On the Prime Geodesic Theorem for Non-Compact Riemann Surfaces

We use B. Randol’s method to improve the error term in the prime geodesic theorem for a noncompact Riemann surface having at least one cusp. The case considered is a general one, corresponding to a Fuchsian group of the first kind and a multiplier system with a weight on it.


Introduction
The Selberg trace formula, introduced by A. Selberg in 1956, describes the spectrum of 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 Riemann surfaces with the error term ( ) 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 ( ) . 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 [8], Th. 3.1) that give the remainder ( ) . Thanks to new integral representations 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 ( ) error term in a non-compact, finite volume case up to ( ) 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].

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 ( )  of the first kind, containing 1 1 n ≥ 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 ( ) and denote by v the multiplier system of the weight m ∈  for Γ .Let ψ be an ir- reducible r r × unitary representation on Γ and ( ) ( ) ( ) , T ∈ Γ .For an r dimensional vector space V over  we consider an essentially self-adjoint oper- ator on the space m  of all twice continuously differentiable functions , such that f and ( ) m f ∆ are square integrable on ℑ , and satisfy the equality be the set of parabolic transformations corresponding to 1 n cusps of Γ .

Selberg Zeta Function
Re 1 s > .Analytic considerations given in ( [5], pp.499-501) yield that the Selberg zeta function in this setting satisfies the functional equation with the fudge factor Here, φ denotes the hyperbolic scattering determinant.It can be represented in the form ( ) where the coefficients n a and n g depend on the group Γ (see, [5], p. 437).Here, n 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

Counting Functions
( ) We shall spend the rest of this section to derive a representation of ( ) (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.
 denote all zeros of the hyperbolic scattering determinant in Z s for each zero Without loss of generality we may assume that . By the Cauchy residue theorem one has and Arguing as in [5] (p.474) and [4] (pp. 105-108), we easily find that the sum of the first eight integrals on the right hand side of ( 5) is Re 1 s > , we obtain that the sum of the first eight integrals on the hand side of ( 6) is . Following [5] (p.474) and [4] (p. 85, Prop.5.7), we obtain that the ninth resp.the third integral on the right hand side of ( 5) resp.( 6) are ( ) A O x − .Now, if we take 2 k = , ( 5) and ( 6) will give us and Bearing in mind location of the poles of ( ) ( ) ], p. 439, Th. 2.16; or [5], p. 498, Th. 5.3) and the fact that 1 m ≤ , we may assume without loss of generality that Calculating residues and passing to the limit , T A → +∞ → +∞ in ( 7) and ( 8) we get and The implied constants on the right sides of ( 9) and ( 10) depend solely on Γ , m and W. With Equations ( 4), ( 3), ( 9) and ( 10) yield where the first sum ranges over the finite set of poles s of

Prime Geodesic Theorem
In our setting, the prime geodesic counting function is defined by where the sum on the right is taken over all primitive hyperbolic classes 0 h P P ∈ Γ with respect to Γ (see, [5], p. 473, [11], p. 13).
not depend on the choice of a representative of the parabolic class { } j T and can be considered as a matrix from r r ×  .By j m we will denote the multiplicity of 1 as an eigen-value of the matrix be the degree of singularity of W. We mention that oper- ator m −∆  has both the discrete and continuous spectrum in the case * 1 1 n ≥ , and only the discrete spectrum in the case * 1 0 n = .The discrete spectrum will be denoted as Following ([5], p. 468), we may also assume T l γ ± ≠ , l ∈  , where, 1 ρ − , 1 ρ − are the zeros of the Selberg zeta function ( ) the second sum ranges over the set of poles s of the same functions with ( ) Im 0 s > , and the third sum ranges over the finite set of their and the implied constant depends solely on Γ , m and W.Proof. Following[6] (p.245) and[15] (p.11), for a positive number 0 d > , we define the second difference operator 2