text:
\section{Introduction}
As it is well known, there is a lot of literature on Selberg zeta functions. Most of papers, starting with the Selberg's one \cite{73Bunke}, discuss the Riemann surface case (two-dimensional case), or at most the finite volume case.

Quite opposite, only a few of papers treat more general settings: Gangolli \cite{24Bunke} (higher dimensional spherical case), Scott \cite{70Bunke} (three-dimensional hyperbolic manifolds case), Wakayama \cite{82Bunke} (general, locally symmetric spaces case), Gon-Park \cite{GonPark} (hyperbolic manifolds with cusps case), Gangolli-Warner \cite{25Bunke} (spherical rank one case), Moscovici-Stanton \cite{50Bunke} (locally symmetric manifolds case), Deitmar \cite{16Bunke} (higher rank locally symmetric spaces case). There are also papers like: Williams \cite{92Bunke}, Schuster \cite{69Bunke}, Parnovskij \cite{54Bunke}, etc.

For the followers of the Selberg's original work, who first obtained a meromorphic continuation of the logarithmic derivative of the Selberg zeta function, it was only possible to show that a certain power of the zeta function is meromorphic. The problem was circumvented by Fried \cite{22Bunke}, who established the connection with hyperbolic dynamics, and expressed the Selberg zeta function in terms of the Ruelle zeta function \cite{64Bunke}, which was known to be a meromorphic one.

Following the approach via the trace formula, Bunke and Olbrich \cite{Bunke} investigated the Selberg zeta function associated with a locally homogeneous vector bundle over the unit sphere bundle of a compact locally symmetric space of rank one. They associated two eliptic operators, one over the locally symmetric space and one over its compact dual symmetric space, and identified the spectrum of both operators together with the set of singularities of the zeta function. The main part of the Chapter 3 in \cite{Bunke} is devoted to the analytical continuation of the Selberg zeta function to all of $\mathbb{C}$, the description of its singularities, its functional equation, and its representation via regularized determinants. The meromorphic continuation of the Selberg zeta function (and not of an integer power) is obtained by showing that all residues of the logarithmic derivative are integers (see, \cite[p. 112, Prop. 3.14]{Bunke}).

Regarding the Ruelle zeta function, the authors in \cite{Bunke} concluded the Subsection 3.1 with the observation due to Fried \cite{22Bunke} that the Ruelle zeta function is a product of the Selberg zeta functions (see also, \cite{21Bunke} and \cite{23Bunke}).

The key role in this research will be played by these zeta functions. The necessary notation and normalization will also follow from Bunke-Olbrich's work.

Let $Y$ $=$ $\Gamma$ $\backslash$ $G$ $/$ $K$ be a compact, $n$-dimensional ($n$ even), locally symmetric Riemannian manifold with strictly negative sectional curvature, where $G$ is a connected semi-simple Lie group of real rank one, $K$ is a maximal compact subgroup of $G$, and $\Gamma$ is a discrete co-compact torsion-free subgroup of $G$.

Through the rest of the paper, we shall assume that the Riemannian metric over $Y$ (induced from the Killing form) is normalized such that the sectional curvature of $Y$ varies between $-4$ and $-1$.

Denote by $X$ the universal covering of $Y$.

As it is known, $X$ is a Riemannian symmetric space of rank one, so it is either a real $H\mathbb{R}^{k}$ or a complex $H\mathbb{C}^{m}$, or a quaternionic hyperbolic space $H\mathbb{H}^{m}$, or the hyperbolic Cayley plane $H\mathbb{C}a^{2}$.

We have, $n$ $=$ $k$, $2m$, $4m$ and $16$, respectively.

Put $\rho$ $=$ $\frac{1}{2}\sum\limits_{\alpha\in\Phi^{+}\br{\mathfrak{g},\mathfrak{a}}}\dim\br{\mathfrak{n}_{\alpha}}\alpha$ to be the half-sum of the positive roots, where $\Phi^{+}\br{\mathfrak{g},\mathfrak{a}}$ is a system of positive roots, $\mathfrak{n}$ $=$ $\sum\limits_{\alpha\in\Phi^{+}\br{\mathfrak{g},\mathfrak{a}}}\mathfrak{n}_{\alpha}$ is the sum of the root spaces, $\mathfrak{g}$ is the Lie algebra of $G$ with the Cartan decomposition $\mathfrak{g}$ $=$ $\mathfrak{k}$ $\oplus$ $\mathfrak{p}$, and $\mathfrak{a}$ is a maximal abelian subspace of $\mathfrak{p}$.

It follows that $\rho$ $=$ $\frac{1}{2}\br{k-1}$, $m$, $2m+1$ and $11$ when $n$ $=$ $k$, $2m$, $4m$ and $16$, respectively.

Let $\mathfrak{g}$ $=$ $\mathfrak{k}$ $\oplus$ $\mathfrak{a}$ $\oplus$ $\mathfrak{n}$ and $G$ $=$ $KAN$ be the Iwasawa decompositions.

Denote by $M$ the centralizer of $\mathfrak{a}$ in $K$ with the Lie algebra $\mathfrak{m}$.

The fact that $\Gamma$ is a co-compact and torsion-free subgroup of $G$ yields that there are only two types of conjugacy classes: the class of the identity $1\in\Gamma$, and classes of hyperbolic elements.

We put $C\Gamma$ to be the set of all conjugacy classes $\SqBr{\gamma}$ in $\Gamma$.

It is known that each hyperbolic $g\in G$ is conjugate to some $am$ $\in$ $A^{+}M$, where $A^{+}$ $=$ $\exp\br{\mathfrak{a}^{+}}$ for $\mathfrak{a}^{+}$ the positive Weyl chamber in $\mathfrak{a}$ (see, \cite{GangolliLength,25Bunke}).

By \cite[pp. 96-99]{Bunke}, the Selberg zeta function $Z_{S,\chi}\br{s,\sigma}$ is defined for $s$ $\in$ $\mathbb{C}$, $\RE\br{s}$ $>$ $\rho$ by the infinite product
\begin{align*}
&Z_{S,\chi}\br{s,\sigma}\\
=&\prod\limits_{\substack{1\neq\SqBr{g}\in C\Gamma\\\textrm{primitive}}}\prod\limits_{k=0}^{\infty}\det\br{1-\br{\sigma\br{m}\otimes\chi\br{g}\otimes S^{k}\br{\Ad\br{ma}_{\bar{\mathfrak{n}}}}}e^{-\br{s+\rho}l\br{g}}},
\end{align*}
where $S^{k}$ is the $k$-th symmetric power of an endomorphism, $\sigma$ and $\chi$ are finite-dimensional unitary representations of $M$ and $\Gamma$, $l\br{g}$ is the length of $g$, and $\bar{\mathfrak{n}}$ $=$ $\theta\mathfrak{n}$ with $\theta$ the Cartan involution of $\mathfrak{g}$.

For $s$ $\in$ $\mathbb{C}$, $\RE\br{s}$ $>$ $2\rho$, the Ruelle zeta function $Z_{R,\chi}\br{s,\sigma}$ is defined by the infinite product
\begin{align*}
Z_{R,\chi}\br{s,\sigma}=\prod\limits_{\substack{1\neq\SqBr{g}\in C\Gamma\\\textrm{primitive}}}\det\br{1-\br{\sigma\br{m}\otimes\chi\br{g}}}^{\br{-1}^{n-1}}.
\end{align*}

In view of Fried's observation noted above, we have that there are sets\\ $I_{p}$ $=$ $\set{\br{\tau,\lambda}\,:\,\tau\in\hat{M},\,\lambda\in\mathbb{R}}$, such that
\begin{align*}
Z_{R,\chi}\br{s,\sigma}=\prod\limits_{p=0}^{n-1}\prod\limits_{\br{\tau,\lambda}\in I_{p}}Z_{S,\chi}\br{s+\rho-\lambda,\tau\otimes\sigma}^{\br{-1}^{p}},
\end{align*}
where $\hat{M}$ (\,$\hat{\Gamma}$\,) stands for the unitary dual of $M$ (\,$\Gamma$\,).

Generally, by a prime geodesic theorem we mean an asymptotic formula for the lengths of the closed geodesics on a Riemannian manifold.

In his most recent research \cite{MDPI1}, the author improved the error term in DeGeorge's more than forty-year-old prime geodesic theorem \cite{DeGe}, up to\\ $O\br{x^{2\rho-\frac{\rho}{n}}\br{\log x}^{-1}}$, and then further reduced it in the Gallagherian sense \cite{Gal1,Gal2}, i.e., outside a set of finite logarithmic measure to\\ $O\br{x^{2\rho-\rho\frac{4n+1}{4n^{2}+1}}\br{\log x}^{\frac{n-1}{4n^{2}+1}-1}\br{\log\log x}^{\frac{n-1}{4n^{2}+1}+\varepsilon}}$, $\varepsilon$ $>$ $0$.

It was proven that there is a set $E$ of finite logarithmic measure, such that
\begin{align*}
\pi_{\Gamma}\br{x}=&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}\sum\limits_{\substack{\alpha\in S_{p,\tau,\lambda}^{\mathbb{R}}\\2\rho-\rho\frac{4n+1}{4n^{2}+1}<\alpha\leq 2\rho}}\li\br{x^{\alpha}}+\\
&O\br{x^{2\rho-\rho\frac{4n+1}{4n^{2}+1}}\br{\log x}^{\frac{n-1}{4n^{2}+1}-1}\br{\log\log x}^{\frac{n-1}{4n^{2}+1}+\varepsilon}}
\end{align*}
as $x$ $\rightarrow$ $\infty$, $x$ $\notin$ $E$, where $S_{p,\tau,\lambda}^{\mathbb{R}}$ is the set of real singularities of\\ $Z_{S,\chi}\br{s+\rho-\lambda,\tau\otimes\sigma}$, and $\pi_{\Gamma}\br{x}$ is the function counting prime geodesics on $Y$ of length not larger than $\log x$.

The result was obtained with the use of Chebyshev type counting function $\psi_{2n}\br{x}$ of order $2n$.

Writing the $O$-term in the form
\begin{align*}
O\br{x^{2\rho-\rho\frac{2\cdot\br{2n}+1}{2n\cdot\br{2n}+1}}\br{\log x}^{\frac{n-1}{2n\cdot\br{2n}+1}-1}\br{\log\log x}^{\frac{n-1}{2n\cdot\br{2n}+1}+\varepsilon}},
\end{align*}
one easily comes to the conjecture that the $O$-term actually depends on $\psi_{2n}\br{x}$, or more generally, on the selection of $\psi_{j}\br{x}$, $j$ $\in$ $\mathbb{N}$.

Finding possible answers to these questions, as well as a possible answer to the question of whether or not the above $O$-term could be further reduced by varying $\psi_{j}\br{x}$, $j$ $\in$ $\mathbb{N}$, represent two of the three main motives to conduct this research at all.

The third motive comes from the knowledge that Hejhal \cite{Hejhal1,Hejhal2}, made use of the function $\psi_{1}\br{x}$ $=$ $\psi_{n-1}\br{x}$ to derive the classical prime geodesic theorems for compact hyperbolic surfaces and generic hyperbolic surfaces of finite volume ($n$ $=$ $2$), and the fact that Park \cite{Park} did the same thing using the function $\psi_{d-1}\br{x}$ $=$ $\psi_{n-1}\br{x}$ in the real hyperbolic manifolds with cusps case ($n$ $=$ $d$).

In view of the motives, we derive new explicit formulas for $\psi_{j}\br{x}$, $j$ $\geq$ $n$ (Theorem 1), and conditional asymptotic formula for $\psi_{n-1}\br{x}$ (Theorem 4). Hence, we derive the corresponding Gallagherian estimates for $\psi_{0}$ (Theorems 2 and 5). From there, we obtain improved prime geodesic theorems (Theorems 3 and 6) in a standard way, making use of lines in \cite[p. 102]{Park}. The results, combined, state that for $j$ $\geq$ $n-1$ and $\varepsilon$ $>$ $0$, there is a set $E$ of finite logarithmic measure, such that
\begin{equation}\label{trokuttokut}
\begin{aligned}
\pi_{\Gamma}\br{x}=&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}\sum\limits_{\substack{\alpha\in S_{p,\tau,\lambda}^{\mathbb{R}}\\2\rho-\rho\frac{2j+1}{2nj+1}<\alpha\leq 2\rho}}\li\br{x^{\alpha}}+\\
&O\br{x^{2\rho-\rho\frac{2j+1}{2nj+1}}\br{\log x}^{\frac{n-1}{2nj+1}-1}\br{\log\log x}^{\frac{n-1}{2nj+1}+\varepsilon}}
\end{aligned}
\end{equation}
as $x$ $\rightarrow$ $\infty$, $x$ $\in$ $\mathbb{R}$ $\backslash$ $E$.

This statement confirms the correctness of our conjecture, and, as it will be explained in detail below, gives a positive answer to the second of our questions, justifying also the use of the function $\psi_{n-1}\br{x}$.

Note that the result (\ref{trokuttokut}) is fully in line with the best known results obtained in the hyperbolic, finite volume, Riemann surface with cusps case, hyperbolic three manifolds case, and higher dimensional, real hyperbolic manifolds with cusps case.

Indeed, for a non-compact, hyperbolic Riemann surface with cusps, Theorem 3.1 in \cite[p. 28]{AvdAnalMath}, states that for $\varepsilon$ $>$ $0$, there is a set $E$ of finite logarithmic measure, such that
\begin{align*}
\pi_{\Gamma}\br{x}=\sum\limits_{\frac{7}{10}<\alpha\leq 1}\li\br{x^{\alpha}}+O\br{x^{\frac{7}{10}}\br{\log x}^{\frac{1}{5}-1}\br{\log\log x}^{\frac{1}{5}+\varepsilon}}
\end{align*}
as $x$ $\rightarrow$ $\infty$, $x$ $\in$ $\mathbb{R}$ $\backslash$ $E$, where $\alpha$ is a zero of the attached Selberg zeta function.

Obviously, the $O$-term follows form our (\ref{trokuttokut}) for the selection $j$ $=$ $n-1$, $n$ $=$ $2$ and $\rho$ $=$ $\frac{1}{2}\br{n-1}$ $=$ $\frac{1}{2}$.

By Theorem 1.2 in \cite[p. 691]{Erata}, for a hyperbolic three manifold, and $\varepsilon$ $>$ $0$, there is a set $E$ of finite logarithmic measure, so that
\begin{align*}
\pi_{\Gamma}\br{x}=\li\br{x^{2}}+\sum\limits_{n=1}^{M}\li\br{x^{s_{n}}}+O\br{x^{\frac{21}{13}}\br{\log x}^{-\frac{11}{13}}\br{\log\log x}^{\frac{2}{13}+\varepsilon}}
\end{align*}
as $x$ $\rightarrow$ $\infty$, $x$ $\in$ $\mathbb{R}$ $\backslash$ $E$, where $s_{1}$, $s_{2}$, ..., $s_{M}$ are the real zeros of the associated Selberg zeta function contained in the interval $\br{1,2}$.

Clearly, the error term coincides with our (\ref{trokuttokut}) for $j$ $=$ $n-1$, $n$ $=$ $3$ and $\rho$ $=$ $\frac{1}{2}\br{n-1}$ $=$ $1$.

Moreover, for a $d$-dimensional manifold with cusps, and $\varepsilon$ $>$ $0$, Theorem 2 in \cite[p. 3021]{AvdZenan} asserts that there exists a set $E$ of finite logarithmic measure, such that
\begin{equation}\label{delta1}
\begin{aligned}
\pi_{\Gamma}\br{x}=&\sum\limits_{d-1-\frac{1}{2}\br{d-1}\frac{2d+1}{2d^{2}+1}<s_{n}\br{k}\leq d-1}\li\br{x^{s_{n}\br{k}}}+\\
&O\br{x^{d-1-\frac{1}{2}\br{d-1}\frac{2d+1}{2d^{2}+1}}\br{\log x}^{\frac{d-1}{2d^{2}+1}-1}\br{\log\log x}^{\frac{d-1}{2d^{2}+1}+\varepsilon}}
\end{aligned}
\end{equation}
as $x$ $\rightarrow$ $\infty$, $x$ $\in$ $\mathbb{R}$ $\backslash$ $E$, where $\br{s_{n}\br{k}-k}\br{2d_{0}-k-s_{n}\br{k}}$ is a small eigenvalue in $\SqBr{0,\frac{3}{4}d_{0}^{2}}$ of $\Delta_{k}$ on $\pi_{\sigma_{k},\lambda_{n}\br{k}}$ with $s_{n}\br{k}$ $=$ $d_{0}$ $+$ $\II{}\lambda_{n}\br{k}$ or $s_{n}\br{k}$ $=$ $d_{0}$ $-$ $\II{}\lambda_{n}\br{k}$ in $\spbr{\frac{3}{2}d_{0},2d_{0}}$, $\Delta_{k}$ is the Laplacian acting on the space of $k$-forms over the manifold, $d_{0}$ $=$ $\frac{d-1}{2}$, and $\pi_{\sigma_{k},\lambda_{n}\br{k}}$ is the principal series representation.

The $O$-term in this result also follows from (\ref{trokuttokut}) for $j$ $=$ $n$, $n$ $=$ $d$ and $\rho$ $=$ $\frac{1}{2}\br{n-1}$ $=$ $\frac{1}{2}\br{d-1}$. Note that our result (\ref{trokuttokut}) improves the $O$-term in (\ref{delta1}) even more for the choice $j$ $=$ $n-1$, $n$ $=$ $d$ and $\rho$ $=$ $\frac{1}{2}\br{n-1}$ $=$ $\frac{1}{2}\br{d-1}$.

The classical (unconditional) prime geodesic theorems have been considered by many authors over the years, not always for the same underlying locally symmetric space. The authors like: DeGeorge \cite{DeGe}, Gangolli \cite{GangolliLength}, Gangolli-Warner \cite{25Bunke}, Park \cite{Park}, Hejhal \cite{Hejhal1,Hejhal2}, Randol \cite{Randol}, Deitmar-Pavey \cite{SL4}, Deitmar \cite{20MDPI1,23MDPI1}, Sarnak \cite{21MDPI1}, Iwaniec \cite{12MDPI1}, Luo-Sarnak \cite{26MDPI1}, Cai \cite{27MDPI1}, Soundararajan-Young \cite{28MDPI1}, Friedman-Jorgenson-Kramer \cite{18MDPI1}, and many others.

We conclude this section with the observation that taking $\psi_{j}\br{x}$, $j$ $\geq$ $n$, does not yield a better result than the one obtained in \cite[p. 9, Th. 2]{MDPI1} (see, Section 7). The mentioned result, however, coincides with the best known results in the Riemann surface case \cite{Randol}, hyperbolic three manifolds case \cite{Erata}, and real hyperbolic manifolds with cusps case \cite{AvdZenan}.

\section{Preliminary material}
We introduce only the necessary notation based on \cite{Bunke} and \cite{MDPI1} (see also, \cite{JMAA,Slovaka}).

For the sake of simplicity, we write $\gamma$ for an element of $C\Gamma$, and $\gamma_{0}$ for a primitive element.

In particular, if $\gamma$ and $\gamma_{0}$ appear in the same formula, it is understood that $\gamma_{0}$ is the primitive element underlying $\gamma$.

We fix $\sigma$ $=$ $1$, $\chi$ $=$ $1$, and reduce the notation by omitting to write them in the sequel.

Put $\psi_{0}\br{x}$ to be the sum $\sum\limits_{1\neq\SqBr{\gamma}\in C\Gamma,\,N\br{\gamma}\leq x}\Lambda\br{\gamma}$, where $N\br{\gamma}$ $=$ $e^{l\br{\gamma}}$, $\Lambda\br{\gamma}$ $=$ $\log N\br{\gamma_{0}}$, and $\psi_{j}\br{x}$ $=$ $\int\limits_{0}^{x}\psi_{j-1}\br{t}dt$, $j$ $\in$ $\mathbb{N}$.

Let $\mathcal{T}$ be the set of all $\tau$ $\in$ $\hat{M}$ occuring in the representation
\begin{equation}\label{4zvijezde}
\begin{aligned}
Z_{R}\br{s}=\prod\limits_{p=0}^{n-1}\prod\limits_{\br{\tau,\lambda}\in I_{p}}Z_{S}\br{s+\rho-\lambda,\tau}^{\br{-1}^{p}}.
\end{aligned}
\end{equation}

\section{Results}
\vspace{5mm}

\subsection{Explicit formulas for $\psi_{j}\br{x}$, $j$ $\geq$ $n$}
In this section we derive new explicit formulas for the counting functions $\psi_{j}\br{x}$, $j$ $\geq$ $n$.

The obtained explicit formulas will be applied in the proof of refined Gallagherian prime geodesic theorem attached to locally symmetric spaces described in this research.

The main result of the section is the following theorem.
\begin{theorem}\label{Th1}
Let $Y$ be as above. If $j$ $\geq$ $n$, then
\begin{equation}\label{cetiri}
\begin{aligned}
\psi_{j}\br{x}=&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}
\sum\limits_{\rho<\alpha\leq 2\rho}\frac{x^{\alpha+j}}{\prod\limits_{k=0}^{j}\br{\alpha+k}}+\\
&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\substack{\br{\tau,\lambda}\in I_{p}\\ \lambda=2\rho}}
\sum\limits_{\RE\br{\alpha}=\rho}\frac{x^{\alpha+j}}{\prod\limits_{k=0}^{j}\br{\alpha+k}},
\end{aligned}
\end{equation}
where $\alpha$ is a singularity of the Selberg zeta function $Z_{S}\br{s+\rho-\lambda,\tau}$.
\end{theorem}
\begin{proof}
Reasoning in the same way as in \cite[p. 98]{Park}, we obtain for $c$ $>$ $2\rho$ (see, Theorems A and B in \cite[pages 18 and 31]{Ingham}, and the relation (3.4) in \cite[p. 97]{Bunke} with $\sigma$ $=$ $1$, $\chi$ $=$ $1$, even $n$)
\begin{align*}
\psi_{j}\br{x}=\frac{1}{2\pi\II{}}\int\limits_{c-\II{}\infty}^{c+\II{}\infty}-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}\frac{x^{s+j}}{s\br{s+1}...\br{s+j}}ds.
\end{align*}

Suppose that there is no pole of the integrand of $\psi_{j}\br{x}$ over boundary\\ of the closed domain $R\br{T}$ given by $R\br{T}$ $=$ $\set{s\in\mathbb{C}\,:\,\abs{s}\leq T,\,\RE\br{s}\leq\rho}$ $\cup$ $\set{s\in\mathbb{C}\,:\,\rho\leq\RE\br{s}\leq c,\,-\tilde{T}\leq\IM\br{s}\leq\tilde{T}}$, where $T$ $\gg$ $0$ and $\tilde{T}$ $=$ $\sqrt{T^{2}-\rho^{2}}$.

For a fixed $0$ $<$ $\varepsilon$ $<$ $c$ $-$ $\rho$, the Cauchy residue theorem gives us
\begin{equation}\label{pet}
\begin{aligned}
&\frac{1}{2\pi\II{}}\int\limits_{c-\II{}\tilde{T}}^{c+\II{}\tilde{T}}-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}ds=
\end{aligned}
\end{equation}
\begin{align*}
&\frac{-1}{2\pi\II{}}\br{\int\limits_{c+\II{}\tilde{T}}^{\rho+\varepsilon+\II{}\tilde{T}}+\int\limits_{\rho+\varepsilon+\II{}\tilde{T}}^{\rho+\II{}\tilde{T}}+\int\limits_{C_{T}}+
\int\limits_{\rho-\II{}\tilde{T}}^{\rho+\varepsilon-\II{}\tilde{T}}+
\int\limits_{\rho+\varepsilon-\II{}\tilde{T}}^{c-\II{}\tilde{T}}}\br{-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}}ds\\
&+\sum\limits_{\alpha\in R\br{T}}\RES_{s=\alpha}\br{-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}},
\end{align*}
where $C_{T}$ denotes the circular part of the boundary of $R\br{T}$ with the anti-clockwise orientation.

First, we estimate the integrals $\int\limits_{\rho+\varepsilon+\II{}\tilde{T}}^{\rho+\II{}\tilde{T}}\cdot ds$ and $\int\limits_{\rho-\II{}\tilde{T}}^{\rho+\varepsilon-\II{}\tilde{T}}\cdot ds$ on the right hand side of (\ref{pet}).

Denote by $N_{S,p}^{\tau,\lambda}\br{y}$ the number of singularities $\rho_{S,p}^{\tau,\lambda}$ $=$ $-\rho$ $+$ $\lambda$ $+$ $\II{}\gamma_{S,p}^{\tau,\lambda}$ of $Z_{S}\br{s+\rho-\lambda,\tau}$ on the interval $-\rho$ $+$ $\lambda$ $+$ $\II{}x$, $0$ $<$ $x$ $\leq$ $y$. By Theorem 3.15 in \cite[p. 113]{Bunke} and Theorem 9.1 in \cite[p. 89]{Duistermat}, $N_{S,p}^{\tau,\lambda}\br{y}$ $=$ $C_{1}y^{n}$ $+$ $O\br{y^{n-1}\br{\log x}^{-1}}$ for some explicitly known constant $C_{1}$. Since the factor $\br{\log x}^{-1}$ does not improve our result, we take
\begin{equation}\label{sest}
\begin{aligned}
N_{S,p}^{\tau,\lambda}\br{y}=C_{1}y^{n}+O\br{y^{n-1}}.
\end{aligned}
\end{equation}

Furthermore, by Theorem 4.1, claim ($b$) in \cite[p. 314]{Slovaka}, for $\delta$ $>$ $0$, and $t$ $\gg$ $0$, such that $\II{}t$ is not a singularity of $Z_{S}\br{s,\tau}$ for all $\tau$ $\in$ $\mathcal{T}$,
\begin{equation}\label{sedam}
\begin{aligned}
\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}=O\br{t^{n-1+\delta}}+\sum\limits_{p=0}^{n-1}\br{-1}^{p}\sum\limits_{\substack{\br{\tau,\lambda}\in I_{p}\\ \lambda=2\rho}}
\sum\limits_{\abs{t-\gamma_{S,p}^{\tau,\lambda}}\leq 1}\frac{1}{s-\rho_{S,p}^{\tau,\lambda}}
\end{aligned}
\end{equation}
for $s$ $=$ $\sigma_{1}$ $+$ $\II{}t$, $\rho$ $\leq$ $\sigma_{1}$ $<$ $\frac{1}{4}t$ $-$ $\rho$, and
\begin{equation}\label{osam}
\begin{aligned}
\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}=O\br{\frac{1}{\eta}t^{n-1+\delta}}
\end{aligned}
\end{equation}
for $s$ $=$ $\sigma_{1}$ $+$ $\II{}t$, $\rho$ $+$ $\eta$ $\leq$ $\sigma_{1}$ $<$ $\frac{1}{4}t$ $-$ $\rho$, where $0$ $<$ $\eta$ $\leq$ $2\rho$.

Note that Park derived a variant of the equation (\ref{sedam}) and a variant of the equation (\ref{osam}) in the case of real hyperbolic manifolds with cusps (see, equations (3.9) and (3.10) in \cite[p. 99]{Park}).

Now, reasoning in the same way as Park did, that is, applying the relations (\ref{sedam}) and (\ref{sest}), we obtain (Cf. \cite[p. 99, relation (3.11)]{Park} and \cite[p. 368, relation (2)]{Koreja})
\begin{equation}\label{devet}
\begin{aligned}
\frac{1}{2\pi\II{}}\int\limits_{\rho+\II{}\tilde{T}}^{\rho+\varepsilon+\II{}\tilde{T}}-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}ds
=O\br{x^{j+\rho+\varepsilon}T^{-j-2+n+\delta}}.
\end{aligned}
\end{equation}

The integral $\int\limits_{\rho-\II{}\tilde{T}}^{\rho+\varepsilon-\II{}\tilde{T}}\cdot ds$ can be treated in the same way, and gives us
\begin{equation}\label{deset}
\begin{aligned}
\frac{1}{2\pi\II{}}\int\limits_{\rho+\varepsilon-\II{}\tilde{T}}^{\rho-\II{}\tilde{T}}-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}ds
=O\br{x^{j+\rho+\varepsilon}T^{-j-2+n+\delta}}.
\end{aligned}
\end{equation}

In order to estimate the integrals $\int\limits_{c+\II{}\tilde{T}}^{\rho+\varepsilon+\II{}\tilde{T}}\cdot ds$ and $\int\limits_{\rho+\varepsilon-\II{}\tilde{T}}^{c-\II{}\tilde{T}}\cdot ds$, which appear in (\ref{pet}), we apply the asymptotics (\ref{osam}).

It follows immediately that
\begin{equation}\label{jedanaest}
\begin{aligned}
\frac{1}{2\pi\II{}}\int\limits_{\rho+\varepsilon+\II{}\tilde{T}}^{c+\II{}\tilde{T}}-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}ds
=O\br{\varepsilon^{-1}x^{c+j}T^{-j-2+n+\delta}}
\end{aligned}
\end{equation}
and
\begin{equation}\label{dvanaest}
\begin{aligned}
\frac{1}{2\pi\II{}}\int\limits_{c-\II{}\tilde{T}}^{\rho+\varepsilon-\II{}\tilde{T}}-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}ds
=O\br{\varepsilon^{-1}x^{c+j}T^{-j-2+n+\delta}}.
\end{aligned}
\end{equation}

The remaining integral on the right hand side of (\ref{pet}) is $\int\limits_{C_{T}}\cdot ds$.

Recall the following facts.

By Corollary 4.2 in \cite[p. 530]{JMAA}, a meromorphic extension over $\mathbb{C}$ of the Ruelle zeta function $Z_{R}\br{s}$ can be expressed as
\begin{equation}\label{trinaest}
\begin{aligned}
Z_{R}\br{s}=\frac{Z_{R}^{1}\br{s}}{Z_{R}^{2}\br{s}},
\end{aligned}
\end{equation}
where $Z_{R}^{1}\br{s}$ and $Z_{R}^{2}\br{s}$ are entire functions of order at most $n$ over $\mathbb{C}$ (Cf. \cite[p. 91, Th. 1.1]{Park}).

Moreover, according to Fried's known result (see, \cite[p. 509, Prop. 7]{22Bunke}), there is a constant $C$ $>$ $0$ such that for arbitrarily large choices of $r$
\begin{equation}\label{cetrnaest}
\begin{aligned}
\int\limits_{r}\abs{\frac{Z^{'}\br{s}}{Z\br{s}}}\abs{ds}\leq Cr^{n}\log r,
\end{aligned}
\end{equation}
where $Z\br{s}$ is a ratio of two nonzero entire functions of order not larger than $n$.

Following Park's reasoning in the case of real hyperbolic manifolds with cusps \cite[p. 99]{Park}, i.e., using the assertions (\ref{trinaest}) and (\ref{cetrnaest}), we estimate

\begin{equation}\label{petnaest}
\begin{aligned}
\frac{1}{2\pi\II{}}\int\limits_{C_{T}}-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}ds
=&O\br{x^{\rho+j}T^{-j-1}\int\limits_{C_{T}}\abs{\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}}\abs{ds}}\\
=&O\br{x^{\rho+j}T^{-j-1}\int\limits_{\abs{s}=T}\abs{\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}}\abs{ds}}\\
=&O\br{x^{\rho+j}T^{-j-1+n}\log T}.
\end{aligned}
\end{equation}

Finally, we analyze the sum given by residues over $R\br{T}$ in (\ref{pet}).

Bearing in mind the fact that the Ruelle zeta function $Z_{R}\br{s}$ is a product of the corresponding Selberg zeta functions (see, relation (\ref{4zvijezde})), and the fact that the singularities are precisely described by Theorem 3.15 in \cite[p. 113]{Bunke}, we deduce (Cf. \cite[p. 110]{Park}, \cite[p. 369]{Koreja} and \cite[p. 744, Th. 4.6]{GonPark} in the real hyperbolic manifolds with cusps case),
\begin{equation}\label{sesnaest}
\begin{aligned}
&\sum\limits_{\alpha\in R\br{T}}\RES_{s=\alpha}\br{-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}}\\
=&\sum\limits_{\rho<\alpha\leq 2\rho}\RES_{s=\alpha}\br{-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}}+
\sum\limits_{\substack{\alpha\in R\br{T}\\ \RE\br{\alpha}=\rho}}\RES_{s=\alpha}\br{.}\\
&+\sum\limits_{\substack{\alpha\in R\br{T}\\ \RE\br{\alpha}<\rho}}\RES_{s=\alpha}\br{.}\\
=&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}\sum\limits_{\rho<\alpha\leq 2\rho}
\RES_{s=\alpha}\br{\frac{Z_{S}^{'}\br{s+\rho-\lambda,\tau}}{Z_{S}\br{s+\rho-\lambda,\tau}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}}
\end{aligned}
\end{equation}
\begin{align*}
&+\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\substack{\br{\tau,\lambda}\in I_{p}\\ \lambda=2\rho}}\sum\limits_{\substack{\alpha\in R\br{T}\\ \RE\br{\alpha}=\rho}}
\RES_{s=\alpha}\br{\frac{Z_{S}^{'}\br{s+\rho-\lambda,\tau}}{Z_{S}\br{s+\rho-\lambda,\tau}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}}+
\end{align*}
\begin{align*}
&\sum\limits_{\substack{\alpha\in R\br{T}\\ \RE\br{\alpha}<\rho}}\RES_{s=\alpha}\br{-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}}.
\end{align*}

Consider the first sum on the right hand side of (\ref{sesnaest}).

For arbitrarily selected and then fixed $p$ $\in$ $\set{0,1,...,n-1}$ and $\br{\tau,\lambda}$ $\in$ $I_{p}$, let's take a closer look at $\RES_{s=\alpha}\br{\frac{Z_{S}^{'}\br{s+\rho-\lambda,\tau}}{Z_{S}\br{s+\rho-\lambda,\tau}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}}$, where $\rho$ $<$ $\alpha$ $\leq$ $2\rho$.

Clearly, all such $\alpha$'s are the poles of $\frac{Z_{S}^{'}\br{s+\rho-\lambda,\tau}}{Z_{S}\br{s+\rho-\lambda,\tau}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}$, i.e., they are the poles of $\frac{Z_{S}^{'}\br{s+\rho-\lambda,\tau}}{Z_{S}\br{s+\rho-\lambda,\tau}}$. This means that they are simple poles of $\frac{Z_{S}^{'}\br{s+\rho-\lambda,\tau}}{Z_{S}\br{s+\rho-\lambda,\tau}}$, that is, they are the singularities of $Z_{S}\br{s+\rho-\lambda,\tau}$.

Let $\alpha$ be a singularity of $Z_{S}\br{s+\rho-\lambda,\tau}$, with $\rho$ $<$ $\alpha$ $\leq$ $2\rho$.

Following Hejhal's reasoning in the compact Riemann surface case \cite[pp. 88-89]{Hejhal1}, we write
\begin{align*}
\frac{Z_{S}^{'}\br{s+\rho-\lambda,\tau}}{Z_{S}\br{s+\rho-\lambda,\tau}}=\frac{o_{\alpha}}{s-\alpha}\br{1+\sum\limits_{i=1}^{+\infty}a_{i}\br{\alpha}\br{s-\alpha}^{i}},
\end{align*}
where $o_{\alpha}$ is the order of $\alpha$, and $a_{i}\br{\alpha}$'s are the corresponding coefficients.

We obtain,
\begin{align*}
&\RES_{s=\alpha}\br{\frac{Z_{S}^{'}\br{s+\rho-\lambda,\tau}}{Z_{S}\br{s+\rho-\lambda,\tau}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}}\\
=&\lim\limits_{s\rightarrow\alpha}\br{s-\alpha}\frac{Z_{S}^{'}\br{s+\rho-\lambda,\tau}}{Z_{S}\br{s+\rho-\lambda,\tau}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}=
o_{\alpha}\frac{x^{\alpha+j}}{\prod\limits_{k=0}^{j}\br{\alpha+k}}.
\end{align*}

We carry out the same argumentation in the case of the second sum on the right-hand side of (\ref{sesnaest}).

Hence,
\begin{equation}\label{sedamnaest}
\begin{aligned}
&\sum\limits_{\alpha\in R\br{T}}\RES_{s=\alpha}\br{-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}}\\
=&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}\sum\limits_{\rho<\alpha\leq 2\rho}\frac{x^{\alpha+j}}{\prod\limits_{k=0}^{j}\br{\alpha+k}}+\\
&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\substack{\br{\tau,\lambda}\in I_{p}\\ \lambda=2\rho}}\sum\limits_{\substack{\alpha\in R\br{T}\\ \RE\br{\alpha}=\rho}}\frac{x^{\alpha+j}}{\prod\limits_{k=0}^{j}\br{\alpha+k}}+\\
&\sum\limits_{\substack{\alpha\in R\br{T}\\ \RE\br{\alpha}<\rho}}\RES_{s=\alpha}\br{-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}},
\end{aligned}
\end{equation}
where, as usual, we omit to write $o_{\alpha}$'s for the sake of clarity.

In the same way Theorem 1.1 in \cite[p. 91]{Park} asserts that the Ruelle zeta function in the real hyperbolic manifolds with cusps case is a meromorphic function of order equal to the dimension of the underlying locally symmetric space, now, the equality (\ref{trinaest}) asserts that the Ruelle zeta function $Z_{R}\br{s}$ is a meromorphic function of order not larger than $n$.

To deal with the poles $\alpha$ $\in$ $R\br{T}$, $\RE\br{\alpha}$ $<$ $\rho$ of $-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}$ in (\ref{sedamnaest}), we proceed in exactly the same way as in \cite[p. 100]{Park}. Thus, we define $Z\br{s}$ to be a meromorphic function of order not larger than $n$ obtained by removing the singularities of $Z_{R}\br{s}$ in the right half plane $\RE\br{s}$ $\geq$ $\rho$ using the canonical product. It follows that
\begin{align*}
&\sum\limits_{\substack{\alpha\in R\br{T}\\ \RE\br{\alpha}<\rho}}\RES_{s=\alpha}\br{-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}}\\
=&\sum\limits_{\substack{\alpha\in R\br{T}\\ \RE\br{\alpha}<\rho}}\RES_{s=\alpha}\br{-\frac{Z^{'}\br{s}}{Z\br{s}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}}=
\frac{1}{2\pi\II{}}\int\limits_{C^{'}}-\frac{Z^{'}\br{s}}{Z\br{s}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}ds,
\end{align*}
where $C^{'}$ $=$ $C_{T}$ $\cup$ $\set{s\in\mathbb{C}\,:\,\RE\br{s}=\rho,\,-\tilde{T}\leq\IM\br{s}\leq\tilde{T}}$ is taken with the anti-clockwise orientation.

Suppose that
\begin{align*}
C^{''}=&\set{s\in\mathbb{C}\,:\,\abs{s}=T,\,\RE\br{s}\geq\rho}\cup\\
&\set{s\in\mathbb{C}\,:\,\RE\br{s}=\rho,\,-\tilde{T}\leq\IM\br{s}\leq\tilde{T}}\\
=&C_{T}^{c}\cup \set{s\in\mathbb{C}\,:\,\RE\br{s}=\rho,\,-\tilde{T}\leq\IM\br{s}\leq\tilde{T}}
\end{align*}
is also taken with the anti-clockwise orientation.

We may write
\begin{align*}
&\sum\limits_{\substack{\alpha\in R\br{T}\\ \RE\br{\alpha}<\rho}}\RES_{s=\alpha}\br{-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}}\\
=&\frac{1}{2\pi\II{}}\int\limits_{C^{'}}-\frac{Z^{'}\br{s}}{Z\br{s}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}ds+\frac{x^{\rho+j}}{2\pi}\cdot0\\
=&\frac{1}{2\pi\II{}}\int\limits_{C^{'}}-\frac{Z^{'}\br{s}}{Z\br{s}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}ds+
\frac{x^{\rho+j}}{2\pi}\int\limits_{C^{''}}-\frac{Z^{'}\br{s}}{Z\br{s}}\frac{ds}{\prod\limits_{k=0}^{j}\br{s+k}}.
\end{align*}

Applying the same reasoning as in the derivation of (\ref{petnaest}), we deduce
\begin{align*}
&\abs{\frac{1}{2\pi\II{}}\int\limits_{C^{'}}-\frac{Z^{'}\br{s}}{Z\br{s}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}ds+
\frac{x^{\rho+j}}{2\pi}\int\limits_{C^{''}}-\frac{Z^{'}\br{s}}{Z\br{s}}\frac{ds}{\prod\limits_{k=0}^{j}\br{s+k}}}\\
\leq&\frac{x^{\rho+j}}{2\pi}\int\limits_{C^{'}}\abs{\frac{Z^{'}\br{s}}{Z\br{s}}}\frac{\abs{ds}}{\prod\limits_{k=0}^{j}\abs{s+k}}+
\frac{x^{\rho+j}}{2\pi}\int\limits_{C^{''}}\abs{\frac{Z^{'}\br{s}}{Z\br{s}}}\frac{\abs{ds}}{\prod\limits_{k=0}^{j}\abs{s+k}}=
\end{align*}
\begin{align*}
&\frac{x^{\rho+j}}{2\pi}\int\limits_{C_{T}}\abs{\frac{Z^{'}\br{s}}{Z\br{s}}}\frac{\abs{ds}}{\prod\limits_{k=0}^{j}\abs{s+k}}+
\frac{x^{\rho+j}}{2\pi}\int\limits_{\rho-\II{}\tilde{T}}^{\rho+\II{}\tilde{T}}\abs{\frac{Z^{'}\br{s}}{Z\br{s}}}\frac{\abs{ds}}{\prod\limits_{k=0}^{j}\abs{s+k}}+\\
&\frac{x^{\rho+j}}{2\pi}\int\limits_{C_{T}^{c}}\abs{\frac{Z^{'}\br{s}}{Z\br{s}}}\frac{\abs{ds}}{\prod\limits_{k=0}^{j}\abs{s+k}}+
\frac{x^{\rho+j}}{2\pi}\int\limits_{\rho+\II{}\tilde{T}}^{\rho-\II{}\tilde{T}}\abs{\frac{Z^{'}\br{s}}{Z\br{s}}}\frac{\abs{ds}}{\prod\limits_{k=0}^{j}\abs{s+k}}\\
=&\frac{x^{\rho+j}}{2\pi}\int\limits_{\abs{s}=T}\abs{\frac{Z^{'}\br{s}}{Z\br{s}}}\frac{\abs{ds}}{\prod\limits_{k=0}^{j}\abs{s+k}}
\leq C_{1}x^{\rho+j}T^{-j-1}\int\limits_{\abs{s}=T}\abs{\frac{Z^{'}\br{s}}{Z\br{s}}}\abs{ds}\\
\leq&C_{2}x^{\rho+j}T^{-j-1+n}\log T.
\end{align*}

Therefore,
\begin{equation}\label{osamnaest}
\begin{aligned}
\sum\limits_{\substack{\alpha\in R\br{T}\\ \RE\br{\alpha}<\rho}}\RES_{s=\alpha}\br{-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}}
=O\br{x^{\rho+j}T^{-j-1+n}\log T}.
\end{aligned}
\end{equation}

It is clear that
\begin{align*}
\frac{1}{2\pi\II{}}\int\limits_{c+\II{}\tilde{T}}^{c+\II{}\infty}-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}ds
=O\br{x^{c+j}T^{-j}}
\end{align*}
and
\begin{align*}
\frac{1}{2\pi\II{}}\int\limits_{c-\II{}\infty}^{c-\II{}\tilde{T}}-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}ds
=O\br{x^{c+j}T^{-j}}.
\end{align*}

Hence,
\begin{equation}\label{devetnaest}
\begin{aligned}
\frac{1}{2\pi\II{}}\int\limits_{c-\II{}\tilde{T}}^{c+\II{}\tilde{T}}-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}\frac{x^{s+j}}{\prod\limits_{k=0}^{j}\br{s+k}}ds
=\psi_{j}\br{x}+O\br{x^{c+j}T^{-j}}.
\end{aligned}
\end{equation}

Taking into account (\ref{devet})-(\ref{dvanaest}), (\ref{petnaest}), (\ref{sedamnaest})-(\ref{devetnaest}), and passing to the limit $T$ $\rightarrow$ $\infty$ in (\ref{pet}), we end up with
\begin{align*}
\psi_{j}\br{x}=&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}\sum\limits_{\rho<\alpha\leq 2\rho}\frac{x^{\alpha+j}}{\prod\limits_{k=0}^{j}\br{\alpha+k}}+\\
&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\substack{\br{\tau,\lambda}\in I_{p}\\ \lambda=2\rho}}\sum\limits_{\RE\br{\alpha}=\rho}\frac{x^{\alpha+j}}{\prod\limits_{k=0}^{j}\br{\alpha+k}}.
\end{align*}

This completes the proof.
\end{proof}

We remark that in the case $j$ $=$ $n-1$, the $O$-terms in the relations (\ref{devet})-(\ref{dvanaest}) and (\ref{devetnaest}) all vanish during the process $T$ $\rightarrow$ $\infty$.

Notice, however, that the $O$-term in (\ref{petnaest}) is $O\br{x^{\rho+n-1}\log T}$ for $j$ $=$ $n-1$. Hence, it does not tend to zero as $T$ approaches to infinity.

This means that the method we made use of in the proof of Theorem \ref{Th1} cannot be applied in the same form to obtain the formula analogous to the formula (\ref{cetiri}) for the function $\psi_{n-1}\br{x}$.

\subsection{Explicit formulas for $\psi_{0}\br{x}$ I}
We derive the asymptotics of $\psi_{0}\br{x}$ from the asymptotics of $\psi_{j}\br{x}$, $j$ $\geq$ $n$ (see, Theorem \ref{Th1}).
\begin{theorem}\label{Th2}
Let $Y$ be as above. If $j$ $\geq$ $n$, then, for $\varepsilon$ $>$ $0$, there exists a set $E$ of finite logarithmic measure, such that
\begin{align*}
\psi_{0}\br{x}=&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}
\sum\limits_{2\rho-\rho\frac{2j+1}{2nj+1}<\alpha\leq2\rho}\frac{x^{\alpha}}{\alpha}+\\
&O\br{x^{2\rho-\rho\frac{2j+1}{2nj+1}}\br{\log x}^{\frac{n-1}{2nj+1}}\br{\log\log x}^{\frac{n-1}{2nj+1}+\varepsilon}},
\end{align*}
as $x$ $\rightarrow$ $\infty$, $x$ $\in$ $\mathbb{R}$ $\backslash$ $E$, where $\alpha$ is a singularity of the Selberg zeta function $Z_{S}\br{s+\rho-\lambda,\tau}$.
\end{theorem}
\begin{proof}
We write the second sum on the right-hand side of (\ref{cetiri}) as follows
\begin{equation}\label{dvadeset}
\begin{aligned}
&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\substack{\br{\tau,\lambda}\in I_{p}\\ \lambda=2\rho}}
\sum\limits_{\substack{\RE\br{\alpha}=\rho\\ \abs{\IM\br{\alpha}}\leq Y}}\frac{x^{\alpha+j}}{\prod\limits_{k=0}^{j}\br{\alpha+k}}+\\
&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\substack{\br{\tau,\lambda}\in I_{p}\\ \lambda=2\rho}}
\sum\limits_{\substack{\RE\br{\alpha}=\rho\\ Y<\abs{\IM\br{\alpha}}\leq W}}\frac{x^{\alpha+j}}{\prod\limits_{k=0}^{j}\br{\alpha+k}}+\\
&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\substack{\br{\tau,\lambda}\in I_{p}\\ \lambda=2\rho}}
\sum\limits_{\substack{\RE\br{\alpha}=\rho\\ \abs{\IM\br{\alpha}}>W}}\frac{x^{\alpha+j}}{\prod\limits_{k=0}^{j}\br{\alpha+k}}.
\end{aligned}
\end{equation}

Following lines of \cite[p. 10]{MDPI1}, we focus our attention to the second sum in (\ref{dvadeset}).

Let $E_{p,\tau}^{i}$ denote the set
\begin{align*}
\set{x\in\sppbr{e^{i},e^{i+1}}\,:\,\abs{\sum\limits_{\substack{\RE\br{\alpha}=\rho\\ Y<\abs{\IM\br{\alpha}}\leq W}}\frac{x^{\alpha+j}}{\prod\limits_{k=0}^{j}\br{\alpha+k}}}>x^{\gamma}\br{\log x}^{\beta}\br{\log\log x}^{\beta+\varepsilon}}.
\end{align*}

It is understood that the element $\alpha$ which appears in the definition of the set $E_{p,\tau}^{i}$ is a singularity of the Selberg zeta function $Z_{S}\br{s+\rho-\lambda,\tau}$ corresponding to $p$ $\in$ $\set{0,1,...,n-1}$ and $\br{\tau,\lambda}$ $\in$ $I_{p}$, $\lambda$ $=$ $2\rho$.

Estimating the logarithmic measure $\mu^{\times}E_{p,\tau}^{i}$ $=$ $\int\limits_{E_{p,\tau}^{i}}\frac{dx}{x}$ of the set $E_{p,\tau}^{i}$ in the same way as in \cite[pp. 10-11]{MDPI1}, we obtain that $\mu^{\times}E_{p,\tau}^{i}$ is
\begin{equation}\label{dvadesetjedan}
\begin{aligned}
&O\br{\frac{e^{2\br{\rho+j-\gamma}i}}{i^{2\beta}\br{\log i}^{2\beta+2\varepsilon}}
\int\limits_{e^{i}}^{e^{i+1}}
\abs{\sum\limits_{\substack{\RE\br{\alpha}=\rho\\ Y<\abs{\IM\br{\alpha}}\leq W}}
\frac{x^{\II\IM\br{\alpha}}}{\prod\limits_{k=0}^{j}\br{\alpha+k}}}^{2}\frac{dx}{x}}\\
=&O\br{\frac{e^{2\br{\rho+j-\gamma}i}}{i^{2\beta}\br{\log i}^{2\beta+2\varepsilon}}
\int\limits_{-\frac{1}{4\pi}}^{\frac{1}{4\pi}}
\abs{\sum\limits_{\substack{\RE\br{\alpha}=\rho\\ Y<\abs{\IM\br{\alpha}}\leq W}}
\frac{e^{\II\IM\br{\alpha}\br{i+\frac{1}{2}}}}{\prod\limits_{k=0}^{j}\br{\alpha+k}}e^{2\pi\II\IM\br{\alpha}u}}^{2}du}.
\end{aligned}
\end{equation}

The Gallagher lemma \cite[p. 78, Lemma 1]{9MDPI1} implies that the integral on the right-hand side of (\ref{dvadesetjedan}) is dominated by
\begin{equation}\label{dvadesetdva}
\begin{aligned}
\int\limits_{-\infty}^{+\infty}\br{\sum\limits_{\substack{t\leq\abs{\IM\br{\alpha}}\leq t+1\\Y<\abs{\IM\br{\alpha}}\leq W}}\frac{1}{\prod\limits_{k=0}^{j}\abs{\alpha+k}}}^{2}dt.
\end{aligned}
\end{equation}

However, the integral in (\ref{dvadesetdva}) is $O\br{\frac{1}{Y^{2j+3-2n}}}$ according to Weyl's asymptotic law $N_{S,p}^{\tau,\lambda}\br{t}$ $=$ $C_{1}t^{n}$ $+$ $O\br{t^{n-1}}$.

Hence, $\mu^{\times}E_{p,\tau}^{i}$ $=$ $O\br{\frac{e^{2\br{\rho+j-\gamma}i}}{Y^{2j+3-2n}i^{2\beta}\br{\log i}^{2\beta+2\varepsilon}}}$.

For $Y$ $\sim$ $e^{\frac{\br{2\rho+2j-2\gamma}i}{2j+3-2n}}i^{\frac{1-2\beta}{2j+3-2n}}\br{\log i}^{\frac{1-2\beta}{2j+3-2n}}$, it follows that\\ $\mu^{\times}E_{p,\tau}^{i}$ $=$ $O\br{\frac{1}{i\br{\log i}^{1+2\varepsilon}}}$.

Consequently, $\mu^{\times}\bigcup\limits_{p}\bigcup\limits_{\substack{\br{\tau,\lambda}\in I_{p}\\ \lambda=2\rho}}E_{p,\tau}^{i}$ $=$ $O\br{\frac{1}{i\br{\log i}^{1+2\varepsilon}}}$.

Let $E$ denote the set $\bigcup\limits_{i}\bigcup\limits_{p}\bigcup\limits_{\substack{\br{\tau,\lambda}\in I_{p}\\ \lambda=2\rho}}E_{p,\tau}^{i}$.

The logarithmic measure of the set $E$ is finite according to the Maclaurin-Cauchy test (Cf. \cite[p. 80]{9MDPI1})
\begin{align*}
\mu^{\times}E\ll\sum\limits_{i}\frac{1}{i\br{\log i}^{1+2\varepsilon}}<\infty.
\end{align*}

Now, by the definition of $E$, it holds for any $x$ $\in$ $\mathbb{R}$ $\backslash$ $E$ that
\begin{align*}
\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\substack{\br{\tau,\lambda}\in I_{p}\\ \lambda=2\rho}}\sum\limits_{\substack{\RE\br{\alpha}=\rho\\ Y<\abs{\IM\br{\alpha}}\leq W}}\frac{x^{\alpha+j}}{\prod\limits_{k=0}^{j}\br{\alpha+k}}=O\br{x^{\gamma}\br{\log x}^{\beta}\br{\log\log x}^{\beta+\varepsilon}}.
\end{align*}

Therefore, for $x$ $\in$ $\mathbb{R}$ $\backslash$ $E$
\begin{equation}\label{dvadesettri}
\begin{aligned}
&d^{-j}\Delta_{j}^{+}\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\substack{\br{\tau,\lambda}\in I_{p}\\ \lambda=2\rho}}\sum\limits_{\substack{\RE\br{\alpha}=\rho\\ Y<\abs{\IM\br{\alpha}}\leq W}}\frac{x^{\alpha+j}}{\prod\limits_{k=0}^{j}\br{\alpha+k}}\\
=&O\br{\frac{x^{\gamma}\br{\log x}^{\beta}\br{\log\log x}^{\beta+\varepsilon}}{d^{j}}},
\end{aligned}
\end{equation}
where the operator $\Delta_{j}^{+}$ is introduced classically by
\begin{align*}
\Delta_{j}^{+} f\br{x}=\int\limits_{x}^{x+d}\int\limits_{t_{j}}^{t_{j}+d}...\int\limits_{t_{2}}^{t_{2}+d}f^{\br{j}}\br{t_{1}}dt_{1}...dt_{j}
\end{align*}
for at least $j$ times differentiable function $f$, and a constant $d$ which will be fixed later.

However, we require the bound $d$ $=$ $O\br{x}$.

It is easily seen that the application of the operator $d^{-j}\Delta_{j}^{+}$ to the first and the second sum in (\ref{dvadeset}) yields the bounds (Cf. \cite[p. 8, equation (11)]{MDPI1})
\begin{equation}\label{dvadesetcetiri}
\begin{aligned}
O\br{x^{\rho}Y^{n-1}}
\end{aligned}
\end{equation}
and
\begin{equation}\label{dvadesetpet}
\begin{aligned}
O\br{\frac{x^{\rho+j}}{d^{j}W^{j+1-n}}},
\end{aligned}
\end{equation}
respectively.

Since $\Delta x^{r}$ $=$ $d^{2n}r\br{r-1}...\br{r-\br{2n-1}}\tilde{x}^{r-2n}$ for some $\tilde{x}$ $\in$ $\SqBr{x,x+jd}$, the first sum in (\ref{cetiri}) gives us
\begin{equation}\label{dvadesetsest}
\begin{aligned}
&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}\sum\limits_{\rho<\alpha\leq2\rho}
d^{-j}\Delta_{j}^{+}\frac{x^{\alpha+j}}{\prod\limits_{k=0}^{j}\br{\alpha+k}}\\
=&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}\sum\limits_{\rho<\alpha\leq2\rho}
\frac{x^{\alpha}}{\alpha}+O\br{x^{2\rho-1}d}.
\end{aligned}
\end{equation}

Now, combining the relations (\ref{cetiri}), (\ref{dvadeset}), (\ref{dvadesettri})-(\ref{dvadesetsest}) together with the fact $\psi_{0}\br{x}$ $\leq$ $d^{-j}\Delta_{j}^{+}\psi_{j}\br{x}$, we end up with
\begin{equation}\label{dvadesetsedam}
\begin{aligned}
\psi_{0}\br{x}\leq &\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}\sum\limits_{\rho<\alpha\leq2\rho}\frac{x^{\alpha}}{\alpha}+O\br{x^{2\rho-1}d}+
O\br{x^{\rho}Y^{n-1}}\\
&+O\br{\frac{x^{\gamma}\br{\log x}^{\beta}\br{\log\log x}^{\beta+\varepsilon}}{d^{j}}}+O\br{\frac{x^{\rho+j}}{d^{j}W^{j+1-n}}}
\end{aligned}
\end{equation}
for $x$ $\in$ $\mathbb{R}$ $\backslash$ $E$.

Following lines of \cite[p. 12]{MDPI1}, we compare the error terms $O\br{x^{2\rho-1}d}$, $O\br{x^{\rho}Y^{n-1}}$ and $O\br{\frac{x^{\gamma}\br{\log x}^{\beta}\br{\log\log x}^{\beta+\varepsilon}}{d^{j}}}$ on the right hand side of (\ref{dvadesetsedam}).

Hence, we require that $d$ $=$ $x^{1-\rho}Y^{n-1}$, while the equality $O\br{x^{2\rho-1}d}$ $=$\\ $O\br{\frac{x^{\gamma}\br{\log x}^{\beta}\br{\log\log x}^{\beta+\varepsilon}}{d^{j}}}$ is comfortably satisfied for $x^{2\rho-1}d$ $=$ $\frac{x^{\gamma}\br{\log x}^{\beta}\br{\log\log x}^{\beta}}{d^{j}}$, that is, for $d$ $=$ $x^{\frac{\gamma-2\rho+1}{j+1}}\br{\log x}^{\frac{\beta}{j+1}}\br{\log\log x}^{\frac{\beta}{j+1}}$.

Comparing the corresponding exponents of $x$ and $\log x$ (with respect to our selection $Y$ $\sim$ $x^{\frac{2\rho+2j-2\gamma}{2j+3-2n}}\br{\log x}^{\frac{1-2\beta}{2j+3-2n}}\br{\log\log x}^{\frac{1-2\beta}{2j+3-2n}}$), we obtain the following system of linear equations:
\begin{align*}
\frac{\gamma-2\rho+1}{j+1}=&1-\rho+\br{n-1}\frac{2\rho+2j-2\gamma}{2j+3-2n},\\
\frac{\beta}{j+1}=&\br{n-1}\frac{1-2\beta}{2j+3-2n}.
\end{align*}

The solution is given by:
\begin{align*}
\gamma=&\frac{2\br{n-\rho}j^{2}+\br{4n-3}\rho j+\rho+j}{2nj+1},\\
\beta=&\frac{\br{n-1}\br{j+1}}{2nj+1}.
\end{align*}

Hence,
\begin{align*}
d=&x^{1-\rho\frac{2j+1}{2nj+1}}\br{\log x}^{\frac{n-1}{2nj+1}}\br{\log\log x}^{\frac{n-1}{2nj+1}},\\
Y\sim&x^{\frac{2\rho j}{2nj+1}}\br{\log x}^{\frac{1}{2nj+1}}\br{\log\log x}^{\frac{1}{2nj+1}}.
\end{align*}

Substituting the obtained $d$ and $Y$ into (\ref{dvadesetsedam}), we arrive at
\begin{align*}
\psi_{0}\br{x}\leq&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}\sum\limits_{\rho<\alpha\leq 2\rho}\frac{x^{\alpha}}{\alpha}+\\
&O\br{x^{2\rho-\rho\frac{2j+1}{2nj+1}}\br{\log x}^{\frac{n-1}{2nj+1}}\br{\log\log x}^{\frac{n-1}{2nj+1}+\varepsilon}}\\
=&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}\sum\limits_{2\rho-\rho\frac{2j+1}{2nj+1}<\alpha\leq2\rho}\frac{x^{\alpha}}{\alpha}+\\
&O\br{x^{2\rho-\rho\frac{2j+1}{2nj+1}}\br{\log x}^{\frac{n-1}{2nj+1}}\br{\log\log x}^{\frac{n-1}{2nj+1}+\varepsilon}}
\end{align*}
as $x$ $\rightarrow$ $\infty$, $x$ $\in$ $\mathbb{R}$ $\backslash$ $E$.

In a similar way,
\begin{align*}
\psi_{0}\br{x}\geq&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}\sum\limits_{2\rho-\rho\frac{2j+1}{2nj+1}<\alpha\leq2\rho}\frac{x^{\alpha}}{\alpha}+\\
&O\br{x^{2\rho-\rho\frac{2j+1}{2nj+1}}\br{\log x}^{\frac{n-1}{2nj+1}}\br{\log\log x}^{\frac{n-1}{2nj+1}+\varepsilon}}
\end{align*}
as $x$ $\rightarrow$ $\infty$, $x$ $\in$ $\mathbb{R}$ $\backslash$ $E$.

This completes the proof.
\end{proof}

\subsection{Gallagherian prime geodesic theorem I}
An immediate consequence of Theorem \ref{Th2} is the following theorem (Cf. \cite[p. 102]{Park}).
\begin{theorem}\label{Th3}\,(Gallagherian Prime Geodesic Theorem)\,
Let $Y$ be as above. If $j$ $\geq$ $n$, then, for $\varepsilon$ $>$ $0$, there is a set $E$ of finite logarithmic measure, so that
\begin{equation}\label{dvadesetosam}
\begin{aligned}
\pi_{\Gamma}\br{x}=&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}
\sum\limits_{2\rho-\rho\frac{2j+1}{2nj+1}<\alpha\leq2\rho}\li\br{x^{\alpha}}+\\
&O\br{x^{2\rho-\rho\frac{2j+1}{2nj+1}}\br{\log x}^{\frac{n-1}{2nj+1}-1}\br{\log\log x}^{\frac{n-1}{2nj+1}+\varepsilon}},
\end{aligned}
\end{equation}
as $x$ $\rightarrow$ $\infty$, $x$ $\in$ $\mathbb{R}$ $\backslash$ $E$, where $\alpha$ is a singularity of the Selberg zeta function $Z_{S}\br{s+\rho-\lambda,\tau}$.
\end{theorem}
\vspace{2mm}

Let's have a closer look at the exponent $2\rho$ $-$ $\rho\frac{2j+1}{2nj+1}$ of $x$ in the $O$-term of Theorem \ref{Th3}.

It is understood that we assume that $n$ $\geq$ $2$.

Hence, the inequality $2\rho$ $-$ $\rho\frac{2j+1}{2nj+1}$ $<$ $2\rho$ $-$ $\rho\frac{2\br{j+1}+1}{2n\br{j+1}+1}$ is valid being equivalent to the inequality $2n$ $+$ $1$ $>$ $3$ which is obviously correct.

Thus, the sequence $\set{2\rho-\rho\frac{2j+1}{2nj+1}}_{j}$ is strictly increasing, with the limit $2\rho$ $-$ $\frac{\rho}{n}$. This means that the optimal size of the error term in (\ref{dvadesetosam}) is\\ $O\br{x^{2\rho-\rho\frac{2n+1}{2n^{2}+1}}\br{\log x}^{\frac{n-1}{2n^{2}+1}-1}\br{\log\log x}^{\frac{n-1}{2n^{2}+1}+\varepsilon}}$, and is achieved for $j$ $=$ $n$. Note that the limit $2\rho$ $-$ $\frac{\rho}{n}$ is nothing else but the optimal exponent of $x$ in DeGeorge's prime geodesic theorem \cite[p. 2, relation (1)]{MDPI1}. It is also clear that the $O$-term in (\ref{dvadesetosam}) improves the $O$-term in Theorem \ref{Th3} \cite[p. 9]{MDPI1} for $n$ $\leq$ $j$ $<$ $2n$.

Since the sequence $\set{2\rho-\rho\frac{2j+1}{2nj+1}}_{j}$ is strictly increasing, and its terms for $j$ $\geq$ $n$ are obtained by the application of the explicit formulas for the counting functions $\psi_{j}\br{x}$, $j$ $\geq$ $n$ (Theorem \ref{Th1}), the question that naturally arises is the possibility of further reduction of the error term in (\ref{dvadesetosam}) via the function $\psi_{n-1}\br{x}$. As we already noted in the remark at the end of Subsection 3.1, the method applied in the proof of Theorem \ref{Th1} cannot be copied to give us the desired form (\ref{cetiri}) for $\psi_{n-1}\br{x}$.

However, recall that Koyama \cite[p. 79, relation (5)]{9MDPI1}, applied one of Hejhal's explicit formulas for $\psi_{1}\br{x}$ (see, Theorem 6.16 \cite[p. 110]{Hejhal1} and proof of Theorem 3.4 \cite[p. 474]{Hejhal2}), to obtain the Gallagherian prime geodesic theorem for compact Riemann surfaces and generic hyperbolic surfaces of finite volume ($n$ $=$ $2$, $\rho$ $=$ $\frac{1}{2}\br{n-1}$ $=$ $\frac{1}{2}$). For the same reason, we consider $\psi_{n-1}\br{x}$ in the next section.

\subsection{Explicit formulas for $\psi_{n-1}\br{x}$}
Following Hejhal's approach in the compact Riemann surface case \cite[pp. 104-110]{Hejhal1}, we consider the counting function $\psi_{n-1}\br{x}$.

We obtain the following conditional result.
\begin{theorem}\label{Th4}
Let $Y$ be as above, and $\varepsilon_{1}$ $>$ $0$, $\delta$ $>$ $0$. Suppose $T$ $\gg$ $0$ and $A$ $\gg$ $2\rho$ are chosen so that $\II{}T$ is not a singularity of $Z_{S}\br{s,\tau}$, $\tau$ $\in$ $\mathcal{T}$, and $-A$ is not a pole of $-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}$. If the integral $\int\limits_{-A+\II{}T}^{\rho+\II{}T}-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}\frac{x^{s+n-1}}{\prod\limits_{k=0}^{n-1}\br{s+k}}ds$ is $O\br{\frac{x^{2\rho+\varepsilon_{1}+n-1}}{\varepsilon_{1}T^{1-\delta}}}$, then
\begin{equation}\label{dvadesetdevet}
\begin{aligned}
\psi_{n-1}\br{x}=&\sum\limits_{p=0}^{n-1}\alpha_{n-1-k}x^{n-1-k}\log x+\sum\limits_{k=0}^{n-1}\beta_{n-1-k}x^{n-1-k}+\\
&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}\sum\limits_{\alpha\in S_{p,\tau,\lambda}^{\mathbb{R}}}\frac{x^{\alpha+n-1}}{\prod\limits_{k=0}^{n-1}\br{\alpha+k}}+\\
&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}\sum\limits_{\substack{\alpha\in S_{p,\tau,\lambda}^{-\rho+\lambda}\\ \abs{\IM\br{\alpha}}\leq T}}\frac{x^{\alpha+n-1}}{\prod\limits_{k=0}^{n-1}\br{\alpha+k}}+O\br{\frac{x^{2\rho+\varepsilon_{1}+n-1}}{\varepsilon_{1}T^{1-\delta}}},
\end{aligned}
\end{equation}
where $S_{p,\tau,\lambda}^{\mathbb{R}}$ is the set of real singularities of $Z_{S}\br{s+\rho-\lambda,\tau}$ not containing the integers $0$, $-1$, ..., $-\br{n-1}$, $S_{p,\tau,\lambda}^{-\rho+\lambda}$ is the set of non-real singularities of $Z_{S}\br{s+\rho-\lambda,\tau}$, and $\alpha_{k}$, $\beta_{k}$, $k$ $\in$ $\set{0,1,...,n-1}$ are some explicitly computable constants.
\end{theorem}
\begin{proof}
By the Cauchy residue theorem applied to the rectangle $R\br{A,T}$ given by vertices: $2\rho$ $+$ $\varepsilon_{1}$ $-$ $\II{}T$, $2\rho$ $+$ $\varepsilon_{1}$ $+$ $\II{}T$, $-A$ $+$ $\II{}T$, $-A$ $-$ $\II{}T$, we immediately have that
\begin{equation}\label{trideset}
\begin{aligned}
&\frac{1}{2\pi\II{}}\int\limits_{2\rho+\varepsilon_{1}-\II{}T}^{2\rho+\varepsilon_{1}+\II{}T}-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}
\frac{x^{s+n-1}}{\prod\limits_{k=0}^{n-1}\br{s+k}}ds\\
=&\frac{1}{2\pi\II{}}\int\limits_{\rho+\varepsilon_{1}+\II{}T}^{2\rho+\varepsilon_{1}+\II{}T}+
\frac{1}{2\pi\II{}}\int\limits_{\rho+\II{}T}^{\rho+\varepsilon_{1}+\II{}T}+
\frac{1}{2\pi\II{}}\int\limits_{-A+\II{}T}^{\rho+\II{}T}+
\frac{1}{2\pi\II{}}\int\limits_{-A-\II{}T}^{-A+\II{}T}\\
&-\frac{1}{2\pi\II{}}\int\limits_{-A-\II{}T}^{\rho-\II{}T}-\frac{1}{2\pi\II{}}\int\limits_{\rho-\II{}T}^{\rho+\varepsilon_{1}-\II{}T}
-\frac{1}{2\pi\II{}}\int\limits_{\rho+\varepsilon_{1}-\II{}T}^{2\rho+\varepsilon_{1}-\II{}T}+
\end{aligned}
\end{equation}
\begin{align*}
&\sum\limits_{\alpha\in R\br{A,T}}\RES_{s=\alpha}\br{-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}\frac{x^{s+n-1}}{\prod\limits_{k=0}^{n-1}\br{s+k}}}.
\end{align*}

As in the derivation of (\ref{devetnaest}), we obtain
\begin{equation}\label{tridesetjedan}
\begin{aligned}
\frac{1}{2\pi\II{}}\int\limits_{2\rho+\varepsilon_{1}-\II{}T}^{2\rho+\varepsilon_{1}+\II{}T}-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}
\frac{x^{s+n-1}}{\prod\limits_{k=0}^{n-1}\br{s+k}}ds=\psi_{n-1}\br{x}+O\br{\frac{x^{2\rho+\varepsilon_{1}+n-1}}{T^{n-1}}}.
\end{aligned}
\end{equation}

By (\ref{osam}),
\begin{equation}\label{tridesetdva}
\begin{aligned}
\frac{1}{2\pi\II{}}\int\limits_{\rho+\varepsilon_{1}+\II{}T}^{2\rho+\varepsilon_{1}+\II{}T}-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}
\frac{x^{s+n-1}}{\prod\limits_{k=0}^{n-1}\br{s+k}}ds=O\br{\frac{x^{2\rho+\varepsilon_{1}+n-1}}{\varepsilon_{1}T^{1-\delta}}}
\end{aligned}
\end{equation}
and
\begin{equation}\label{tridesettri}
\begin{aligned}
-\frac{1}{2\pi\II{}}\int\limits_{\rho+\varepsilon_{1}-\II{}T}^{2\rho+\varepsilon_{1}-\II{}T}-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}
\frac{x^{s+n-1}}{\prod\limits_{k=0}^{n-1}\br{s+k}}ds=O\br{\frac{x^{2\rho+\varepsilon_{1}+n-1}}{\varepsilon_{1}T^{1-\delta}}}.
\end{aligned}
\end{equation}

Reasoning in the same way as in the derivation of (\ref{devet}) and (\ref{deset}), we have
\begin{equation}\label{tridesetcetiri}
\begin{aligned}
\frac{1}{2\pi\II{}}\int\limits_{\rho+\II{}T}^{\rho+\varepsilon_{1}+\II{}T}-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}
\frac{x^{s+n-1}}{\prod\limits_{k=0}^{n-1}\br{s+k}}ds=O\br{\frac{x^{\rho+\varepsilon_{1}+n-1}}{T^{1-\delta}}}
\end{aligned}
\end{equation}
and
\begin{equation}\label{tridesetpet}
\begin{aligned}
-\frac{1}{2\pi\II{}}\int\limits_{\rho-\II{}T}^{\rho+\varepsilon_{1}-\II{}T}-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}
\frac{x^{s+n-1}}{\prod\limits_{k=0}^{n-1}\br{s+k}}ds=O\br{\frac{x^{\rho+\varepsilon_{1}+n-1}}{T^{1-\delta}}}.
\end{aligned}
\end{equation}

Note that the $O$-terms in (\ref{tridesetjedan})-(\ref{tridesetpet}) are all dominated by $O\br{\frac{x^{2\rho+\varepsilon_{1}+n-1}}{\varepsilon_{1}T^{1-\delta}}}$.

We calculate the residues in (\ref{trideset}) in the same way as in \cite[pp. 6-7]{MDPI1}. We obtain,
\begin{equation}\label{tridesetsest}
\begin{aligned}
&\sum\limits_{\alpha\in R\br{A,T}}\RES_{s=\alpha}\br{-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}\frac{x^{s+n-1}}{\prod\limits_{k=0}^{n-1}\br{s+k}}}\\
=&\sum\limits_{k=0}^{n-1}\alpha_{n-1-k}x^{n-1-k}\log x+\sum\limits_{k=0}^{n-1}\beta_{n-1-k}x^{n-1-k}+\\
&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}\sum\limits_{\substack{\alpha\in S_{p,\tau,\lambda}^{\mathbb{R}}\\ \alpha>-A}}
\frac{x^{\alpha+n-1}}{\prod\limits_{k=0}^{n-1}\br{\alpha+k}}+\\
&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}\sum\limits_{\substack{\alpha\in S_{p,\tau,\lambda}^{-\rho+\lambda}\\ \abs{\IM\br{\alpha}}\leq T}}
\frac{x^{\alpha+n-1}}{\prod\limits_{k=0}^{n-1}\br{\alpha+k}}.
\end{aligned}
\end{equation}

Combining the relations (\ref{trideset})-(\ref{tridesetsest}) together with the assumption\\ $\int\limits_{-A+\II{}T}^{\rho+\II{}T}-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}\frac{x^{s+n-1}}{\prod\limits_{k=0}^{n-1}\br{s+k}}ds$ $=$ $O\br{\frac{x^{2\rho+\varepsilon_{1}+n-1}}{\varepsilon_{1}T^{1-\delta}}}$, we arrive at

\begin{align*}
\psi_{n-1}\br{x}=&\sum\limits_{k=0}^{n-1}\alpha_{n-1-k}x^{n-1-k}\log x+\sum\limits_{k=0}^{n-1}\beta_{n-1-k}x^{n-1-k}+\\
&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}\sum\limits_{\substack{\alpha\in S_{p,\tau,\lambda}^{\mathbb{R}}\\ \alpha>-A}}
\frac{x^{\alpha+n-1}}{\prod\limits_{k=0}^{n-1}\br{\alpha+k}}+
\end{align*}
\begin{align*}
&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}\sum\limits_{\substack{\alpha\in S_{p,\tau,\lambda}^{-\rho+\lambda}\\ \abs{\IM\br{\alpha}}\leq T}}
\frac{x^{\alpha+n-1}}{\prod\limits_{k=0}^{n-1}\br{\alpha+k}}+\\
&O\br{\frac{x^{2\rho+\varepsilon_{1}+n-1}}{\varepsilon_{1}T^{1-\delta}}}+
\frac{1}{2\pi\II{}}\int\limits_{-A-\II{}T}^{-A+\II{}T}-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}
\frac{x^{s+n-1}}{\prod\limits_{k=0}^{n-1}\br{s+k}}ds.
\end{align*}

Passing to the limit $A$ $\rightarrow$ $+\infty$, (Cf. also, \cite[p. 85, Prop. 5.7]{Hejhal1} and \cite[p. 244]{Randol}), we obtain the assertion.

This completes the proof.
\end{proof}

\subsection{Explicit formulas for $\psi_{0}\br{x}$ II}
Now, we derive the asymptotics of $\psi_{0}\br{x}$ from the asymptotics of $\psi_{n-1}\br{x}$ (see, Theorem \ref{Th4}).
\begin{theorem}\label{Th5}
Suppose that the assumptions of Theorem \ref{Th4} hold. For $\varepsilon$ $>$ $0$, there is a set $E$ of finite logarithmic measure, such that
\begin{align*}
\psi_{0}\br{x}=&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}
\sum\limits_{\substack{\alpha\in S_{p,\tau,\lambda}^{\mathbb{R}}\\2\rho-\rho\frac{2\br{n-1}+1}{2n\br{n-1}+1}<\alpha\leq2\rho}}\frac{x^{\alpha}}{\alpha}+\\
&O\br{x^{2\rho-\rho\frac{2\br{n-1}+1}{2n\br{n-1}+1}}\br{\log x}^{\frac{n-1}{2n\br{n-1}+1}}\br{\log\log x}^{\frac{n-1}{2n\br{n-1}+1}+\varepsilon}},
\end{align*}
as $x$ $\rightarrow$ $\infty$, $x$ $\in$ $\mathbb{R}$ $\backslash$ $E$.
\end{theorem}
\begin{proof}
We write the third sum on the right-hand side of (\ref{dvadesetdevet}) as follows
\begin{equation}\label{tridesetsedam}
\begin{aligned}
&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}\sum\limits_{\substack{\alpha\in S_{p,\tau,\lambda}^{-\rho+\lambda}\\ \abs{\IM\br{\alpha}}\leq Y}}
\frac{x^{\alpha+n+1}}{\prod\limits_{k=0}^{n-1}\br{\alpha+k}}+\\
&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}\sum\limits_{\substack{\alpha\in S_{p,\tau,\lambda}^{-\rho+\lambda}\\ Y<\abs{\IM\br{\alpha}}\leq T}}\frac{x^{\alpha+n+1}}{\prod\limits_{k=0}^{n-1}\br{\alpha+k}}.
\end{aligned}
\end{equation}

Reasoning in the same way as in the proof of Theorem \ref{Th3} in \cite[pp. 10-11]{MDPI1}, we obtain that $\mu^{\times}E_{p,\tau,\lambda}^{i}$ $=$ $\int\limits_{E_{p,\tau,\lambda}^{i}}\frac{dx}{x}$ $=$ $O\br{\frac{e^{2\br{\rho+n-1-\gamma}i}}{Yi^{2\beta}\br{\log i}^{2\beta+2\varepsilon}}}$, where the sets $E_{p,\tau,\lambda}^{i}$ have the same form as before, with $j$ replaced by $i$, $2n$ replaced with $n-1$, $W$ $=$ $T$, and $\alpha$ in the expression $x^{\alpha}\br{\log x}^{\beta}\br{\log\log x}^{\beta+\varepsilon}$ replaced with $\gamma$.

The relations (15) and (16) in \cite{MDPI1} are adjusted with respect to adapted notation.

In particular,
\begin{align*}
\int\limits_{-\infty}^{+\infty}\br{\sum\limits_{\substack{t\leq\abs{\IM\br{\alpha}}\leq t+1\\Y<\abs{\IM\br{\alpha}}\leq T}}\frac{1}{\prod\limits_{k=0}^{n-1}\abs{\alpha+k}}}^{2}dt=O\br{\frac{1}{Y}}.
\end{align*}

Put $Y$ $\sim$ $e^{2\br{\rho+n-1-\gamma}i}i^{1-2\beta}\br{\log i}^{1-2\beta}$.

It follows that $\mu^{\times}E$ $<$ $\infty$, where $E$ $=$ $\bigcup\limits_{i}\bigcup\limits_{p}\bigcup\limits_{\br{\tau,\lambda}\in I_{p}}E_{p,\tau,\lambda}^{i}$.

We obtain:
\begin{equation}\label{tridesetosam}
\begin{aligned}
&d^{-\br{n-1}}\Delta_{n-1}^{+}\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}\sum\limits_{\substack{\alpha\in S_{p,\tau,\lambda}^{-\rho+\lambda}\\ Y<\abs{\IM\br{\alpha}}\leq T}}\frac{x^{\alpha+n-1}}{\prod\limits_{k=0}^{n-1}\br{\alpha+k}}\\
=&O\br{\frac{x^{\gamma}\br{\log x}^{\beta}\br{\log\log x}^{\beta+\varepsilon}}{d^{n-1}}}
\end{aligned}
\end{equation}
for $x$ $\in$ $\mathbb{R}$ $\backslash$ $E$,
\begin{equation}\label{tridesetdevet}
\begin{aligned}
d^{-\br{n-1}}\Delta_{n-1}^{+}\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}\sum\limits_{\substack{\alpha\in S_{p,\tau,\lambda}^{-\rho+\lambda}\\ \abs{\IM\br{\alpha}}\leq Y}}\frac{x^{\alpha+n-1}}{\prod\limits_{k=0}^{n-1}\br{\alpha+k}}
=O\br{x^{\rho}Y^{n-1}},
\end{aligned}
\end{equation}
\begin{equation}\label{cetrdeset}
\begin{aligned}
&d^{-\br{n-1}}\Delta_{n-1}^{+}\br{\sum\limits_{k=0}^{n-1}\alpha_{n-1-k}x^{n-1-k}\log x+\sum\limits_{k=0}^{n-1}\beta_{n-1-k}x^{n-1-k}}\\
=&O\br{\log x},
\end{aligned}
\end{equation}
\begin{equation}\label{cetrdesetjedan}
\begin{aligned}
&d^{-\br{n-1}}\Delta_{n-1}^{+}\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}\sum\limits_{\alpha\in S_{p,\tau,\lambda}^{\mathbb{R}}}
\frac{x^{\alpha+n-1}}{\prod\limits_{k=0}^{n-1}\br{\alpha+k}}\\
=&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}\sum\limits_{\substack{\alpha\in S_{p,\tau,\lambda}^{\mathbb{R}}\\ 0<\alpha\leq2\rho}}
\frac{x^{\alpha}}{\alpha}+O\br{x^{2\rho-1}d}.
\end{aligned}
\end{equation}

Combining the relations (\ref{dvadesetdevet}), (\ref{tridesetsedam})-(\ref{cetrdesetjedan}) for $x$ $\in$ $\mathbb{R}$ $\backslash$ $E$ together with the fact that\\ $\psi_{0}\br{x}$ $\leq$ $d^{-\br{n-1}}\Delta_{n-1}^{+}\psi_{n-1}\br{x}$, we end up with
\begin{equation}\label{cetrdesetdva}
\begin{aligned}
\psi_{0}\br{x}\leq&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}\sum\limits_{\substack{\alpha\in S_{p,\tau,\lambda}^{\mathbb{R}}\\ 0<\alpha\leq2\rho}}
\frac{x^{\alpha}}{\alpha}+O\br{x^{2\rho-1}d}+\\
&O\br{x^{\rho}Y^{n-1}}+O\br{\frac{x^{\gamma}\br{\log x}^{\beta}\br{\log\log x}^{\beta+\varepsilon}}{d^{n-1}}}+\\
&O\br{\frac{x^{2\rho+\varepsilon_{1}+n-1}}{\varepsilon_{1}d^{n-1}T^{1-\delta}}}.
\end{aligned}
\end{equation}

We compare the error terms $O\br{x^{2\rho-1}d}$, $O\br{x^{\rho}Y^{n-1}}$ and\\ $O\br{\frac{x^{\gamma}\br{\log x}^{\beta}\br{\log\log x}^{\beta+\varepsilon}}{d^{n-1}}}$ in the same way as earlier.

Note that it is enough to replace $j$ with $n-1$ in the corresponding relations in the proof of Theorem \ref{Th2}.

We obtain the system:
\begin{align*}
\frac{\gamma-2\rho+1}{n}=&1-\rho+2\br{n-1}\br{\rho+n-1-\gamma},
\end{align*}
\begin{align*}
\frac{\beta}{n}=&\br{n-1}\br{1-2\beta},
\end{align*}
its solution:
\begin{align*}
\gamma=&\frac{2\br{n-\rho}\br{n-1}^{2}+\br{4n-3}\br{n-1}\rho+\rho+\br{n-1}}{2n\br{n-1}+1},\\
\beta=&\frac{\br{n-1}n}{2n\br{n-1}+1},
\end{align*}
and:
\begin{align*}
d=&x^{1-\rho\frac{2\br{n-1}+1}{2n\br{n-1}+1}}\br{\log x}^{\frac{n-1}{2n\br{n-1}+1}}\br{\log\log x}^{\frac{n-1}{2n\br{n-1}+1}},\\
Y\sim&x^{\frac{2\rho\br{n-1}}{2n\br{n-1}+1}}\br{\log x}^{\frac{1}{2n\br{n-1}+1}}\br{\log\log x}^{\frac{1}{2n\br{n-1}+1}}.
\end{align*}

Substituting the obtained $d$ and $Y$ into (\ref{cetrdesetdva}), we finally have that for $x$ $\in$ $\mathbb{R}$ $\backslash$ $E$
\begin{align*}
\psi_{0}\br{x}\leq&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}\sum\limits_{\substack{\alpha\in S_{p,\tau,\lambda}^{\mathbb{R}}\\ 2\rho-\rho\frac{2\br{n-1}+1}{2n\br{n-1}+1}<\alpha\leq 2\rho}}\frac{x^{\alpha}}{\alpha}+\\
&O\br{x^{2\rho-\rho\frac{2\br{n-1}+1}{2n\br{n-1}+1}}\br{\log x}^{\frac{n-1}{2n\br{n-1}+1}}\br{\log\log x}^{\frac{n-1}{2n\br{n-1}+1}+\varepsilon}}.
\end{align*}

Since the opposite inequality holds also true, the assertion follows.

This completes the proof.
\end{proof}

Now, we are in position to improve the result given by Theorem \ref{Th3}.

\subsection{Gallagherian prime geodesic theorem II}
As in the case of Theorem \ref{Th3}, the direct consequence of Theorem \ref{Th5} is the following theorem.
\begin{theorem}\label{Th6}\,(Gallagherian Prime Geodesic Theorem)\,
Suppose that the assumptions of Theorem \ref{Th4} hold. For $\varepsilon$ $>$ $0$, there is a set $E$ of finite logarithmic measure, such that
\begin{align*}
\pi_{\Gamma}\br{x}=&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}
\sum\limits_{\substack{\alpha\in S_{p,\tau,\lambda}^{\mathbb{R}}\\2\rho-\rho\frac{2\br{n-1}+1}{2n\br{n-1}+1}<\alpha\leq2\rho}}\li\br{x^{\alpha}}+\\
&O\br{x^{2\rho-\rho\frac{2\br{n-1}+1}{2n\br{n-1}+1}}\br{\log x}^{\frac{n-1}{2n\br{n-1}+1}-1}\br{\log\log x}^{\frac{n-1}{2n\br{n-1}+1}+\varepsilon}},
\end{align*}
as $x$ $\rightarrow$ $\infty$, $x$ $\in$ $\mathbb{R}$ $\backslash$ $E$.
\end{theorem}

\subsection{Prime geodesic theorem}
In \cite{MDPI1}(see, Theorem 2), we derived the prime geodesic theorem by applying the counting function $\psi_{2n}\br{x}$.

In this section, we prove that taking the function $\psi_{j}\br{x}$ with $j$ $\geq$ $n$ (see, Theorem \ref{Th1}), does not yield a better result.

Indeed (Cf. \cite[pp. 370-371]{Koreja}), by (\ref{cetiri}),
\begin{align*}
\psi_{0}\br{x}\leq& d^{-j}\Delta_{j}^{+}\psi_{j}\br{x}\\
=&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}
\sum\limits_{\rho<\alpha\leq 2\rho}d^{-j}\Delta_{j}^{+}\frac{x^{\alpha+j}}{\prod\limits_{k=0}^{j}\br{\alpha+k}}+\\
&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\substack{\br{\tau,\lambda}\in I_{p}\\ \lambda=2\rho}}
\sum\limits_{\RE\br{\alpha}=\rho}d^{-j}\Delta_{j}^{+}\frac{x^{\alpha+j}}{\prod\limits_{k=0}^{j}\br{\alpha+k}}\\
=&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}
\sum\limits_{\rho<\alpha\leq 2\rho}\frac{x^{\alpha}}{\alpha}+O\br{x^{2\rho-1}d}+\\
&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\substack{\br{\tau,\lambda}\in I_{p}\\ \lambda=2\rho}}O\br{x^{\rho}\int\limits_{\rho}^{K}t^{-1}dN_{S,p}^{\tau,\lambda}\br{t}}+\\
&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\substack{\br{\tau,\lambda}\in I_{p}\\ \lambda=2\rho}}O\br{d^{-j}x^{\rho+j}\int\limits_{K}^{+\infty}t^{-j-1}dN_{S,p}^{\tau,\lambda}\br{t}}\\
=&\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}
\sum\limits_{\rho<\alpha\leq 2\rho}\frac{x^{\alpha}}{\alpha}+O\br{x^{2\rho-1}d}+O\br{x^{\rho}K^{n-1}}+\\
&O\br{d^{-j}x^{\rho+j}K^{n-j-1}}.
\end{align*}

Substituting $d$ $=$ $x^{1-\frac{\rho}{n}}$, $K$ $=$ $x^{\frac{\rho}{n}}$, we get
\begin{align*}
\psi_{0}\br{x}\leq\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}
\sum\limits_{2\rho-\frac{\rho}{n}<\alpha\leq 2\rho}\frac{x^{\alpha}}{\alpha}+O\br{x^{2\rho-\frac{\rho}{n}}},
\end{align*}
that is,
\begin{equation}\label{cetrdesettri}
\begin{aligned}
\pi_{\Gamma}\br{x}=\sum\limits_{p=0}^{n-1}\br{-1}^{p+1}\sum\limits_{\br{\tau,\lambda}\in I_{p}}
\sum\limits_{2\rho-\frac{\rho}{n}<\alpha\leq 2\rho}\li\br{x^{\alpha}}+O\br{x^{2\rho-\frac{\rho}{n}}\br{\log x}^{-1}}
\end{aligned}
\end{equation}
as $x$ $\rightarrow$ $\infty$.

Note that one is in position to derive a variant of the asymptotics (\ref{cetrdesettri}) from the asymptotics of $\psi_{n-1}\br{x}$ (Cf. \cite[pp. 98-102]{Park}).

However, such approach yields only a weaker form of the prime geodesic theorem, with $O\br{x^{2\rho-\frac{\rho}{n}}\br{\log x}^{-\frac{1}{2}}}$ in place of $O\br{x^{2\rho-\frac{\rho}{n}}\br{\log x}^{-1}}$ in (\ref{cetrdesettri}).

\section{Discussion}
The error terms in (\ref{dvadesetosam}), i.e., the terms\\ $O\br{x^{2\rho-\rho\frac{2j+1}{2nj+1}}\br{\log x}^{\frac{n-1}{2nj+1}-1}\br{\log\log x}^{\frac{n-1}{2nj+1}+\varepsilon}}$, $j$ $\geq$ $n$, determine the sequence $\set{2\rho-\rho\frac{2j+1}{2nj+1}}_{j}$, whose elements are given by the exponents of $x$, and are such that $2\rho$ $-$ $\rho\frac{2j+1}{2nj+1}$ $<$ $2\rho$ $-$ $\rho\frac{2\br{j+1}+1}{2n\br{j+1}+1}$ for $j$ $\in$ $\mathbb{N}$. The sequence is strictly increasing since the last inequality is equivalent to the inequality $n$ $>$ $1$, which is obviously true. Being interested in $j$ $\geq$ $n$ in Theorem \ref{Th3}, we conclude that the optimal size of the error term in (\ref{dvadesetosam}) is $O\br{x^{2\rho-\rho\frac{2n+1}{2n^{2}+1}}\br{\log x}^{\frac{n-1}{2n^{2}+1}-1}\br{\log\log x}^{\frac{n-1}{2n^{2}+1}+\varepsilon}}$, and is achieved for $j$ $=$ $n$. So, this $O$-term, as well as each of the $O$-terms $O\br{x^{2\rho-\rho\frac{2j+1}{2nj+1}}\br{\log x}^{\frac{n-1}{2nj+1}-1}\br{\log\log x}^{\frac{n-1}{2nj+1}+\varepsilon}}$, $n$ $<$ $j$ $<$ $2n$, improve the $O$-term $O\br{x^{2\rho-\rho\frac{2\cdot\br{2n}+1}{2n\cdot\br{2n}+1}}\br{\log x}^{\frac{n-1}{2n\cdot\br{2n}+1}-1}\br{\log\log x}^{\frac{n-1}{2n\cdot\br{2n}+1}+\varepsilon}}$ derived in our previous study \cite[p. 9, Th. 3]{MDPI1}.

Note that our $O\br{x^{2\rho-\rho\frac{2n+1}{2n^{2}+1}}\br{\log x}^{\frac{n-1}{2n^{2}+1}-1}\br{\log\log x}^{\frac{n-1}{2n^{2}+1}+\varepsilon}}$, $n$ $\geq$ $4$, fully agrees with the best estimate in the $d$-dimensional, real hyperbolic manifolds with cusps case. Indeed, for $n$ $=$ $d$ and $\rho$ $=$ $\frac{1}{2}\br{n-1}$, the $O$-term in \cite[p. 3021, Th. 2]{AvdZenan} is given by
\begin{align*}
&O\br{x^{\alpha_{n}}\br{\log x}^{\beta_{n}-1}\br{\log\log x}^{\beta_{n}+\varepsilon}}\\
=&O\br{x^{\br{n-1}\br{1-\frac{2n+1}{4n^{2}+2}}}\br{\log x}^{\frac{n-1}{2n^{2}+1}-1}\br{\log\log x}^{\frac{n-1}{2n^{2}+1}+\varepsilon}}\\
=&O\br{x^{\br{n-1}-\frac{1}{2}\br{n-1}\frac{2n+1}{2n^{2}+1}}\br{\log x}^{\frac{n-1}{2n^{2}+1}-1}\br{\log\log x}^{\frac{n-1}{2n^{2}+1}+\varepsilon}}\\
=&O\br{x^{2\rho-\rho\frac{2n+1}{2n^{2}+1}}\br{\log x}^{\frac{n-1}{2n^{2}+1}-1}\br{\log\log x}^{\frac{n-1}{2n^{2}+1}+\varepsilon}}.
\end{align*}

As noted earlier, the $O$-terms in (\ref{dvadesetosam}), and hence the elements of the sequence $\set{2\rho-\rho\frac{2j+1}{2nj+1}}_{j}$, stem from the application of the Gallagher-Koyama techniques to the explicit formulas for the counting functions $\psi_{j}\br{x}$, $j$ $\geq$ $n$, given by Theorem \ref{Th1}. This fact, and the fact that the error terms are decreasing when $j$ is decreasing, motivated us to consider the possibility of further reduction of the error term in (\ref{dvadesetosam}) via the function $\psi_{n-1}\br{x}$. Unfortunately, we were not able to derive the explicit formula for $\psi_{n-1}\br{x}$ which would be analogous to (\ref{cetiri}) (see, remark at the end of Subsection 3.1). However, we were in position to adjust Hejhal's methods \cite{Hejhal1,Hejhal2}, to our setting, and thus obtain the explicit formula for $\psi_{n-1}\br{x}$, which is conditional and asymptotic, and is given by Theorem \ref{Th4}. In particular, we required the integral $\int\limits_{-A+\II{}T}^{\rho+\II{}T}-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}\frac{x^{s+n-1}}{\prod\limits_{k=0}^{n-1}\br{s+k}}ds$ to be $O\br{\frac{x^{2\rho+\varepsilon_{1}+n-1}}{\varepsilon_{1}T^{1-\delta}}}$, where $\varepsilon_{1}$ $>$ $0$, $\delta$ $>$ $0$, and $T$ $\gg$ $0$, $A$ $\gg$ $2\rho$ are selected such that $\II{}T$ is not a singularity of $Z_{S}\br{s,\tau}$, $\tau$ $\in$ $\mathcal{T}$, and $-A$ is not a pole of $-\frac{Z_{R}^{'}\br{s}}{Z_{R}\br{s}}$. Note that an analogous condition of the given one is easily verified to be true in the case of compact Riemann surfaces for example ($n$ $=$ $2$, $\rho$ $=$ $\frac{1}{2}\br{n-1}$ $=$ $\frac{1}{2}$). Namely, according to Propositions 6.12-6.14 in \cite[pp. 105-108]{Hejhal1}, and the discussion given on page 108 of the same book, the integral $\int\limits_{-A+\II{}T}^{1+\varepsilon+\II{}T}\frac{Z^{'}\br{s}}{Z\br{s}}\frac{x^{s+1}}{s\br{s+1}}ds$ is $O\br{\frac{x^{2+\varepsilon}}{\varepsilon T}}$, where $Z\br{s}$ is the corresponding Selberg zeta function. The fact that the Ruelle zeta function in our case is a product of the Selberg zeta functions complicates things, so we left the aforementioned condition as an assumption.

As expected, the error term corresponding to $\psi_{n-1}\br{x}$ is\\ $O\br{x^{2\rho-\rho\frac{2\br{n-1}+1}{2n\br{n-1}+1}}\br{\log x}^{\frac{n-1}{2n\br{n-1}+1}-1}\br{\log\log x}^{\frac{n-1}{2n\br{n-1}+1}+\varepsilon}}$.

This is in line with (\ref{dvadesetosam}).

The obtained error term improves the error terms in (\ref{dvadesetosam}).

In particular, it improves the error term derived in \cite[p. 9, Th. 3]{MDPI1}.

If $n$ $=$ $2$, $\rho$ $=$ $\frac{1}{2}\br{n-1}$ $=$ $\frac{1}{2}$, our bound\\ $O\br{x^{2\rho-\rho\frac{2\br{n-1}+1}{2n\br{n-1}+1}}\br{\log x}^{\frac{n-1}{2n\br{n-1}+1}-1}\br{\log\log x}^{\frac{n-1}{2n\br{n-1}+1}+\varepsilon}}$ becomes\\ $O\br{x^{\frac{7}{10}}\br{\log x}^{\frac{1}{5}-1}\br{\log\log x}^{\frac{1}{5}+\varepsilon}}$, which is exactly the bound obtained by moving from the level $\psi_{0}\br{x}$ to the level $\pi_{\Gamma}\br{x}$ in the non-compact, hyperbolic, finite-volume Riemann surface with cusps case (Cf. \cite[p. 28, Th. 3.1]{AvdAnalMath}).

For $n$ $=$ $3$, $\rho$ $=$ $\frac{1}{2}\br{n-1}$ $=$ $1$,\\ $O\br{x^{2\rho-\rho\frac{2\br{n-1}+1}{2n\br{n-1}+1}}\br{\log x}^{\frac{n-1}{2n\br{n-1}+1}-1}\br{\log\log x}^{\frac{n-1}{2n\br{n-1}+1}+\varepsilon}}$ is\\ $O\br{x^{\frac{21}{13}}\br{\log x}^{-\frac{11}{13}}\br{\log\log x}^{\frac{2}{13}+\varepsilon}}$. This is in line with the result in the hyperbolic three manifolds case (see, \cite[p. 691, Th. 1.2]{Erata}).

In the remark on lower dimensions \cite[p. 3025]{AvdZenan}, the authors noted that one is in position to derive the asymptotics of $\psi_{0}\br{x}$ from $\psi_{d-1}\br{x}$ instead of reaching for $\psi_{d}\br{x}$ in the $d$-dimensional, real hyperbolic manifolds with cusps case. Taking into account the results we derived in this research, it becomes pretty obvious that the improved variant of Theorem 2 in \cite{AvdZenan}(the one obtained from $\psi_{d-1}\br{x}$), should state that for $\varepsilon$ $>$ $0$, there is a set $E$ of finite logarithmic measure, such that:
\begin{equation}\label{cetrdesetcetiri}
\begin{aligned}
&\pi_{\Gamma}\br{x}\\
=&\sum\limits_{\br{d-1}-\frac{1}{2}\br{d-1}\frac{2\br{d-1}+1}{2d\br{d-1}+1}<s_{n}\br{k}\leq2d_{0}}\br{-1}^{k}\li\br{x^{s_{n}\br{k}}}+\\
&O\br{x^{\br{d-1}-\frac{1}{2}\br{d-1}\frac{2\br{d-1}+1}{2d\br{d-1}+1}}\br{\log x}^{\frac{d-1}{2d\br{d-1}+1}-1}\br{\log\log x}^{\frac{d-1}{2d\br{d-1}+1}+\varepsilon}}
\end{aligned}
\end{equation}
as $x$ $\rightarrow$ $\infty$, $x$ $\in$ $\mathbb{R}$ $\backslash$ $E$, where $d_{0}$ $=$ $\frac{d-1}{2}$, $\br{s_{n}\br{k}-k}\br{2d_{0}-k-s_{n}\br{k}}$ is a small eigenvalue in $\SqBr{0,\frac{3}{4}d_{0}^{2}}$ of $\Delta_{k}$ on $\pi_{\sigma_{k},\lambda_{n}\br{k}}$ with $s_{n}\br{k}$ $=$ $d_{0}$ $+$ $\II{}\lambda_{n}\br{k}$ or $s_{n}\br{k}$ $=$ $d_{0}$ $-$ $\II{}\lambda_{n}\br{k}$ in $\spbr{\frac{3}{2}d_{0},2d_{0}}$, $\Delta_{k}$ is the Laplacian acting on the space of $k$-forms over $Y$, and $\pi_{\sigma_{k},\lambda_{n}\br{k}}$ is the principal series representation.

Notice that we used the notation $Y$ to temporarily denote the $d$-dimensional, real hyperbolic manifold with cusps in the particular case (\ref{cetrdesetcetiri}).

Also, we point out that the $O$-term $O\br{x^{2\rho-\frac{\rho}{n}}\br{\log x}^{-1}}$ in our unconditional prime geodesic theorem (\ref{cetrdesettri}) does not depend on the choice of the counting function $\psi_{j}\br{x}$, $j$ $\geq$ $n$ (Cf. \cite[p. 6, relation (8)]{MDPI1}).

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Daniel describes a very difficult time in Israel. Constant battles between the Seleucid and Ptolemian empires have wreaked havoc on the people of God. They have been an occupied territory for several years, and now face the threat of annihilation, as the Seleucid king, Antiochus IV, has set his attention on destroying Israel. He outlawed the Jewish religion, ordered the Jews to worship Greek gods. In 168 B.C., his soldiers descended upon Jerusalem, massacred thousands of people, burned the Holy Scriptures, and desecrated the Temple by erecting an altar to Zeus, and sacrificing pigs on the altar. It is with this dark, and seemingly hopeless, backdrop that Daniel declares, "the people who know their God will be strong and carry out great exploits." In other words, the darker the backdrop, the greater opportunity for the child of God to shine.
History records, during this time, a priest by the name of Mattathias, and his five sons, mounted a rebellion against Antiochus and his forces. They soon began to succeed in one battle after another, until they had taken back the city of Jerusalem, and restored worship within the Temple. It was at this time, the events that led to the celebration of Hanukkah took place. The story is told, when the temple was rededicated, they had only enough oil to light the lamp stand for one day, and it would take several days to prepare more oil. They lit the lamp and began preparing more oil. Eight days later, the oil was completed, and the lamp was miraculously still burning. Hanukkah, in many ways, is the celebration of the people of God doing great exploits.
We live in dark days, that seem to be getting darker. Personal freedoms are being lost in the name of tolerance, the rising threat of terrorism impacts almost every aspect of life, and immorality is on the rise, in what seems to be unprecedented levels. This is not the time for the child of God to cower in fear or lean on the arm of the flesh. These are times for "the people who know their God [to be] strong, and carry out great exploits." These are times for us to live more committed to Christ, and to declare with an ever increasing boldness, the simple message of the Gospel. Times like these are the birthplace of heroes.
Daniel finds himself in a place where he desperately needs direction from the Lord. He has seen the spiritual condition of the nation and knows that the only hope for his people is a work of God. He determines to set himself to seeking the Lord. Along with daily prayer and digging into the Scriptures, Daniel chooses to set out upon a voluntary fast.
Fasting is a spiritual exercise that involves denying the body physical things in order to focus attention on spiritual things. Fasting is often associated with both prayer and personal repentance, and is designed to bring us into step with what God wants to do in and through our lives.
Technically, fasting is to restrict yourself from all food for a period of time, but actually, a person can fast from just about anything. Here Daniel has chosen to restrict himself from "pleasant food." Because of his age, it is likely he ate something each day, but limited it to absolute necessity. We can follow his example by setting aside something that tends to occupy a lot of our time, and give that to the Lord. Today, one of the most valuable fasts may be to give up social media for a week and use the time to pray and read the Bible.
Whatever we choose to fast from, we should understand that much of the value of fasting is to train the flesh to say no. The same drives that compel me to eat, also tempt me to do what is forbidden, or to react in an ungodly manner. If I can resist the flesh's desire for a cupcake, I can learn to resist its desire to complain, lash out in anger, or to be right all the time.
Why not choose to fast this week? Perhaps you will set aside food for a few days, or maybe there is something else that you can lay down for a while and give that time to seeking the Lord.
Daniel is one of the greatest heroes in all of Scripture. His personal commitment and public ministry impacted the lives of his companions, co-workers, and even kings. We know he had opportunity to witness to the most powerful and influential men in the world, and it seems some of them were converted. One of the reasons for his success in ministry was his faithfulness. Daniel set himself apart to the Lord and sought to live a godly life. His commitment to the Lord affected every area of his life. Privately, we know he was a man of prayer and the Word. Publicly, we know he was a man of commitment, faithfulness and conviction. When his accusers sought to find some charge to bring against him, they realized that the only way they could find fault was if his religious convictions forbid him to obey an earthly law.
We should seek to follow the example Daniel set. The secret to being found above reproach, when examined by others, is to first carefully examine yourself in light of the Word of God. If we will daily let the light of the Word shine into the dark places of our lives, and make the appropriate changes, we will find ourselves absolved of all charges when examined by others.
Take a few moments to allow God to examine you. If you find you are practicing something He forbids, or holding on to something you need to let go of, then respond appropriately. Lay those things at the cross, where you can be forgiven and they can be removed.
Have you ever wished God would write a message specifically for you? Perhaps hoping His finger would suddenly appear writing upon the clouds. Belshazzar had that experience. On the night before his death, the silence of heaven was broken when a warning was written on the walls of his palace. Sadly, Belshazzar ignored the warnings, did nothing in response to the message, and died that night not prepared to face eternity.
While I do not expect God will write upon the walls of my home, I have found heaven is not as silent as we think. The Bible is filled with page after page of messages for mankind. Like Belshazzar, the question is not whether heaven speaks, but whether we are willing to listen.
If you want to measure up when weighed in eternity's balance, then you must look into the Word of God and live accordingly. The first step is to have our sins forgiven. This happens when we receive Christ and the pardon He offers through the cross. Second, we must decide to follow Him by looking into the Bible and doing what it says. Similarly, when a road map says turn and we turn, so when Scripture says change we change.
Let's not find ourselves in Belshazzar's shoes, face to face with eternity and unprepared to go.


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There will be food stations throughout the center of substantial hot and cold dishes along with local wines catered by The Main Course.
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The death rates from pretty much every major cause — heart disease, cancer, Alzheimer's, diabetes, suicide, sepsis, guns, infant mortality — remain highest in the South, according to updated data from the Centers for Disease Control and Prevention.
Between the lines: Rural Appalachia has higher death rates from drug overdoses. But a lot of the poorest health outcomes in the South reflect longstanding poverty, fewer health care resources and longstanding barriers to care.
Sen. Michael Bennet (D-Colo.) is still considering a 2020 presidential bid after confirming on Friday that he has been issued a clean bill of health following prostate cancer surgery, NBC reports.
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text:
Q: Cairo: Refilling an object(circle) and deleting it I'm drawing some arcs using cairo, Here's the function.
cairo_t* drawCircle(cairo_surface_t *container, int x, int y, int r, int cr, int cg, int   cb, int ca)
{
  cairo_t               *cairoInstance;                                                                                //Create instance
  cairoInstance = cairo_create(container);

  cairo_set_source_rgba(cairoInstance, 0,0,1,0.5);
  cairo_arc(cairoInstance, x, y, r, 0, 2*M_PI);
  cairo_stroke_preserve(cairoInstance);
  cairo_fill(cairoInstance);

  gtk_widget_queue_draw_area(GTK_WIDGET(window), 0, 0, WINDOW_WIDTH, WINDOW_HEIGHT); //Redraws the window

return cairoInstance;
}

First question: When I change the source_rgba later and then fill, it fills over the previous color. Is there a way to reset the fill before filling again with a different color?
Second of all, If I wanted to unrender/delete the arc/circle later on, how do I do that? I tried to use cairo_destroy but it seems to destroy the instance and not delete the actual object.
Thanks!

A: Cairo uses the stencil/paint model so drawing is (usually) done on top of the existing image1.
It doesn't maintain layers or objects. Just like drawing on paper, you can't move or modify the circle after its drawn. It has become a pattern in the pixmap.
To delete the circle, fill the whole image with the original background (cairo_set_source_rgb(cr, 1, 1, 1); cairo_paint(cr)), and redraw everything except the circle.
1. You can change the raster opcode to do other types of compositing.

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Ten Lost Years
STREETS OF FIRE
Crime may be declining elsewhere in Canada, writes CHARLIE GILLIS, but violence burdens the troubled Toronto neighbourhood where Omar Hortley was gunned down
STREETS OF FIRE June 7 2004
The faces in cabinet changed from time to time, as did the priorities. But as successful as the Liberals have been at the polls, much has been neglected over their three successive mandates, in an election campaign series starting this week, Maclean's looks at people and places that have not shared in the country's general well-being over the past decade.
HER VOICE WAVERS, and she pauses at times to stop her tears. But Patricia Fough is ready—determined, actually—to tell her story. It begins in a suburban home in northeast Toronto, with her 21-year-old son, Omar Hortley, rising after dinner and asking if he might go visit some friends. It was a cold evening last January—not the kind for a walk down a windswept Scarborough street. But Omar's mates had arranged to watch a wrestling match on pay-per-view television, and the house where they were meeting was only 2 V2 blocks away. So he pulled on his coat, kissed his mother goodbye and promised to call before he came home.
The next few hours are a blur. At 8 p.m. there was a puzzling call from Hortley's friends, who were still awaiting his arrival; then, her own desperate inquiries with hospitals and police as it became clear Omar was missing; finally, after 11 p.m., a visit from two Toronto police detectives confirming her most sickening fears. Hortley, they said, had been killed by a bullet to the head in a driveby shooting mere minutes after stepping out of his own front door. Because he'd left his wallet at home, they'd been unable to identify him until she showed them his picture.
For Fough, a 43-year-old accounts clerk who was born in Guyana, the loss defied all logic. For 13 years, she'd lived without a hint of trouble in her neighbourhood north of Highway 401, raising a son who seemed the very antithesis of disaffected adolescence. "We were like best friends, brother and sister, you name it," she recalls. "He was my whole life." Far from running with a dangerous crowd, Hortley spent most of his time at home, caring for his grandmother before her death 18 months ago and, later, preparing to attend Durham College in nearby Oshawa. "He was not a gang member," Fough stresses. "He always asked me if he could go out, which I found amazing for a 21-year-old kid. I had never had any trouble with him."
CANADA, the statistics tell us, is not a violent place, and crime is way down the list of issues driving the current federal election campaign—no doubt because Canadians have suffered progressively less of it over the past decade. Between 1992 and 2002, the overall crime rate fell nearly 25 per cent across the country, while even growing cities like Toronto saw general declines in violent crime from their heights in the early 1990s. But Hortley's death and a rash of other shootings in suburban Toronto point to anq emerging hazard on the criminal landscape, one that's been largely ignored as the rest of the country grew peaceful. In the past 12 months, there have been no less than five gunrelated deaths in the neighbourhood known as Malvern, a cluster of winding residential streets in Toronto's northeast end, and still more shootings in northwest neighbourhoods like Rexdale, near Pearson International Airport. Several people have been injured as a result, and in a few cases, stray bullets have flown through the walls of homes, in one instance killing someone inside.
The effect has been to create a series of nogo zones around the perimeter of the city, formerly peaceful bedroom communities burdened by a distinctly un-suburban sense of violence, poverty and despair. While Malvern was once a thriving blue-collar neighbourhood, the most recent census figures suggest some 19 per cent of its mostly Caribbean, East Asian and South Asian families now live below the poverty line—almost three times the 1981 level. That decline epitomizes a city-wide shift in poverty from the old city toward the inner suburbs, where there has been a sixfold increase in the number of higher poverty districts, according to a report published recently by the United Way of Greater Toronto and the Canadian Council on Social Development. Not surprisingly, there is little confidence among young people in these areas that things are going to change: at a community meeting in Rexdale last March to discuss the growing
problems of gang violence, teenagers dismissed the event as a publicity stunt, and warned that most people hope to leave the first chance they get. "If things continue the way they are and I have a family," said 17-year-old Ankit Kadakia, a student at North Albion Collegiate, where the meeting was held, "I would move out of the area for their protection."
MALVERN was once a thriving blue-collar community, but now 19 per cent of families live below the poverty line
This bleak outlook prevails despite a flurry of official initiatives, including a strategy announced by Mayor David Miller to explore ways of preventing violence. The police, too, have been busy, arresting some 64 suspected members of a gang called the Malvern Crew on May 12 in a sweep that won widespread applause. But enthusiasm over such victories has been tempered by the slow grind of bureaucracy, or the frustrating vagaries of the justice system. So far, Miller's initiative has produced an advisory panel, a "community safety secretariat" and a variety of well-intentioned plans, but little in the way of concrete results. The police, meanwhile, saw 10 of the first 15 alleged gang members who were processed in court released on bail. As if to mock the whole process, several of the accused mooned and gave the middle finger to the media on their way out of court; one invited a photographer to try taking pictures on the suspect's "home turf."
Hortley's murder in January, one of five gun-related deaths in the neighbourhood in the last 12 months, remains unsolved
PATRICIA FOUGH has certainly placed her faith elsewhere. In the days following Omar's killing, the grief-stricken mother found solace at Rhema Christian Ministries, a 2,000-strong, largely black congregation running services out of a warehouse complex in northeast Toronto. At the time, she was near distraction from mental images of her son falling to the sidewalk, and haunting thoughts about his final seconds. "I'd wonder whether he'd cried out," she says, "whether he'd asked for me before he died." The shooters, she knew, were still at large: while they paused long enough to spray several bullets in Hortley's direction, nobody came forward to identify them. And while Fough had her husband (not Omar's father) to support her, the church offered her a place to grieve, pray and call for solace at practically any hour of the day. "Just hearing that voice," she says, "would make me feel better."
Discovering Rhema proved as fateful for the church as it did for Fough. Inspired by the woman's plight, its founder, Orim Meikle, a rhetorically gifted, born-again pastor, made troubled neighbourhoods his church's keystone cause. In February, he held a funeral for Omar in Rhema's cavernous sanctuary, and has since made no secret of his belief that secular authorities are out of their depth. "Government should govern," he says in his spartan office, "and the church should make sure that the social structures—the fibres of family and community—are in place."
The message is a familiar one from Christian churches working in troubled neighbourhoods. The absence of religion in modern life has broken down the family unit, they say, driving young people out of the house toward gangs and drugs and sexual permissiveness. But Meikle, who is black, has added an overtly racial dimension to the theory, which he's happy to rub in the face of the politically correct. Born in Jamaica and raised in Toronto, he frequently calls on black people to assume greater personal responsibility for their troubles, and to exercise will against their appetites. Compare the black community to other ethnic groups, he argues, and its failures stand in bold relief: "Whether it's Asians who come and do menial labour to make their way, or [East] Indians who come and live in one house until everyone can buy a house," he says, "they're doing some things and they're advancing, while we're kind of laissez-faire, sitting back." Such talk is bound to ruffle feathers, Meikle acknowledges, but tiptoeing around the issue isn't going to change anything. "Plain speech," he says, "is what people want to hear."
To prove his seriousness, Meikle has taken his message—and his congregation—directly to the street. On May 1, he led more than 600 supporters on a march through the residential avenues of Malvern, and had planned another such demonstration for last weekend in Regent Park, a downtown Toronto housing project. The church has also started a basketball program at one of Malvern's high schools in hopes of keeping youths away from gangs, and is in the process of purchasing homes in the neighbourhood, which will become hallway houses for gang members emerging from prison. All three initiatives have been well-received, inside and outside Malvern, with onlookers applauding the church's activism at a time when secular agencies appear paralyzed by indecision.
In the meantime, Rhema can claim another kind of victory in the spiritual restoration of Patricia Fough. Last week, with her emotions in check and a determination to speak out, she swept gracefully into an anteroom at the church's sprawling facility, where in a lengthy interview she gave credit to her fellow congregants for rushing to her aid in the days and weeks following the shooting. There was a time, Fough acknowledged, when she wished to hide from the world, when the mere thought of Omar was enough to knock her flat. "But I know my son would not want me to shut myself away," she says. "I can almost hear him saying it: 'Mom, you've got to go on. I'm OK, and I'm always there with you. I'm not going to be there in flesh, but I'll always be there in spirit.' "
charlie.gillis@macleans.rogers.com
A CANNES SCRAPBOOK
June 2004 By Brian D. Johnson
MARTINIZING THE MESSAGE
LANG'S LOVE LETTER
June 2004 By LIANNE GEORGE
JAROME AND THE RED 'C'
June 2004 By JAMES DEACON
AFTER THE ARMY LEFT
HARPER'S ELECTION TO LOSE
June 2004 By Peter C. Newman
A STEEL CITY'S BLUES
June 2004 By Sault Ste. Marie

redpajama_set_name: RedPajamaCommonCrawl


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AU Optronics Absorbs Quanta Display
CIO	 Oct 2, 2006 8:00 am PDT
Taiwan's AU Optronics, the world's third-largest maker of liquid crystal display (LCD) screens, completed its acquisition of Quanta Display, an important step forward for much-needed consolidation in the industry.
Although users have benefited for years from intermittent oversupply in the LCD industry, which has helped keep prices low, companies in the industry have started to combine to shore up the supply situation. The deal between the two Taiwanese LCD panel makers has resulted in a larger, more viable company in a much better position to take on the two market leaders in South Korea.
AU Optronics paid for Quanta in shares of stock, a deal valued at approximately US$2.2 billion when it was announced in April.
The companies gained approval for the takeover from regulators and shareholders prior to closing the deal. It immediately turns AU Optronics into a more potent rival for South Korea's Samsung Electronics and LG.Philips LCD. Prior to the deal, AU ran a distant third to the two Korean companies, but now its output capacity and market share put it far closer to the companies, although it remains in third place.
K.Y. Lee will remain the chairman of AU Optronics, and H.B. Chen will remain the company's president. The company did not plan to lay off any workers after the deal. AU Optronics' global workforce increased to more than 42,000 from 29,000 before the acquisition. The LCD industry is growing so fast in Taiwan that companies have had to offer ever greater incentives to attract enough workers for their factories.
-Dan Nystedt, IDG News Service (Taipei Bureau)
Sony to Produce 3 Millionth Made-in-Japan LCD TV
LCD TV Monitor Panel Prices on the Rise
Sharp Develops Triple-View LCD
Check out our CIO News Alerts and Tech Informer pages for more updated news coverage.


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Bulgaria's business climate indicator unchanged m/m in August 2013
Written by The Sofia Globe staff on August 28, 2013 in Business - No comments
Bulgaria's business climate indicator remained unchanged in August 2013 compared with July, which had been the first month this year since February to see an increase.
This emerges from figures released by the National Statistical Institute on August 28.
In February 2013, the business climate indicator rose by 5.7 percentage points. But the following months all saw declines – 1.8 percentage points in March, 0.6 in April, 1.1 in May and 0.7 in June. July 2013 saw an increase of 1.4 percentage points.
This pattern appears to track the political turmoil of the year, from the "cost of living" protests in February, the resignation of the Borissov centre-right government that took effect in mid-March, and the May elections that resulted in the second-ranked Bulgarian Socialist Party coming to power.
The National Statistical Institute said that in August 2013, the overall business climate indicator kept the same level as in July, with a reduction among the indicators seen only in industry.
The composite indicator "business climate in industry" decreased by 4.7 percentage points in
comparison with July because of more moderate assessments and expectations among managers about the business situation of enterprises.
At the same time, present production activity was assessed as slightly improved, but expectations about the next three months were more reserved, according to the NSI.
"The main obstacles for business development in the branch remain the uncertain economic environment and insufficient domestic demand, pointed out respectively by 57.2 per cent and 45.1 per cent of the industrial entrepreneurs," the institute said.
(Photo: Vangelis Thomaidis)
Bulgaria's exports in January – November 2021 up 23.3% y/y
Unemployment in Bulgaria in November 2021 was 5% – Eurostat