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On the Prime Geodesic Theorem for Non-Compact Riemann Surfaces

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On the Prime Geodesic Theorem for Non-Compact Riemann Surfaces 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 904 M. Avdispahić, Dž. Gušić 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) 905 M. Avdispahić, Dž. Gušić 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 906 M. Avdispahić, Dž. Gušić 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 .
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