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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.

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 [

mann surfaces with the error term

Using basically the same method as in [

B. Randol [

[

sentations of the logarithmic derivative of the Selberg zeta function (cf. [

Whereas the authors in [

Let X be a non-compact Riemann surface regarded as a quotient

and denote by v the multiplier system of the weight

on the space

The operator

ator

Let

with the fudge factor

Here,

where the coefficients

The logarithmic derivative of the Selberg zeta function

where

tive element

Lemma 1. For

where

Proof.

We shall spend the rest of this section to derive a representation of

Let us recall the following theorem given in ( [

Theorem 1. If the Dirichlet’s series

By Lemma 1,

We have,

Therefore, substituting ω = 1,

Then,

Now, put

and

for

For

Assume

Without loss of generality we may assume that

and

Arguing as in [

first eight integrals on the right hand side of (5) is

Prop. 5.7), we obtain that the ninth resp. the third integral on the right hand side of (5) resp. (6) are

and

Bearing in mind location of the poles of

Calculating residues and passing to the limit

and

The implied constants on the right sides of (9) and (10) depend solely on

where the first sum ranges over the finite set of poles s of

with

In our setting, the prime geodesic counting function is defined by

where the sum on the right is taken over all primitive hyperbolic classes

Theorem 2. For

holds true, where,

Proof. Following [

Here, d is a constant which will be fixed later. By the mean value theorem, we have

for some

Reasoning as in [

Since (14) holds true, one can easily deduce that

Similarly,

In order to estimate

By (14) it is evident that

On the other hand, the mean value theorem (13) gives us

Let

Thus,

Similarly,

Observe that

Let us write

where

Putting

Since the left sides of Equations (20), (21) are

Returning to (15), we conclude that inequality

holds true. Following ( [

Hence,

Arguing as in [

We thank the Editor and the referee for their comments.

Avdispahić, M. and Gušić, Dž. (2016) On the Prime Geodesic Theorem for Non-Compact Riemann Surfaces. Advances in Pure Mathematics, 6, 903- 914. http://dx.doi.org/10.4236/apm.2016.612068