Advances in Pure Mathematics, 2016, 6, 903-914 http://www.scirp.org/journal/apm ISSN Online: 2160-0384 ISSN Print: 2160-0368
On the Prime Geodesic Theorem for Non-Compact Riemann Surfaces
Muharem Avdispahić, Dženan Gušić
Department of Mathematics, Faculty of Sciences and Mathematics, University of Sarajevo, Sarajevo, Bosnia and Herzegovina
How to cite this paper: Avdispahić, M. and Abstract Gušić, Dž. (2016) On the Prime Geodesic Theorem for Non-Compact Riemann Sur- We use B. Randol’s method to improve the error term in the prime geodesic theorem faces. Advances in Pure Mathematics, 6, 903- for a noncompact Riemann surface having at least one cusp. The case considered is a 914. general one, corresponding to a Fuchsian group of the first kind and a multiplier http://dx.doi.org/10.4236/apm.2016.612068 system with a weight on it. Received: October 23, 2016 Accepted: November 19, 2016 Keywords Published: November 22, 2016 Selberg Trace Formula, Selberg Zeta Function, Prime Geodesic Theorem Copyright © 2016 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International 1. Introduction License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ The Selberg trace formula, introduced by A. Selberg in 1956, describes the spectrum of Open Access the hyperbolic Laplacian in terms of geometric data involving the lengths of geodesics on a Riemann surface. Motivated by analogy between this trace formula and the explicit formulas of number theory relating the zeroes of the Riemann zeta function to prime numbers, Selberg [1] introduced a zeta function whose analytic properties are encoded in the Selberg trace formula. By focusing on the Selberg zeta function, H. Huber ([2], p. 386; [3], p. 464), proved an analogue of the prime number theorem for compact Rie- 3 1 − 4 mann surfaces with the error term Ox(log x) 2 that agrees with Selberg’s one. Using basically the same method as in [4], D. Hejhal ([5], p. 475), established also the prime geodesic theorem for non-compact Riemann surfaces with the remainder 3 1 − 4 Ox(log x) 2 . However, in the compact case there exist several different proofs (see, B. Randol [6], p. 245; P. Buser [7], p. 257, Th. 9.6.1; M. Avdispahić and L. Smajlović
DOI: 10.4236/apm.2016.612068 November 22, 2016 M. Avdispahić, Dž. Gušić
3 − 4 1 [8], Th. 3.1) that give the remainder Ox(log x) . Thanks to new integral repre- sentations of the logarithmic derivative of the Selberg zeta function (cf. [9], p. 185; [10], p. 128), M. Avdispahić and L. Smajlović ([11], p. 13) were in position to improve 3 1 3 − − 4 4 1 Ox(log x) 2 error term in a non-compact, finite volume case up to Ox(log x) . Whereas the authors in [8] and [11] approached the prime number theorem in various settings via explicit formulas for the Jorgenson-Lang fundamental class of functions, our main goal is to obtain this improvement for non-compact Riemann surfaces with cusps following a more direct method of B. Randol [6].
2. Preliminaries
Let X be a non-compact Riemann surface regarded as a quotient Γ \ of the upper half-plane by a finitely-generated Fuchsian group Γ⊆PSL( 2, ) of the first
kind, containing n1 ≥ 1 cusps. Let ℑ denote the fundamental region of Γ . We shall assume that the fundamental region ℑ of Γ has a finite non-Euclidean area ℑ . We put ab az+ b Γ= ∈SL(2,) : ∈Γ cd cz+ d
and denote by v the multiplier system of the weight m ∈ for Γ . Let ψ be an ir- reducible rr× unitary representation on Γ and WT( ) =ψ ( T) vT( ) , T ∈Γ. For an r dimensional vector space V over we consider an essentially self-adjoint oper- ator
2 ∂∂ ∂ ∆=m y 22 + −imy ∂∂xy ∂x
on the space m of all twice continuously differentiable functions fV: → , such
that f and ∆m ( f ) are square integrable on ℑ , and satisfy the equality
m (cz+ d ) ab f( Sz) = m W( S) f( z), for all z ∈ and = ∈Γ. cz+ d cd
The operator −∆m has the unique self-adjoint extension −∆m to the space m , a 2 dense subspace of L (Γ \ ) . Let Tj , jn= 1, , 1 be the set of parabolic transforma-
tions corresponding to n1 cusps of Γ . WT( j ) does not depend on the choice of a
representative of the parabolic class {Tj } and can be considered as a matrix from rr× . By m j we will denote the multiplicity of 1 as an eigen-value of the matrix n1 * WT( j ) , and nm1 = ∑ j will be the degree of singularity of W. We mention that oper- j=1 * ator −∆m has both the discrete and continuous spectrum in the case n1 ≥ 1, and only * the discrete spectrum in the case n1 = 0 . The discrete spectrum will be denoted as λ =λλ < < < λ →∞ { n}n>0 ( 0 01 n ). The continuous spectrum is expressed through
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zeros (or equivalently poles) of the hyperbolic scattering determinant (see, [12]).
3. Selberg Zeta Function
Let PΓh denotes the set of Γ -conjugacy classes of a primitive hyperbolic element P0 in Γ , and Γh denotes the set of Γ -conjugacy classes of a hyperbolic element P in Γ that satisfy property Tr(P) > 2 . Assume that m ≤ 1 . We define the Selberg zeta function associated to the pair ( Γ,W ) by ∞ − −−sk ZΓ,Wr( s) =∏∏ det (I WP( 00) NP( ) ). Pk0h∈=PΓ 0
ZsΓ,W ( ) is absolutely convergent for Re(s) > 1. Analytic considerations given in ([5], pp. 499-501) yield that the Selberg zeta function in this setting satisfies the func- tional equation
ZΓΓ,,WW( s) Ψ=( sZ) (1 − s) with the fudge factor 1 s η′ Ψ=(s) φη( s) ⋅∫ ( uu)d. (1) 2 1 η 2 Here, φ denotes the hyperbolic scattering determinant. It can be represented in the form
n* 1 1 πΓ(ss) Γ−∞ 2 an φ (s) = ∑ 2s , mm n=1 g Γ+ss Γ− n 22 where the coefficients an and gn depend on the group Γ (see, [5], p. 437). Here, * n1 denotes the degree of singularity of W (see Section 2). An explicit expression for the fudge factor η in the Equation (1) is given in ([5], p. 501, Equation (5.10)).
The logarithmic derivative of the Selberg zeta function ZsΓ,W ( ) is given by Zs′ () Λ (P) Γ,W = ∑ s Tr(WP( )) , ZsΓ,W () P∈Γ h NP( )
log NP( 0 ) where NP( ) denotes the norm of the class P and Λ=(P) −1 for a primi- 1− NP( ) = n tive element P0 such that PP0 for some n ∈ . We will omit the indices in ZΓ,W in the sequel.
4. Counting Functions ψ n ( xW, )
Lemma 1. For Re(s) > 1,
Zs′′( ) −s Zs( +1) =∑ Λ+1 (P)Tr ( WP( )) NP( ) , Zs( ) P∈Γ h Zs( +1)
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n where Λ=10(P) log NP( ) for a primitive element P0 such that PP= 0 for some n ∈ . Proof.
Zs′( ) −−−1 =Λ−1 s ∑ 1 (P) Tr( WP( )) ( 1 NP( ) ) NP( ) Zs( ) P∈Γh
−−ss−1 =Λ∑∑11(P) Tr ( WP( )) NP( ) +Λ( P) Tr ( WP( )) ( NP( ) −1) NP( ) PP∈Γhh∈Γ −1 =Λ −ss+Λ − −11−+( ) ∑∑11(P) Tr ( WP( )) NP( ) ( P) Tr( WP( )) ( 1 NP( ) ) NP( ) . PP∈Γhh∈Γ
We shall spend the rest of this section to derive a representation of ψ 2 ( xW, ) in the form (11) bellow. We choose not to write it in a separate statement because of the length of expressions involved. However, it will serve as a base for the proof of the prime geodesic theorem in Section 5. Let us recall the following theorem given in ([13], p. 51, Th. 40). −λn −s Theorem 1. If the Dirichlet’s series f( s) =Σ=Σ aen alnn is summable (lk, ) for s = β and c > 0 , c > β , then
ci+∞ − k 1 Γ(ks +Γ1) ( ) ωωks−= ω ∑ann( l) ∫ fs( ) d.s (2) ln <ω 2πici−∞ Γ( ks ++1 )
By Lemma 1,
Zs′′( ) Zs( +1) −s −=Λ∑ 1 (P)Tr ( WP( )) NP( ) . Zs( ) Zs( +1) P∈Γ h
We have,
ci+∞ 1 Zs′′( ) Zs( +1) −1 −−11s ∫ −s( s ++1d) ( s k) xs 2πici−∞ Zs( ) Zs( +1) −s 11ci+∞NP( ) Γ( k +Γ1) ( s) = Λ ∫ ∑ 1 (P)Tr ( WP( )) ds . ki!2π P∈Γ xΓ( k ++1 s) ci−∞h
−s NP( ) = Λ Therefore, substituting ω = 1, fs( ) ∑ P∈Γ 1 ( P)Tr ( WP( )), and hence h x NP( ) a= Λ ( P)Tr ( WP( )) , ln = in (2), we get n 1 x
ci+∞ 1 Zs′′( ) Zs( +1) −1 −−11s ∫ −s( s ++1d) ( s k) xs 2πici−∞ Zs( ) Zs( +1) −s 1 NP( ) =Λ−∑ 1 (P)Tr( WP( )) 1 kx! NP( ) ≤1 x −s 1 NP( ) =Λ−∑ 1 (P)Tr( WP( )) 1 . kx! NP( )≤ x
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Then,
ci+∞ 1 Zs′( ) − −−11 ∫ s1 ( s++1d) ( s k) xss 2πici−∞ Zs( ) k 1 NP( ) =Λ−∑ 1 (P)Tr( WP( )) 1 (3) kx! NP( )≤ x ci+∞ 1 Zs′( +1) −−11 +∫ s−1 ( s++1) ( s k) xssd. 2πici−∞ Zs( +1)
Now, put
ψ 01( xW, ) =∑ Λ ( P)Tr ( W( P)) NP( )≤ x and x ψψ= jj( xW,) ∫ −1 ( tW ,d) t 0 for j = 1, 2,. Using ([14], p. 12, Th. 1.3.5), it is easy to get that
1 j ψ j ( xW, ) =Λ−∑ 1 (P)Tr ( W( P))( x N( P)) . (4) j! NP( )≤ x
1 11 1 For 0 <<λ , let s=−=− ir i λ −, nK= 1, 2, , , be the zeros of Zs( ) n 4 nn22 n 4 1 in ,1 . Let ρk , kM= 0,1, , e denote all zeros of the hyperbolic scattering de- 2 1 terminant in ,1 . 2 1 1 Assume T > 2 , Tlr±≠n , l ∈ , where snn= + ir and snr= − ir for 2 2 1 1 rinn=−−λ , λn > . Following ([5], p. 468), we may also assume Tl±≠γ , l ∈ , 4 4 where, 1− ρ , 1− ρ are the zeros of the Selberg zeta function Zs( ) for each zero 1 3 ρ=++ ηγi , η ≥ 0 , γ > 0 , of the hyperbolic scattering determinant φ. Let A ≥ 2 0 2 1 m be a large constant such that A0 ∉ , A0 +∉ , A0 ±∉ . We put AA=0 +1 . 2 2 Without loss of generality we may assume that c =1 + ε , ε > 0 , x ≥ 2 . Let R( AT,) =−[ Ac ,,] ×−[ T T] . By the Cauchy residue theorem one has c+ iT 1 Zs′( ) −−11 ∫ s−1 ( s++1d) ( s k) xss 2πic− iT Zs( ) 11 11 −+εε ++ +−εε −− −+ iT iT iT iT −− 1 A00 iT 22 c+ iT 22 A iT −−A iT =∫∫∫∫∫∫∫∫ +++++++ (5) 2πi −+ −+ 11− 11−− A iT A00 iT −+εεiT ++ iT c iT +−εεiT −− iT A iT 22 22 −+A iT 1 Zs′( ) −−11 ++ −1 ++s ∫ ∑ Ressz= s (s1) ( sk) x 2πi−−A iT z∈ R( AT, ) Zs( )
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and
c+ iT 1 Zs′( +1) −−11 ∫ s−1 ( s++1d) ( sk) xss 2πic− iT Zs( +1)
11 11 −+εε ++ +−εε −− −+ iT iT iT iT −− 1 A00 iT 22 c+ iT 22 A iT −−A iT =∫∫∫∫∫∫∫∫ +++++++ (6) 2πi −+A iT −+ A iT 11 c− iT 11−−A iT 00−+εεiT ++ iT +−εεiT −− iT 22 22
−+A iT 1 Z′(s +1) −−11 ++ −1 ++s ∫ ∑ Ressz= s( s1.) ( sk) x 2πi −−A iT z∈ R( AT, ) Zs( +1)
Arguing as in [5] (p. 474) and [4] (pp. 105-108), we easily find that the sum of the x1+ε first eight integrals on the right hand side of (5) is O k . Similarly, taking into ac- εT Zs′( ) count that is bounded for Re(s) > 1, we obtain that the sum of the first eight Zs( ) x1+ε integrals on the right hand side of (6) is O k . Following [5] (p. 474) and [4] (p. 85, εT Prop. 5.7), we obtain that the ninth resp. the third integral on the right hand side of (5) resp. (6) are Ox( − A ) . Now, if we take k = 2 , (5) and (6) will give us
1 c+ iT Zs′( ) xs ∫ ds 2πic− iT Zs( ) ss( ++12)( s ) (7) xx1+ε Zs′( ) s =++Ox− A O Res ( ) 2 ∑ sz= εT z∈ R( AT, ) Zs( ) ss( ++12)( s )
and
1 c+ iT Zs′( +1) xs ∫ ds 2πic− iT Zs( +1) ss( ++ 12)( s ) (8) xx1+ε Zs′( +1) s =++Ox− A O Res . ( ) 2 ∑ sz= εT z∈ R( AT, ) Zs( +1) ss( ++ 12)( s )
Zs′( ) Bearing in mind location of the poles of given in ([5], p. 439, Th. 2.16; or [5], Zs( ) p. 498, Th. 5.3) and the fact that m ≤ 1 , we may assume without loss of generality that
mm51 mm 5 −−1 ,1 −+ ∈− , − ,3 −− ,3 −+ ∉−A , − . 2 2 42 2 2 2
Calculating residues and passing to the limit TA→ +∞, → +∞ in (7) and (8) we get
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+∞ −ρ 11ci Zs′( ) xs KK xxsnns Me x1 k =+ ++ ∫ dsx ∑∑∑ 2πici−∞ Zs( ) ss( ++12) ( s ) 6 nnk=1 ssnn( ++12) ( s n ) =1 ssnn( ++ 12) ( s n ) =0 ( 1 −ρρρk) ( 2 − k) ( 3 − k) m mm − −−1 x2 xx 22 ++AA + B 01mm m mmm 0 mmm − −+1 −+ 2 −− 1 − −+1 + 12 + 22 2 222 222 m −+1 11 2 − x 8 1822* *1−− *1 +B1 +Tr Ir −Φ x − nx1 − gnx 11 − nx 1 log x mmm 15 2 3 −+11 + 222 ** ssnn 3 nn11 xx +− +hx1 +log +∑∑+ (9) ++ ++ 2 22 rrnn>0 ssnn( 12) ( s n ) >0 ssnn( 12) ( s n )
1 1 −−ηγi −+ηγi x 2 x 2 + ∑ + ∑ γ >0 1 35 γ >0 135 −−ηγi −−ηγii −− ηγ −+ ηγ iii −+ ηγ −+ ηγ 2 22 222
mm −− −+ 2235 * * 22−− nn xx88**22 31 −21 −2 +A2 +B2 +nx11 − nx ++ f 1 x +xxlog m m m m mm 3 15 2 2 2 −−21 −− − −+ 21 −+ 2 2 2 2 22 and
m −− +∞ − − 1 1 ci Zs′( +1) xs KK xxsnn11s x2 = ++ ∫ dsA∑∑ 0 2πi−∞ Zs( +1) ss( ++ 12) ( s ) nn=11(s−+11) ss( ) = ( s −+ 11) ss ( ) m mm ci n nn n nn −−11 − −+ 2 22 m −+ 1 1 −ρ xx2 81 − Me k 2 ′ *1−− *1 +B0 −Tr Ixr −Φ +∑ −−gnx11 nx 1 log x mmm 32 k =0 (−−ρρ) ( +12) ( −+ρ ) −+11 + k kk 222
1 −−−ηγi ssnn−−11 2 3′ 11 xx x +− +hx1 +log +∑∑∑+ + 2 22 rr>0(s−+11) ss( ) >> 00( s −+ 11) ss ( ) γ 113 (10) nnn nn n nn −−−ηγiii −− ηγ −− ηγ 222
1 m −−+ηγi −−2 2 2 x x + ∑ + A1 γ >0 11 3 m mm −−+ηγii −+ ηγ −+ηγi −−−−−21 22 2 2 22
m −+ 2 ** 35 x 2 3nn 88−− ′ 11−−22 *22 * +B1 ++f1 x + xlog x + nx11 − nx . m mm 2 2 2 3 15 −+21 −+ 2 22 The implied constants on the right sides of (9) and (10) depend solely on Γ , m and W. With k = 2 in (3), j = 2 in (4), Equations (4), (3), (9) and (10) yield
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1 2 ψ 21( xW, ) =Λ−∑ (P) Tr ( W( P)) ( x N( P)) 2! NP( )≤ x 1 ci+∞Zs′′( ) xxss++22ci+∞Zs( +1) = ∫∫ddss− 2πici−∞ Zs( ) ss( ++12) ( s) ci−∞ Zs( + 1) ss( ++ 12) ( s ) ++ −ρ 1 KKxxssnn22 Me x 3 k =3 + ++ x ∑∑∑ 6 nnk=110ssnn( ++12) ( s n ) = ssnn( ++ 12) ( s n ) = ( 1 −ρρρk) ( 2 − k) ( 3 − k) m mm −+2 12−+ x 2 xx22 + A ++AB 0 mmm 10 mmm mmm − −+1 −+2 −− 1 − −+1 + 12 + 222 222 222 m 1+ 2 53 * x 8 1822** 3 n1 2 +B1 +Tr Ir −Φ x − nx1 − gnx 11 + − + h1 x mmm 15 2 3 2 2 −+11 + 222 m −+1 * K sn +11K sn + 2 n1 2 x xx +−xxlog ∑ −−∑ A0 2 n=1 (s−+11) ss( ) n=1 ( s −+ 11) ss ( ) m mm n nn n nn −−11 − −+ 2 22 m + 1 3 −+ρ 2 Me k 2 xx8 112 2 −B0 +Tr Ir −Φ x − xxlog −∑ mmm 3 22 k =0 (−ρρ) ( −+12) ( −+ ρ) −+11 + kk k 222 (11) ssnn++22 ′′*231 xx +gnx11 −− + h1 x +∑∑+ ++ ++ 22 rn >>00ssnn( 12) ( s n ) rn ssnn( 12) ( s n ) 55 −−ηγii−+ηγ xx22 ++∑∑ γγ>>00135 135 −−ηγηγηγiii −− −− −+ ηγηγηγ iii −+ −+ 222 222 3 −−ηγi xxssnn++11 x 2 −−−∑∑∑ rr>>00(s−+11) ss( ) ( s −+ 11) ss ( ) γ >0 113 nnn nn n nn −−−ηγiii −− ηγ −− ηγ 222
3 m −+ηγi − 22 xx −+∑ A2 γ >0 113 m mm −−+ηγiii −+ ηγ −+ ηγ −−21 −− − 222 222 m m − 2 * * 2 x 3 n1 n1 x +Bf21+++−log xA1 m mm 222 m mm −+21 −+ −−−−−21 2 22 2 22
m ** 2 nn x 3 ′ 11 −B11−+fx −log m mm 2 22 −+21 −+ 2 22 1 =x3 +++∑∑∑, 6 I II III
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where the first sum ranges over the finite set of poles s of Zs′′( ) xxssZs( +1) , Zs( ) ss( ++12)( s) Zs( + 1) ss( ++ 12)( s ) 5 with Re(s) >− , Im(s) = 0 , the second sum ranges over the set of poles s of the 4 same functions with Im(s) > 0 , and the third sum ranges over the finite set of their 5 poles s with Re(s) <− . 4
5. Prime Geodesic Theorem
In our setting, the prime geodesic counting function is defined by
π 00( xW,) = ∑ Tr( W( P)) , x≥ 1, NP( 0 )≤ x where the sum on the right is taken over all primitive hyperbolic classes P0h∈ΓP with respect to Γ (see, [5], p. 473, [11], p. 13). Theorem 2. For x ≥ 2 , the formula
K 3 −1 π =sn + 4 0 ( x, W) ∑ li( x) O x(log x) n=0
11 1 holds true, where, s =+−λ for 0 ≤<λ , and the implied constant de- nn24 n 4 pends solely on Γ , m and W. Proof. Following [6] (p. 245) and [15] (p. 11), for a positive number d > 0 , we de- + fine the second difference operator ∆2 by xd+ yd+ ∆=+ ′′ 2 f( x) ∫∫ f( t)dd ty . (12) xy
Here, d is a constant which will be fixed later. By the mean value theorem, we have
+−θθ22 ∆=2 xdθθ( −1) x (13) for some x ∈+[ xx,2 d] . It is easy to verify that + ∆2 fx( ) = fx( +22 d) − fx( ++ d) fx( ) . (14)
Reasoning as in [5] (p. 475), we may assume without loss of generality that ψ 0 ( xW, ) is non-decreasing. Hence, (12) implies
−+2 ψ0( xW, ) ≤∆ d22 ψψ( xW,) ≤0( x + 2, dW) . (15)
+++ Since (14) holds true, one can easily deduce that ∆=∆22CfxC( ) fx( ) , ∆=2C 0 , 13 ∆=+ −+22∆= −+22∆=44 = −+2∆= 2 x 0 , d2 xO(1) , d2 xlog x Ox log x Ox , d2 log xO( 1) ,
−+231 d∆=+2 x x Od( ) . Thus, (13) and finiteness of the sums contained in ∑ on 6 I the right hand side (11) yield
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1 K xsn 3 −+23∆ +=++ + 4 d2 x∑∑ x Od( ) O x . (16) 6 I1n= sn Similarly,
3 − −+2∆= 2 d2 ∑ Ox. (17) III −+2∆ In order to estimate d 2 ∑ II , we will first consider
sn +2 −+2 x d ∆2 ∑ . rn >0 ssnn( ++12)( s n )
By (14) it is evident that
sn +2 5 x −3 −+22∆=− 2 d 2 Od sn x . (18) ssnn( ++12)( s n ) On the other hand, the mean value theorem (13) gives us
sn +2 1 x −1 −+2∆=2 d 2 Osn x . (19) ssnn( ++12)( s n ) 1 Let Nt( ) be the number of roots of Zs( ) on the critical line + ix in the inter- 2 r ℑ val 0 <≤xt. It is known ([5], p. 477, Th. 3.8) that Nt( ) ~ t2 . Taking M > 1 4π
and following ([3], pp. 463-464; [6], p. 246), we use (19) resp. (18) in the sums over sn , 1 < (below) to get 2 n n n n
sn +2 −+2 x d ∆2 ∑ rn >0 ssnn( ++12)( s n )
ssnn++22 −+22xx−+ ≤∆∑∑dd22 + ∆ 1 ss++12 s ≤< ss++12 s <
sn +2 −+2 x +∆∑ d 2 sMn > ssnn( ++12)( s n )
11 5 −− − 2211−2 2 3 ≤+Cx11∑∑ snn Cx s + Cd 2 x ∑ s n 1 ≤< > <
Thus,
sn +2 11 5 −+2 x −21 − d∆ =++OxOMxOdxM22 2 . (20) 2 ∑ ++ rn >0 ssnn( 12)( s n )
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Similarly,
sn +2 11 5 −+2 x −21 − d∆ =++OxOMxOdxM22 2 . (21) 2 ∑ ++ rn >0 ssnn( 12)( s n ) = 2 Observe that ∑ γ ≤t1 Ot( ) (see, [5], p. 437, Prop. 2.13). Thus, application of −+2∆ d 2 to the third and the fourth sum in ∑ II gives us 11 5 22−−21 2 OxOMxOdxM++ . Let us write ∑∑∑= − , (22) II 1 2 where ∑1 denotes the sum of the first four sums in ∑ II and ∑ 2 denotes the sum of the last four sums in ∑ II . Now, Equations (11), (16), (17), (20), (21) and (22) give us
K sn 31 −+ x 2∆ψ =+++42 + d22( x, W) x O( d) ∑ O x O Mx n=1 sn (23) 5 +−2122 − −∆ −+ O d xM d 2 ∑. 2
1 3 Putting Mx= 4 , dx= 4 , the Equation (23) becomes
K xsn 3 −+22∆ψ =+ +4 −∆−+ d22( xW,.) x∑∑ O x d 2 (24) n=12sn 3 4 Since the left sides of Equations (20), (21) are Ox for such choice of M and d, 3 1 − d−+2∆= Ox4 d−+2∆= Ox4 we get 2 ∑1 . Now, it is obvious that 2 ∑1 . Finally, Equ- ation (24) gives us
K sn 3 −+ x 2∆ψ =++4 d22( xW,.) x∑ O x n=1 sn Returning to (15), we conclude that inequality K xsn 3 ψ ≤+ + 4 0 ( xW, ) x∑ O x n=1 sn holds true. Following ([15], p. 11), we analogously obtain that K xsn 3 ++4 ≤ψ x∑ O x0 ( xW,.) n=1 sn Hence, K xsn 3 ψ =++4 0 ( xW,.) x∑ O x (25) n=1 sn
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Arguing as in [5] (p. 475) and [4] (p. 113), one immediately sees that equality (25) proves the theorem.
Acknowledgements
We thank the Editor and the referee for their comments.
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[4] Hejhal, D. (1973) The Selberg Trace Formula for PSL( 2, ) , Vol. I. Lecture Notes in Ma- thematics, Volume 548. Springer-Verlag, Berlin-Heidelberg.
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