + p0 n+ρ n+ τλ I p Cpτλ 4 τλ I p Suppoe < h 2 We introduce the operator 2n + 2n f i 2n i i0 S pτλ c p τ λ c p τ λ 2 f + 2n i h 22 If f i at leat 2n time differentiable function then +h + 2n f t 2n+h t 2n t 2+h t 2 f 2n t dt dt 2n 23 The mean value theorem applied to 23 yield + 2n f h2n f 2n 24 where [ + 2nh] Since ψ 0 i nondecreaing we obtain ψ 0 h 2n + 2n ψ 2n ψ 0 + 2nh 25 Now 2 22 and the fact that h 2 imply n h 2n + 2n ψ 2n p0 + n p+ p0 + n p0 + n p0 p+ p+ p+ τλ I p B pτλ τλ I p Cpτλ 3 τλ I p Cpτλ 4 τλ I p S pτλ h 2n n+ρ n+ h 2n + 2n c p τ λ h 2n + 2n c p τ λ h 2n + 2n c p τ λ h 2n + 2n c p τ λ 26 Conider the um over B pτλ on the right hand ide of 26 Let B pτλ 0 Suppoe that 0 I pτλ Then 5 24 and the fact: n log n n! log + n! n l l n n n! yield h 2n + 2n c 0 p τ λ o pτλ 0 log pτλ0 + o pτλ 0 a pτλ 0 27 where pτλ0 [ + 2nh] If 0 I pτλ then ISSN:

9 h 2n + 2n c 0 p τ λ Z S ρ λ τ Z S ρ λ τ 28 c p τ λ o pτλ +2n + + 2n by 6 Let B pτλ j Suppoe that j I pτλ Since k log n n k k! n k! n k 0 for 0 k < n k N we get h 2n + 2n c j p τ λ o pτλ j j where pτλ j [ + 2nh] If j I pτλ then pτλ j j and k n 29 h 2n + 2n c j p τ λ 0 30 We derive two etimate for h 2n + 2n c p τ λ Firtly by 22 h 2n + 2n c p τ λ h 2n o pτλ ++2n Since h 2 we obtain 2n i0 i 2n i + 2n i h +2n h 2n + 2n c p τ λ O h 2n 2n +2n 34 Now and the fact that h 2 imply B pτλ h 2n + 2n c p τ λ O log 3 Conider the um over Cpτλ 3 on the right hand ide of 26 Let Cpτλ 3 By 4 and 24 h 2n + 2n c p τ λ o pτλ opτλ pτλ pτλ opτλ n+ρ n+ pτλ where pτλ [ + 2nh] Hence h 2 and the fact that i a finite et yield C 3 pτλ Secondly by 23 h 2n + 2n c p τ λ +h t 2n+h t 2+h h 2n opτλ t dt dt 2n t 2n t 2 h +h t 2n+h t 2+h 2n o pτλ t dt dt 2n t 2n Hence by the mean value theorem and the fact that h 2 h 2n + 2n c p τ λ O 35 t 2 h 2n + 2n c n+ρ p τ λ O n+ 32 Let M > Now uing 34 and 35 we deduce C 3 pτλ Similarly the um over Cpτλ 4 on the right hand ide of 26 i a finite one We have h 2n + 2n c pτλ pτλ p τ λ opτλ pτλ pτλ pτλ for pτλ Cpτλ 4 where pτλ [ + 2nh] Hence reaoning a in [20 p 246] and [9 p 0] we obtain h 2n + 2n c p τ λ C 4 pτλ pτλ n+ρ n+ ] where pτλ i counted o pτλ pτλ pτλ pτλ h Finally we etimate the um over S pτλ S pτλ By 4 33 time in the lat um in 26 Let ISSN: O O S pτλ S pτλ < M h 2n + 2n c p τ λ h 2n + 2n c p τ λ + S pτλ < M M t dn pτλ t S pτλ >M h 2n + 2n c p τ λ h 2n +2n h 2n +2n + S pτλ >M O M n h 2n +2n M n M 2n 36 t 2n dn pτλ t where N pτλ t O t n denote the number of ingularitie of Z S + ρ λ τ on the interval ρ + λ + i 0 < t Combining and 36 we obtain

10 h 2n + 2n ψ 2n n p0 p+ h pτλ τλ I p pτλ n+ρ ] pτλ n+ ρ M n h 2n ρ+2n M n n+ρ n+ Subtituting h n+ρ n+ M taking into account 25 we get ψ 0 n p0 p+ ρ n+ pτλ τλ I p pτλ n+ρ ] pτλ n+ n+ρ n+ 37 into 37 and Analogouly ee eg [9 pp 0 02] one prove ψ 0 n p0 p+ pτλ τλ I p pτλ n+ρ ] pτλ n+ n+ρ n+ Combining 38 and 39 we conclude that ψ 0 n p0 p+ pτλ τλ I p pτλ n+ρ ] pτλ n+ n+ρ n Now uing 40 and following line of [9 p 02] we finally obtain π Γ n p0 p+ τλ I p li pτλ n+ρ ] n+ n+ρ n+ log a + Thi complete the proof VI CONCLUDING EMAKS pτλ Let u ummarie the apect in which Theorem 5 repreent an improvement of A already mentioned in the Introduction X i one of the following pace: H k k even k 2 HC m m HH l l HCa 2 Hence n k 2m 4l 6 and ρ 2 k m 2l + repectively Since HC H 2 and HH H 4 ee eg [5] we may aume m 2 and l 2 Now α n + q k 2m 4l repectively Obviouly α The ie of the error term in i O 2n We n+ρ compare thi bound to our bound O n+ log The factor log give to our bound ome advantage However let u have a look at the correponding power of The inequality n + ρ n + 2n alway hold true ince the correponding equivalent inequality n 0 i alway valid Here the equality ign occur only if X H 2 Furthermore the inequalitie n + ρ n ρ 2n are alway true Indeed the left-hand inequality i valid being equivalent to the inequalit + The equality occur only if X H k k 2 k even On the other ide the right-hand inequality hold alo true ince it reduce to n 2 0 Clearly the right-hand inequality become equality only if X H 2 Summariing reult derived above we end up with the concluion that the obtained bound O i of the form O n+ρ n+ log θ log where θ < 3 2 ρ if X HCm m 2 HH l l 2 HCa 2 and θ 3 2 ρ if X Hk k even k 2 Note that our reult coincide with the bet known reult for the compact iemann urface [20] and the real hyperbolic manifold with cup [] Alo note that taking k > 2n in the proof of Theorem 5 doe not yield a better reult EFEENCES [] M Avdipahić and Dž Gušić On the error term in the prime geodeic theorem Bull Korean Math Soc vol 49 pp [2] M Avdipahić and Dž Gušić Order of Selberg and uelle eta function for compact even-dimenional locally ymmetric pace J Math Anal Appl vol 43 pp [3] M Avdipahić Dž Gušić and D Kamber Order of eta function for compact even-dimenional ymmetric pace Bull Hellenic Math Soc vol 59 pp [4] M Avdipahić and L Smajlović On the prime number theorem for a compact iemann urface ocky Mountain J Math vol 39 pp [5] U Bröcker On Selberg eta function topological ero and determinant formula SFB 288 Berlin Tech ep no [6] U Bunke and M Olbrich Selberg eta and theta function A Differential Operator Approach Berlin: Akademie Verlag 995 [7] D L DeGeorge Length pectrum for compact locally ymmetric pace of trictlegative curvature Ann Sci Ec Norm Sup vol 0 pp [8] J J Duitermaat J A C Kolk and V S Varadarajan Spectra of compact locally ymmetric manifold of negative curvature Invent Math vol 52 pp [9] D Fried The eta function of uelle and Selberg I Ann Sci Ec Norm Sup vol 9 pp ISSN:

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