text:
\section{\textnormal{Introduction}}
\end{center}

The zeta functions of Ruelle and Selberg are dynamical functions associated with the geodesic flow of a compact hyperbolic manifold $X$.
They can be represented by infinite products, which involve exponentials
of the lengths of the prime closed geodesics on $X$.

Our geometrical setting consists of
compact hyperbolic manifolds $X$ of odd dimension $d$, obtained as $X:=\Gamma\backslash \H^{d}$, where
$\H^{d}$ denotes the $d$-dimensional real hyperbolic space and
$\Gamma$ a discrete torsion-free subgroup of $\SO^{0}(d,1)$.
Here, we identify  $\H^{d}$ with $\SO^{0}(d,1)/ \SO(d)$.
We set $G:=\SO^{0}(d,1)$ and $K:=\SO(d)$. Let $G=KAN$ be the Iwasawa
decomposition of $G$ and $M$ the centralizer of $A$ in $K$.
Let $\widehat{M}$ be the set of the equivalent classes of unitary irreducible representations of $M$.

We denote by $[\gamma]$ the $\Gamma$-conjugacy class
of $\gamma\in\Gamma$. One can associate to  $[\gamma]\neq e$, a unique closed geodesic
$c_{\gamma}$. We consider the lengths $l(\gamma)$ of
the corresponding closed geodesics
$c_{\gamma}$.
We call the conjugacy class $[\gamma]$ prime if
there exist no $k\in\mathbb{N}$ with $k>1$ and $\gamma_{0}\in\Gamma$ such that $\gamma=\gamma_{0}^{k}$.
The prime geodesics can be viewed as the
geodesics of minimum length.

Let $\sigma\in\widehat M$.
We give here the definition of the zeta functions in terms of
finite dimensional representations $\chi$ of $\Gamma$.
The twisted Selberg zeta function $Z(s;\sigma,\chi)$ is defined by
\begin{equation*}
Z(s;\sigma,\chi):=\prod_{\substack{[\gamma]\neq e,\\ [\gamma]\prim}} \prod_{k=0}^{\infty}\det\Big(\Id-\big(\chi(\gamma)\otimes\sigma(m_\gamma)\otimes S^k(\Ad(m_\gamma a_\gamma)_{\overline{\mathfrak{n}}})\big)e^{-(s+|\rho|)l(\gamma)}\Big),
\end{equation*}
where $\overline{\mathfrak{n}}$ is the sum of the negative root spaces of the system $(\mathfrak{g},\mathfrak{a})$
and $S^k(\Ad(m_\gamma a_\gamma)_{\overline{\mathfrak{n}}})$ denotes the $k$-th
symmetric power of the adjoint map $\Ad(m_\gamma a_\gamma)$ restricted to $\overline{\mathfrak{n}}$.
$Z(s;\sigma,\chi)$ is defined for $s\in\mathbb{C}$ with Re$(s)>c_{1}$, for some constant $c_{1}>0$.

We define also the twisted Ruelle zeta function $R(s;\sigma,\chi)$ by
\begin{equation*}
 R(s;\sigma,\chi):=\prod_{\substack{[\gamma]\neq{e}\\ [\gamma]\prim}}\det(\Id-\chi(\gamma)\otimes\sigma(m_{\gamma})e^{-sl(\gamma)})^{(-1)^{d-1}},
\end{equation*}
where $s\in\mathbb{C}$ with Re$(s)>c_{2}$, for some constant $c_{2}>0$.

The meromorphic continuation of the Ruelle and Selberg zeta function has been long studied by Fried (\cite{Fried}), Bunke and Olbrich (\cite{BO}), Wotzke (\cite{Wo}) and Pfaff
(\cite{Pf}), but only for
specific representations of the subgroup $\Gamma$.
Under certain assumptions for the representation of $\Gamma$ i.e., the representation is orthogonal and acyclic,
Fried (\cite[Theorem 1]{Fried}) managed to prove that the Ruelle zeta function admits a meromorphic continuation to the whole complex plane and moreover that
is regular at zero and its absolute value at zero is equal to the Ray-Singer analytic torsion.
Bunke and Olbrich (\cite{BO}) dealt with the meromorphic continuation of the dynamical zeta functions but in the case of a unitary representation
of $\Gamma$. Wotzke (\cite{Wo}) generalized in his thesis the results of Fried to the case of the induced representations of $\Gamma$, i.e., representations
arising from restrictions of finite dimensional representations of $G=\SO^{0}(d,1)$.
Pfaff (\cite{Pf}) extended these results to the co-finite case using the induced representation as well.

In \cite{Spil}, the existing results for the meromorphic continuation of the dynamical zeta function are generalized to the case of an arbitrary representation of $\Gamma$,
not necessarily unitary.
In this paper, we study further this case, under the condition that the action of the restricted Weyl group $W_{A}$ on $\widehat{M}$
is not trivial, i.e., $w\sigma\neq\sigma$, where $w$
is a non-trivial element of $W_{A}$.
We have then to introduce three more zeta functions the symmterized $S(s,\sigma,\chi)$, super $Z^{s}(s,\sigma,\chi)$ and super Ruelle $R^{s}(s\,\sigma,\chi)$ zeta function
(see Definitions 3.3, 3.4 and 3.5). To be able to prove the meromorphic continuation of the zeta functions, we introduce the twisted Dirac operator, defined in a
similar way as the twisted Bochner-Laplace operator in \cite{M1}.

We give here a short description of the twisted Dirac operator. As before, we consider an arbitrary
finite dimensional representation
$\chi$ of $\Gamma$. Let $E_{\chi}$ be the flat vector bundle over $X$.
Let $\widehat{K}$ be the set of the equivalent classes of unitary irreducible representations of $K$.
Let  $s$ be the spin representation of $K$ and $\tau(\sigma)\in\widehat{K}$.
We define the representation $\tau_{s}(\sigma)$ of $K$ by $\tau_{s}(\sigma):= s\otimes\tau(\sigma)$.
We consider then the  locally homogeneous vector bundle $E_{\tau_{s}(\sigma)}$ over $X$.
We let $D(\sigma)$ be the Dirac operator associated with $\tau_{s}(\sigma)$.
Let $D^{\sharp}_{\chi}(\sigma)$ be the twisted Dirac operator acting on $C^{\infty}(X,E_{\tau_{s}(\sigma)}\otimes E_{\chi})$.
Locally, it is given by the following formula.
\begin{equation*}
\widetilde{D}^{\sharp}_{\chi}(\sigma)=\widetilde{D}(\sigma)\otimes \Id_{V_{\chi}},
\end{equation*}
where $\widetilde{D}^{\sharp}_{\chi}(\sigma)$ and $\widetilde{D}(\sigma)$
are the lifts to $\widetilde{X}$ of
$D^{\sharp}_{\chi}(\sigma)$ and $D(\sigma)$, respectively.
Since the representation of $\Gamma$
is not necessarily unitary, we deal with non self-adjoint operators.
Nevertheless, one can observe by the formula above that the twisted Dirac operator $D^{\sharp}_{\chi}(\sigma)$
has the same principal symbol as ${D}(\sigma)$.
Hence, it has nice spectral properties, i.e., its the spectrum is a discrete subset of a positive cone in $\mathbb{C}$.
Let $D^{\sharp}_{\chi}(\sigma)e^{-tD^{\sharp}_{\chi}(\sigma)}$ be the associated operator acting on the space
of smooth sections of the vector bundle $E_{\tau_{s}(\sigma)}\otimes E_{\chi}$.
By \cite[Lemma 2.4]{M1}, this is an integral operator with smooth kernel and hence of trace class.
We derive a corresponding trace formula for $D^{\sharp}_{\chi}(\sigma)e^{-tD^{\sharp}_{\chi}(\sigma)}$.
\newline
\begin{thm}
 For every $\sigma \in \widehat{M}$,
 \begin{equation*}
 \Tr(D^{\sharp}_{\chi}(\sigma)e^{-t(D^{\sharp}_{\chi}(\sigma))^{2}})=
  \sum_{[\gamma]\neq e}\frac{-2\pi i}{(4\pi t)^{3/2}}\frac{l^{2}(\gamma)\tr(\chi(\gamma)\otimes(\sigma(m_{\gamma})-w\sigma(m_{\gamma})))}
  {n_{\Gamma}(\gamma)D(\gamma)}e^{-l^{2}(\gamma)/4t}.
 \end{equation*}
\end{thm}
This trace formula is the key point to prove the meromorphic continuation of the super zeta function.
\begin{thm}
The super zeta function $Z^{s}(s;\sigma,\chi)$ admits a meromorphic continuation to the whole complex plane $\mathbb{C}$. The singularities
are located at $\{s_{k}^{\pm}=\pm i\lambda_{k}:\lambda_{k}\in \spec(D^{\sharp}_{\chi}(\sigma)), k\in\mathbb{N}\}$
of order  $\pm m_{s}(\lambda_{k})$, where $m_{s}(\lambda_{k})=m(\lambda_{k})-m(-\lambda_{k})\in\mathbb{N}$ and
$m(\lambda_{k})$ denotes the algebraic multiplicity of the eigenvalue $\lambda_{k}$.
\end{thm}
In \cite{Spil}, the differential operator $A^{\sharp}_{\chi}(\sigma)$, which is induced by the twisted Bochner-Laplace operator $\Delta^{\sharp}_{\tau,\chi}$, is introduced.
Moreover, a trace formula for the corresponding heat semi-group $e^{-tA^{\sharp}_{\chi}(\sigma)}$ is proved. In the present case, where $w \sigma\neq \sigma$,
we derive a trace formula for for the operator $e^{-tA^{\sharp}_{\chi}(\sigma)}$, which
is  slightly different.

\begin{thm}
 For every $\sigma \in \widehat{M}$ we have
   \begin{align*}\label{f:trace formula 2}
  \Tr(e^{-tA_{\chi}^{\sharp}(\sigma)})=&2\dim(V_{\chi})\Vol(X)\int_{\mathbb{R}}e^{-t\lambda^{2}}P_{\sigma}(i\lambda)d\lambda\\
  &+\sum_{[\gamma]\neq e}\frac{l(\gamma)}{n_{\Gamma}(\gamma)}L_{sym}(\gamma;\sigma+w\sigma)
  \frac{e^{-l(\gamma)^{{2}}/4t}}{(4\pi t)^{1/2}},
 \end{align*}
where
\begin{equation*}
 L_{sym}(\gamma;\sigma)= \frac{\tr(\sigma(m_{\gamma})\otimes\chi(\gamma))e^{-|\rho|l(\gamma)}}{\det(\Id-\Ad(m_{\gamma}a_{\gamma})_{\overline{\mathfrak{n}})}}.
\end{equation*}
\end{thm}
The trace formula in Theorem 1.3 is again the tool which we use to prove
the meromorphic continuation of the symmetrized zeta function.
\begin{thm}
The symmetrized zeta function $S(s;\sigma,\chi)$ admits a meromorphic continuation to the whole complex plane $\mathbb{C}$. The set of the singularities
equals $\{s_{k}^{\pm}=\pm i{\sqrt{\mu_{k}}}:\mu_{k}\in \spec(A^{\sharp}_{\chi}(\sigma)), k\in\mathbb{N}\}$.
The orders of the singularities are equal to $m(\mu_{k})$, where $m(\mu_{k})\in\mathbb{N}$ denotes the algebraic multiplicity of the eigenvalue $\mu_{k}$.
For $\mu_{0}=0$, the order of the singularity $s_{0}$ is equal to $2m(0)$.
\end{thm}
Finally, we prove the following theorems.
\begin{thm}
The Selberg zeta function $Z(s;\sigma,\chi)$ admits a meromorphic continuation to the whole complex plane $\mathbb{C}$. The set of the singularities
equals to $\{s_{k}^{\pm}=\pm i\lambda_{k}:\lambda_{k}\in \spec(D^{\sharp}_{\chi}(\sigma)),k\in\mathbb{N}\}$.
The orders of the singularities are equal to $\frac{1}{2}(\pm m_{s}(\lambda_{k})+m(\lambda^{2}_{k}))$.
For $\lambda_{0}=0$, the order of the singularity is equal to $m(0)$.
\end{thm}
\begin{thm}
For every $\sigma\in\widehat{M}$, the Ruelle zeta function $R(s;\sigma,\chi)$ admits a meromorphic continuation to the whole complex plane $\mathbb{C}$.
\end{thm}
\textbf{Acknowledgment.} The author wishes
to acknowledge the contribution of her supervisor Werner M\"{u}ller,
as the present paper is part of the author's PhD thesis.

\begin{center}
\section{\textnormal{Preliminaries}}
\end{center}

Let $X$ be a compact hyperbolic locally symmetric manifold with universal
covering the real hyperbolic space $\H^{d}$,
where $d=2n+1$ is an odd integer.
We let $G:=\Spin(d,1)$, $K:=\Spin(d)$ be the universal coverings of $\mathrm{SO}^0(d,1)$ and $\mathrm{SO}(d)$, respectively.
We put  $\widetilde{X}:=G/K$.
We consider the Lie algebras $\mathfrak{g},\mathfrak{k}$ of $G$ and $K$, respectively. Let $ \mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$
be the Cartan decomposition of $\mathfrak{g}$.
Let $\mathfrak{a}$ be a Cartan subalgebra of $\mathfrak{p}$.
Then, there exists a canonical isomorphism $T_{eK}\cong \mathfrak{p}$.
We consider the subgroup $A$ of $G$ with Lie algebra $\mathfrak{a}$. We let $M:=\cent_{K}(A)$ be the centralizer of $A$ in $K$.
Then, $M=\Spin(d-1)$ or $M=\SO(d-1)$. Let $\mathfrak{m}$ be its Lie algebra
and $\mathfrak{b}$ a Cartan subalgebra of $\mathfrak{m}$. Let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$.
Let $\mathfrak{g}_{\mathbb{C}}$, $\mathfrak{h}_{\mathbb{C}}$, $\mathfrak{m}_{\mathbb{C}} $ be the complexifications of $\mathfrak{g}$, $\mathfrak{h}$
and $\mathfrak{m}$, respectively.
We consider the inner product
\begin{equation}
 \langle Y_1, Y_2\rangle_{0}:=\frac{1}{2(d-1)}B(Y_1,Y_2),\quad Y_1,Y_2 \in \mathfrak{g},
\end{equation}
induced by the Killing form $B(Y_{1},Y_{2}):=\Tr (\ad(Y_{1})\circ \ad(Y_{2}))$ on $\mathfrak{g}\times \mathfrak{g}$.
The restriction of $\langle\cdot,\cdot\rangle_{0}$ to $\mathfrak{p}$ defines
an inner product on $\mathfrak{p}$ and
hence induces a $G$-invariant riemannian metric on $\widetilde{X}$, which
has constant curvature $-1$. Then, $\widetilde{X}$, equipped with this metric,
is isometric to $\H^{d}$.
Let $\Gamma\subset G$ be a discrete cocompact torsion free subgroup of $G$. Then,
$X:=\Gamma\backslash \widetilde{X}$
is a locally symmetric compact
hyperbolic manifold of dimension $d$.

Let $G=KAN$
be the standard Iwasawa decomposition of $G$.
Let $\Delta^{+}(\mathfrak{g},\mathfrak{a})$ be the set of positive roots of the system $(\mathfrak{g},\mathfrak{a})$.
Then, $\Delta^{+}(\mathfrak{g},\mathfrak{a})=\{\alpha\}$.
Let $H_{\mathbb{R}}\in\mathfrak{a}$ such that $\alpha(H_{\mathbb{R}})=1$.
We set
\begin{equation}
 A^{+}:=\{\exp(tH_{\mathbb{R}})\colon t\in\mathbb{R}^{+}\},
\end{equation}
\begin{align}
 &\rho:=\frac{1}{2}\sum_{\alpha\in \Delta^+(\mathfrak{g},\mathfrak{a})}\dim(\mathfrak{g}_\alpha)\alpha,\\
&\rho_{\mathfrak{m}}:=\frac{1}{2}\sum_{\alpha\in \Delta^+(\mathfrak{m}_{\mathbb{C}},\mathfrak{b})}\alpha.\\\notag
\end{align}
Let $\widehat{K},\widehat{M}$ be the sets of equivalent classes of
irreducible unitary representations of $K$ and $M$, respectively.
Let $\nu_{\tau}$, $\nu_{\sigma}$ be the highest weights of $\tau\in \widehat{K}$ and $\sigma\in \widehat{M}$,
respectively.
Then, by  \cite[p. 20]{BO}
\begin{align*}
  \nu_{\tau}=(\nu_{1},\ldots,\nu_{n}),
\end{align*}
where $\nu_{1}\geq\ldots\geq\nu_{n}$ and $\nu_{i},i=1,\ldots,n$ are all integers or all half integers
and
\begin{align}
  \nu_{\sigma}=(\nu_{1},\ldots,\nu_{n-1},\nu_{n}),
\end{align}
where $\nu_{1}\geq\ldots\geq\nu_{n-1}\geq\lvert\nu_{n}\rvert$ and $\nu_{i},i=1,\ldots,n$ are all integers or all half integers .

Let $(s,S)$ be the spin representation of $K$ and $(s^{\pm},S^{\pm})$ the half spin representations of $M$ as in \cite[p. 7]{Spil}.
By \cite[p. 20]{BO},
the highest weight $ \nu_{s}$ of $s$ is given by $=(\frac{1}{2},\ldots,\frac{1}{2})$,
and the highest weights $\nu_{s^{\pm}}$ of $s^{\pm}$ are
$\nu_{s^{\pm}}=(\frac{1}{2},\ldots,\pm\frac{1}{2})$.
\newpage

\begin{center}
{\section{\textnormal{Twisted Selberg and Ruelle zeta functions}}}
\end{center}
The twisted Ruelle and Selberg zeta functions are associated with the geodesic flow on the
sphere vector bundle $S(X)=\Gamma\backslash G/M$ of $X$.
By (\cite{GKM}), there is a 1-1 correspondence between the closed geodesics on a manifold $X$ with negative sectional curvature
and the non-trivial conjugacy classes of $\Gamma$.
Since $\Gamma$ is a cocompact subgroup of $G$, we realize every element $\gamma\in\Gamma-\{e\}$ as hyperbolic.
By \cite[Lemma 6.5]{Wa}, there exist a $g\in G$, a $m_{\gamma} \in M$, and an
$a_{\gamma} \in A^{+}$, such that $ g^{-1}\gamma g=m_{\gamma}a_{\gamma}$.
The element $m_{\gamma}$ is determined up to conjugacy classes in $M$, and the element $a_{\gamma}$ depends only on $\gamma$.

Let $c_{\gamma}$ be the closed geodesic on $X$, associated with the conjugacy class $[\gamma]$.
We denote by $l(\gamma)$ the length of $c_{\gamma}$.
An element $\gamma\in\Gamma$ is primitive, if there exists no $n\in\mathbb{N}$ with $n>1$ and $\gamma_{0}\in\Gamma$ such that $\gamma=\gamma_{0}^{n}$.
The prime geodesics on $X$ are those geodesics whose periodic orbits are of minimal length.
They are associated with the primitive elements $\gamma_{0}\in\Gamma$.

Let now $\chi\colon\Gamma\rightarrow \GL(V_{\chi})$ be an arbitrary finite dimensional representation of $\Gamma$ and $\sigma\in \widehat{M}$.

\begin{defi}
The twisted Selberg zeta function $Z(s;\sigma,\chi)$ for
$X$ is defined  by the infinite product
\begin{equation}
Z(s;\sigma,\chi):=\prod_{\substack{[\gamma]\neq{e}\\ [\gamma]\prim}} \prod_{k=0}^{\infty}\det\big(\Id-(\chi(\gamma)\otimes\sigma(m_\gamma)\otimes S^k(\Ad(m_\gamma a_\gamma)|_{\overline{\mathfrak{n}}})) e^{-(s+|\rho|)\lvert l(\gamma)}\big),
\end{equation}
where $s\in \mathbb{C}$, $\overline{\mathfrak{n}}=\theta \mathfrak{n}$ is the sum of the negative root spaces of $\mathfrak{a}$,
$S^k(\Ad(m_\gamma a_\gamma)_{\overline{\mathfrak{n}}})$ denotes the $k$-th
symmetric power of the adjoint map $\Ad(m_\gamma a_\gamma)$ restricted to $\mathfrak{\overline{n}}$, and $\rho$ is as in (2.3).
\end{defi}
By \cite[Proposition 3.4]{Spil}, there exists a positive constant $c$, such that the infinite product in (3.1) converges
absolutely and uniformly on compact subsets of the half-plane Re$(s)>c$.
\begin{defi}
The twisted Ruelle zeta function $ R(s;\sigma,\chi)$ for $X$ is defined by the infinite product
\begin{equation}
 R(s;\sigma,\chi):=\prod_{\substack{[\gamma]\neq{e}\\ [\gamma]\prim}}\det\big(\Id-\chi(\gamma)\otimes\sigma(m_{\gamma})e^{-sl(\gamma)}\big)^{(-1)^{d-1}}.
\end{equation}
\end{defi}
By \cite[Proposition 3.5]{Spil}, there exists a positive constant $r$, such that the infinite product in (3.2) converges
absolutely and uniformly on compact subsets of the half-plane Re$(s)>r$.

Let  $M'=\Norm_{K}(A)$ be the normalizer of $A$ in $K$.
We define the restricted Weyl group by $ W_{A}:=M'/M$.
Then, $W_{A}$ has order 2. Let $w\in W_{A}$ be a non-trivial element of $W_{A}$, and
$m_{w}$ a representative of $w$ in $M'$.
The action of $W_{A}$ on $\widehat{M}$ is defined by
\begin{equation*}
(w\sigma)(m):=\sigma(m_{w}^{-1}mm_{w}),\quad m\in M, \sigma\in\widehat{M}.
\end{equation*}
We have already associated the Selberg and Ruelle zeta functions with irreducible representations $\sigma$ of $M$. These representations are chosen precisely to be
the representations arising from restrictions of representations of $K$.
Let $i^{*}:R(K)\rightarrow R(M)$ be the pullback of the embedding
$i:M\hookrightarrow K$.
We will distinguish the following two cases:
\begin{itemize}
 \item  {\bf case (a)}: \textit{$\sigma$ is invariant under the action of the restricted Weyl group $W_{A}$.}
 \item {\bf case (b)}: \textit{$\sigma$ is not invariant under the action of the restricted Weyl group $W_{A}$.}
\end{itemize}
In case (b), we define the following twisted zeta functions.
\begin{defi}
Let $\chi\colon\Gamma\rightarrow \GL(V_{\chi})$ be a finite dimensional representation of $\Gamma$ and $\sigma\in \widehat{M}$.
The symmetrized zeta function $Z(s;\sigma,\chi)$ for
$X$ is defined  by
 \begin{equation}
 S(s;\sigma,\chi):=Z(s;\sigma,\chi)Z(ws;\sigma,\chi),
\end{equation}
where $w$ is a non-trivial element of the restricted Weyl group $W_{A}$.
\end{defi}
\begin{defi}
Let $\chi\colon\Gamma\rightarrow \GL(V_{\chi})$ be a finite dimensional representation of $\Gamma$ and $\sigma\in \widehat{M}$.
The super zeta function $Z(s;\sigma,\chi)$ for
$X$ is defined  by
 \begin{equation}
 Z^{s}(s;\sigma,\chi):=\frac{Z(s;\sigma,\chi)}{Z(ws;\sigma,\chi)},
\end{equation}
where $w$ is a non-trivial element of the restricted Weyl group $W_{A}$.
\end{defi}
\begin{defi}
Let $\chi\colon\Gamma\rightarrow \GL(V_{\chi})$ be a finite dimensional representation of $\Gamma$ and $\sigma\in \widehat{M}$.
The super Ruelle zeta function $R^{s}(s;\sigma,\chi)$ for
$X$ is defined  by
\begin{equation}
 R^{s}(s;\sigma,\chi):=\frac{R(s;\sigma,\chi)}{R(s;w\sigma,\chi)},
\end{equation}
where $w$ is a non-trivial element of the restricted Weyl group $W_{A}$.
\end{defi}
We compute here the logarithmic derivative of the symmetrized and super zeta function.
\begin{lem}
Let
\begin{equation}
 L_{sym}(\gamma;\sigma):=\frac{\tr(\chi(\gamma)\otimes\sigma(m_{\gamma}))e^{-|\rho|l(\gamma)}}
 {\det{(\Id-\Ad(m_{\gamma}a_{\gamma})_{\overline{n}})}}.
\end{equation}
Then we have
\begin{enumerate}
\item
 The logarithmic derivative of the symmetrized zeta function $S(s;\sigma,\chi)$ is given by
 \begin{equation}\label{f:log der symmetrized}
 L_{S}(s):=\frac{d}{ds}\log(S(s;\sigma,\chi))=\sum_{[\gamma]\neq{e}}\frac{l(\gamma)}{n_{\Gamma}(\gamma)}L_{sym}(\gamma;\sigma+w\sigma)
e^{-sl(\gamma)}.
\end{equation}
\item
 The logarithmic derivative of the super zeta function $Z^{s}(s;\sigma,\chi)$ is given by
 \begin{equation}\label{f:log der super}
 L^{s}(s):=\frac{d}{ds}\log(Z^{s}(s;\sigma,\chi))=\sum_{[\gamma]\neq{e}}\frac{l(\gamma)}{n_{\Gamma}(\gamma)}L_{sym}(\gamma;\sigma-w\sigma)
e^{-sl(\gamma)}.
\end{equation}
\end{enumerate}
\end{lem}

\begin{proof}
\begin{enumerate}
 \item In case (b), for the symmetrized zeta function $S(s;\sigma,\chi)$, we see by equation (3.3)
\begin{align*}
\frac{d}{ds}\log(S(s;\sigma,\chi))&=\frac{d}{ds}\log(Z(s;\sigma,\chi)+\frac{d}{ds}\log(Z(s;w\sigma,\chi)\\
 &=\sum_{[\gamma]\neq{e}}\frac{l(\gamma)}{n_{\Gamma}(\gamma)}\tr(\chi(\gamma)\otimes\sigma(m_\gamma))\frac{e^{-sl(\gamma)}e^{-|\rho|l(\gamma)}}{\det(1-\Ad(m_\gamma a_\gamma)_{\overline{\mathfrak{n}}})}\\
 &+\sum_{[\gamma]\neq{e}}\frac{l(\gamma)}{n_{\Gamma}(\gamma)}\tr(\chi(\gamma)\otimes w\sigma(m_\gamma))\frac{e^{-sl(\gamma)}e^{-|\rho|l(\gamma)}}{\det(1-\Ad(m_\gamma a_\gamma)_{\overline{\mathfrak{n}}})}\\
 &=\sum_{[\gamma]\neq{e}}\frac{l(\gamma)}{n_{\Gamma}(\gamma)}L_{sym}(\gamma;\sigma+w\sigma)e^{-sl(\gamma)}.
\end{align*}

\item In case (b), for the super zeta function $Z^{s}(s;\sigma,\chi)$, we see by equation (3.4)
\begin{align*}
\frac{d}{ds}\log(Z^{s}(s;\sigma,\chi))&=\frac{d}{ds}\log(Z(s;\sigma,\chi)-\frac{d}{ds}\log(Z(s;w\sigma,\chi)\\
 &=\sum_{[\gamma]\neq{e}}\frac{l(\gamma)}{n_{\Gamma}(\gamma)}\tr(\chi(\gamma)\otimes\sigma(m_\gamma))\frac{e^{-sl(\gamma)}e^{|\rho|l(\gamma)}}{\det(1-\Ad(m_\gamma a_\gamma)_{\overline{\mathfrak{n}}})}\\
 &-\sum_{[\gamma]\neq{e}}\frac{l(\gamma)}{n_{\Gamma}(\gamma)}\tr(\chi(\gamma)\otimes w\sigma(m_\gamma))\frac{e^{-sl(\gamma)}e^{|\rho|l(\gamma)}}{\det(1-\Ad(m_\gamma a_\gamma)_{\overline{\mathfrak{n}}})}\\
 &=\sum_{[\gamma]\neq{e}}\frac{l(\gamma)}{n_{\Gamma}(\gamma)}L_{sym}(\gamma;\sigma-w\sigma)e^{-sl(\gamma)}.
\end{align*}
\end{enumerate}
\end{proof}

\begin{center}
\section{\textmd{The twisted Dirac operator}}
\end{center}

Let $\sigma\in\widehat{M}$ be an irreducible representation of $M$ with highest weight $\nu_{\sigma}$ as in (2.5).
We recall that $\nu_{n}$ denotes the last coordinate of  $\nu_{\sigma}$.
Let $s$ be the spin representation of $K$. Since $d-1$ is an even integer, $s$ splits into two irreducible half-spin representations $(s^{+},S^{+})$, $(s^{-},S^{-})$ of $M$.
Let $\Cl(\mathfrak{p})$ be the Clifford algebra of $\mathfrak{p}$ with respect to the inner product $\langle\cdot,\cdot\rangle_{0}$, as in (2.1), restricted to $\mathfrak{p}$.
Let $
 \cdot\colon\mathfrak{p}\otimes S\rightarrow S
$
be the Clifford multiplication on $\mathfrak{p}\otimes S$.
Let $H$ be in $\mathfrak{a}^{+}$, where $\mathfrak{a}^{+}$ is the Lie algebra of $A^{+}$ and $A^{+}$ is defined as in (2.2). Since $M$ centralizes $\mathfrak{a}$,
the Clifford multiplication by $H$ preserves the decomposition $S=S^{+}\oplus S^{-}$. The Clifford multiplication by $H^{2}$ acts as $-\Id$. Then, $H$
acts on $S^{\pm}$ with eigenvalues $\pm i$. Hence, we can consider the Clifford multiplication by $H$
as multiplication by $\pm i \sign(\nu_{n})$.

We consider the connection $\nabla$ in $\Cl(\mathfrak{p})$, induced by the canonical connection in the tangent frame bundle of $X$.
Let $L$ be any bundle of left modules over $\Cl(\mathfrak{p})$ over $\widetilde{X}$, i.e., a spinor bundle over $\widetilde{X}$. We lift the connection $\nabla$ in $L$ and obtain a connection also denoted by $\nabla$.
The Dirac operator $D\colon C^{\infty}(X,L)\rightarrow C^{\infty}(X,L)$  is defined as
\begin{equation*}
 D:C^{\infty}(X,L)\overset{\nabla}\rightarrow
 C^{\infty}(X,T^{*}X\otimes L)\overset{g}\rightarrow C^{\infty}(X,TX\otimes L)
\overset{\cdot}\rightarrow C^{\infty}(X,L),
\end{equation*}
where we identify $TX\cong T^{*}X$ using the riemannian metric, and
$\cdot$ denotes the Clifford multiplication as above.
Locally, it can be described as
\begin{equation*}
 Df\equiv\sum_{i=1}^{d}e_{i}\cdotp\nabla_{e_{i}}f,
\end{equation*}
where $(e_{1},\ldots,e_{d})$ is a local orthonormal frame for $T_{x}X, x\in X$.
The bundle $L$ is a Dirac bundle over $\widetilde{X}$. This means that
\begin{itemize}
 \item the Clifford multiplication by unit vectors in $\Cl(\mathfrak{p})$ is orthogonal i.e.,
\begin{equation*}
 \langle e f_{1},e f_{2}\rangle=\langle f_{1},f_{2}\rangle,
\end{equation*}
for all unit vectors $e\in T_{x}\widetilde{X}, x\in\widetilde{X}$ and all $f_{1},f_{2}\in L_{x}$, where $L_{x}$ denotes the fiber of $L$ over $x\in \widetilde{X}$.
\item the connection $\nabla$ satisfies the product rule
\begin{equation*}
 \nabla(\phi f)=(\nabla\phi)\cdotp f+ \phi\cdotp(\nabla f),
\end{equation*}
for all $\phi\in C^\infty(X,\Cl(\mathfrak{p}))$ and all $f\in C^{\infty}(X,L)$.
\end{itemize}
The operator $D$ is an elliptic (\cite[Lemma 5.1]{LM}), formally self-adjoint (\cite[Proposition 5.3]{LM}) operator of first order.
We want to define twisted Dirac operators acting on smooth sections of vector bundles associated with the representations $\sigma$ of $M$
and arbitrary represenations $\chi$ of $\Gamma$.
\begin{prop}
Let $\sigma\in\widehat{M}$. Then, there exists a unique element  $\tau(\sigma)\in \widehat{K}$ and a splitting
\begin{equation*}
 s \otimes \tau(\sigma)=\tau^{+}(\sigma)\oplus \tau^{-}(\sigma)
\end{equation*}
where $\tau^{+}(\sigma),\tau^{-}(\sigma)\in R(K)$
such that
\begin{equation}
   \sigma+w\sigma=i^{*}(\tau^{+}(\sigma)-\tau^{-}(\sigma))
\end{equation}
\end{prop}
\begin{proof}
 This is proved in \cite[Proposition 1.1, (3)]{BO}.
\end{proof}
We define the representation $\tau_{s}(\sigma)$ of $K$ by
\begin{equation}
 \tau_{s}(\sigma):= s\otimes\tau(\sigma),
\end{equation}
with representation space $V_{\tau_{s}(\sigma)}=S\otimes V_{\tau(\sigma)}$,
where $V_{\tau(\sigma)}$ is the representation space of $\tau(\sigma)$.\\
We consider the homogeneous vector bundle $\widetilde{E}_{\tau(\sigma)}$ over $\widetilde{X}$ given by
\begin{equation*}
 \widetilde{E}_{\tau(\sigma)}=G\times_{\tau(\sigma)}V_{\tau(\sigma)}\rightarrow\widetilde{X}.
\end{equation*}
The vector bundle $\widetilde{E}_{\tau_{s}(\sigma)}:= \widetilde{E}_{\tau(\sigma)}\otimes S$ over $\widetilde{X}$ carries a connection $\nabla^{\tau_{s}(\sigma)}$, defined by the formula
\begin{equation*}
 \nabla^{\tau_{s}(\sigma)}=\nabla^{\tau(\sigma)}\otimes 1+1\otimes\nabla.
\end{equation*}
where $\nabla^{\tau(\sigma)}$ denotes the canonical connection in  $\widetilde{E}_{\tau(\sigma)}$.
We extend the Clifford multiplication by requiring that it acts on $V_{\tau_{s}(\sigma)}=S\otimes V_{\tau(\sigma)}$ as follows.
\begin{equation*}
  e\cdot (\phi\otimes \psi)=(e\cdot \phi)\otimes \psi, \quad e\in\Cl(\mathfrak{p}), \phi\in S, \psi\in V_{\tau(\sigma)}.
\end{equation*}

We define the Dirac operator $\widetilde{D}(\sigma)$ acting on $C^{\infty}(\widetilde{X},V_{\tau_{s}(\sigma)})$ by
\begin{equation*}
 \widetilde{D} (\sigma)f=\sum_{i=1}^{d}e_{i}\cdot \nabla_{e_{i}}^{\tau_{s}(\sigma)}f,
\end{equation*}
where $(e_{1},\ldots,e_{d})$ is local orthonormal frame for $T_{x}{\widetilde{X}}$ and $f\in C^{\infty}(\widetilde{X},V_{\tau_{s}(\sigma)})$.
The space of smooth sections $C^{\infty}(\widetilde{X},V_{\tau_{s}(\sigma)})$ can be identified with $C^{\infty}(G;\tau_{s}(\sigma))$ as in \cite[equation (5.1)]{Spil}.\\

Let now $\chi:\Gamma\rightarrow \GL(V_{\chi})$ be an arbitrary finite dimensional representation of $\Gamma$. Let $E_{\chi}$
be the associated flat vector bundle over $X$.
Let $E_{\tau_{s}(\sigma)}:= \Gamma\backslash \widetilde{E}_{\tau_{s}(\sigma)}$ be the locally homogeneous vector bundle over $X$.
We consider the product vector bundle $E_{\tau_{s}(\sigma)}\otimes E_{\chi}$ over $X$
and we equip this bundle with the product connection $\nabla^{E_{\tau_{s}(\sigma)}\otimes E_{\chi}}$ defined by
\begin{equation*}
 \nabla^{E_{\tau_{s}(\sigma)}\otimes E_{\chi}}=\nabla^{E_{\tau_{s}(\sigma)}}\otimes 1+1\otimes\nabla^{E_{\chi}}.
\end{equation*}
We consider the Clifford multiplication on $(V_{\tau_{s}(\sigma)}\otimes V_{\chi})$ by requiring that it acts only on $V_{\tau_{s}(\sigma)}$, i.e.,
\begin{equation*}
 e\cdot (w\otimes v)=(e\cdot w)\otimes v, \quad e \in\Cl(\mathfrak{p}), w\in V_{\tau_{s}(\sigma)}, v\in V_{\chi}.
\end{equation*}

For our proposal, we introduce the twisted Dirac operator $D^{\sharp}_{\chi}(\sigma)$ associated with $ \nabla^{E_{\tau_{s}(\sigma)}\otimes E_{\chi}}$.
We want to describe it locally. We consider an open subset of $X$ such that $E_{\chi}\lvert_{U}$ is trivial. Let $(v_{j}), j=1,\dots,m$,
be a basis of flat sections of $E_{\chi}\lvert_{U}$, where $m=\rank(E_\chi)$, and $\phi_{j}\in C^{\infty}(U, E_{\tau_{s}(\sigma)}\lvert_{U})$.
Then,
\begin{equation*}
 E_{\tau_{s}(\sigma)}\otimes E_{\chi}\lvert_{U}\cong\bigoplus_{j=1}^{m}E_{\tau_{s}(\sigma)}\lvert_{U},
\end{equation*}
and for each $\phi \in C^{\infty}(U,E_{\tau_{s}(\sigma)} \otimes E_{\chi}\lvert_{U})$,
\begin{equation*}
\phi=\sum_{j=1}^{m}\phi_{j} \otimes v_{j}.
\end{equation*}
The product connection is given by
\begin{equation*}
  \nabla^{E_{\tau_{s}(\sigma)}\otimes E_{\chi}}(\phi)=\sum_{j=1}^{m}(\nabla^{E_{\tau_{s}(\sigma)}})(\phi_{j})\otimes v_{j}.
\end{equation*}
Then the Dirac operator is described as follows.
\begin{align}
D^{\sharp}_{\chi}(\sigma)\phi\notag&=\sum_{i=1}^{d}e_{i}\cdot\nabla_{e_{i}}^{E_{\tau_{s}(\sigma)}\otimes E_{\chi}}(\phi)\\\notag
&=\sum_{i=1}^{d}e_{i}\cdot\big(\sum_{j=1}^{m}(\nabla_{e_{i}}^{{E}_{\tau_{s}(\sigma)}})(\phi_{j})\otimes v_{j}\big)\\
&=\sum_{i=1}^{d} \sum_{j=1}^{m}e_{i}\cdot\big((\nabla_{e_{i}}^{{E}_{\tau_{s}(\sigma)}})(\phi_{j})\otimes v_{j}\big).
\end{align}
We consider the pullbacks $\widetilde{E}_{\tau_{s}(\sigma)}, \widetilde{E}_{\chi}$ to $\widetilde{X}$ of $E_{\tau_{s}(\sigma)},E_{\chi}$, respectively,
then, $\widetilde{E}_{\chi}\cong \widetilde{X}\times V_{\chi}$.
We have
\begin{equation*}
 C(\widetilde{X}, \widetilde{E}_{\tau_{s}(\sigma)}\otimes \widetilde{E}_{\chi})\cong  C(\widetilde{X}, \widetilde{E}_{\tau_{s}(\sigma)})\otimes V_{\chi}.
\end{equation*}
With respect to this isomorphism, it follows from (4.3) that the lift ${\widetilde{D}}^{\sharp}_{\chi}(\sigma)$ of the twisted Dirac operator $D^{\sharp}_{\chi}(\sigma)$
to $\widetilde{X}$ is of the form
\begin{equation}
\widetilde{D}^{\sharp}_{\chi}(\sigma)=\widetilde{D}(\sigma)\otimes \Id_{V_{\chi}}.
\end{equation}
We recall the definition of the operator $A_{\chi}^{\sharp}(\sigma)$ acting on $C^{\infty}(X,E(\sigma)\otimes E_{\chi})$ from \cite[equation 5.26]{Spil}
\begin{equation*}
 A_{\chi}^{\sharp}(\sigma):=\bigoplus_{m_{\tau}(\sigma)\neq 0}A^{\sharp}_{\tau,\chi}+c(\sigma),
\end{equation*}
where
\begin{equation*}
 c(\sigma):=-\lvert \rho \rvert^{2}-\lvert\rho_{m}\rvert^{2}+\lvert \nu_{\sigma}+\rho_{m}\rvert^{2},
\end{equation*}
and $\rho,\rho_{m}$ are defined by (2.3) and (2.4), respectively.
The operator $A_{\tau,\chi}^{\sharp}$
is induced by the twisted Bochner-Laplace operator
$\Delta^{\sharp}_{\tau,\chi}$ and
acts on smooth sections of the twisted vector bundle $E(\sigma)\otimes E_{\chi}$ over $X$ (see \cite[p. 27-28]{Spil}).
We recall here the definition of the vector bundle $E(\sigma)$.
We consider always representations $\sigma$ of $M$ coming from restricions of representations of $K$.
Let $\tau_{\sigma}\in R(K)$ with $\tau_{\sigma}:=\tau^{+}(\sigma)-\tau^{-}(\sigma)$.
By \cite[Proposition 1.1]{BO}, there exists unique integers $m_{\tau}(\sigma)\in\{-1,0,1\}$, which are equal to zero except for finitely many $\tau\in \widehat{K}$,
such that for the case (b) (see p. 8)
\begin{equation}
 \sigma+w\sigma=\sum_{\tau\in\widehat{K}}m_{\tau}(\sigma)i^{*}(\tau).
\end{equation}
Then, the locally homogeneous vector bundle $E(\sigma)$ associated with $\tau$ is of the form
\begin{equation*}
 E(\sigma)=\bigoplus_{\substack{\tau\in\widehat{K}\\m_{\tau}(\sigma)\neq 0}}E_{\tau},
\end{equation*}
where $E_{\tau}$ is the locally homogeneous vector bundle associated with $\tau\in\widehat{K}$.
Therefore, the vector bundle $E(\sigma)$ has a grading $E(\sigma)=E(\sigma)^{+}\oplus E(\sigma)^{-}$.
This grading is defined exactly by the positive or negative sign of $m_{\tau}(\sigma)$.
Let $\widetilde{E}(\sigma)$ be the pullback of $E(\sigma)$ to $\widetilde{X}$.
Then,
\begin{equation*}
 \widetilde{E}(\sigma)=\bigoplus_{\substack{\tau\in\widehat{K}\\m_{\tau}(\sigma)\neq0}}\widetilde{E}_{\tau}.
\end{equation*}
One can easily observe that by Proposition 4.1 (see also \cite[Proposition 5.1 and p. 27-28]{BO}),
that up to a $\mathbb{Z}_{2}$ grading, one can identify the two vector bundles $E(\sigma)$ and $E_{\tau_{s}(\sigma)}$.

We consider the lift $\widetilde{A}_{\chi}^{\sharp}(\sigma)$ of $A_{\chi}^{\sharp}(\sigma)$ to the universal covering $\widetilde{X}$.
Then,
\begin{align}
 \widetilde{A}_{\chi}^{\sharp}(\sigma)\notag&=\bigoplus_{m_{\tau}(\sigma)\neq 0}\widetilde{A}^{\sharp}_{\tau,\chi}+c(\sigma)\\\notag
&=\bigoplus_{m_{\tau}(\sigma)\neq 0}(\widetilde{A}_{\tau}\otimes \Id_{V_{\chi}})+c(\sigma)\\
&=\bigoplus_{m_{\tau}(\sigma)\neq 0}\big(\widetilde{A}_{\tau}+c(\sigma)\big)\otimes \Id_{V_{\chi}}.
\end{align}
The Parthasarathy formula from \cite[equation (1.11)]{BO} states
\begin{equation}
(\widetilde{D}(\sigma))^{2}=\bigoplus_{m_{\tau}(\sigma)\neq 0}\big(\widetilde{A}_{\tau}+c(\sigma)\big)
\end{equation}
If we combine (4.4), (4.6) and (4.7) the Parthasarathy formula generalizes as
\begin{equation}
(D^{\sharp}_{\chi}(\sigma))^{2}=A_{\chi}^{\sharp}(\sigma).
\end{equation}

\begin{center}
{\section{\textnormal{Trace formulas}}}
\end{center}
By (4.4), we get
\begin{equation}
  (\widetilde{D}^{\sharp}_{\chi}(\sigma))^{2}=(\widetilde{D}(\sigma))^{2}\otimes \Id_{V_{\chi}}
\end{equation}
The square of the twisted Dirac operator $(D^{\sharp}_{\chi}(\sigma))^{2}$ acting on smooth sections of $E_{\tau_{s}(\sigma)}\otimes E_{\chi}$ is not a self-adjoint operator in general.
Nevertheless, its principal symbol is given by
\begin{equation*}
 \sigma_{(D^{\sharp}_{\chi}(\sigma))^{2}}(x,\xi)=\lVert\xi\rVert^{2}\otimes\Id_{(V_{\tau_{s}(\sigma)}\otimes V_{\chi})_{x}}, \quad x\in X,\quad \xi\in T_{x}^{*}X,\xi\neq 0.
 \end{equation*}
Therefore, $({D^{\sharp}}_{\chi}(\sigma))^{2}$ is a second order elliptic differential operator with nice spectral properties, i.e., its spectrum is discrete
and contained in a translate of a positive cone $C\subset\mathbb{C}$.
Furthermore, we can define the integral operators $e^{-t({D^{\sharp}}_{\chi}(\sigma))^{2}}$ and $D^{\sharp}_{\chi}(\sigma)e^{-t({D^{\sharp}}_{\chi}(\sigma))^{2}}$, and derive a
corresponding trace formula.
We introduce here a more general setting.

\begin{setting}
Let $E\rightarrow X$ be a complex vector bundle over a smooth compact riemannian manifold $X$ of dimension $d$.
Let $D:C^{\infty}(X,E)\rightarrow C^{\infty}(X,E)$ be an elliptic differential operator of order $m\geq 1$.
Let $\sigma_{D}$ be its principal symbol.
\end{setting}
\begin{defi}
 A spectral cut\index{spectral cut} is a ray
\begin{equation*}
 R_{\theta}:=\{\rho e^ {i\theta}: \rho\in[0,\infty]\},
\end{equation*}
where $\theta\in[0,2\pi)$.
\end{defi}
\begin{defi}
 The angle $\theta$ is a principal angle for an elliptic operator $D$ if
\begin{equation*}
 \spec(\sigma_{D}(x,\xi))\cap R_{\theta}=\emptyset,\quad \forall x\in X,\forall\xi\in T_{x}^{*}X,\xi\neq 0.
\end{equation*}
\end{defi}
\begin{defi}
 We define the solid angle $L_{I}$ associated with a closed interval $I$ of $\mathbb{R}$ by
\begin{equation*}
 L_{I}:=\{\rho e^ {i\theta}: \rho\in(0,\infty), \theta\in I \}.
\end{equation*}
\end{defi}
\begin{defi}
The angle $\theta$ is an Agmon angle\index{Agmon angle} for an elliptic operator $D$, if it is a principal angle for $D$
and there exists $\varepsilon>0$ such that
\begin{equation*}
 \spec(D)\cap L_{[\theta-\varepsilon,\theta+\varepsilon]}=\emptyset.
\end{equation*}
\end{defi}
\begin{lem}
Let $\varepsilon\in(0,\frac{\pi}{2})$ be an angle such that the principal symbol $\sigma_{D}(x,\xi)$ of $D$,
for $\xi\in T_{x}^{*}X,\xi\neq 0$ does not take values in $ L_{[-\varepsilon,\varepsilon]}$.
Then, the spectrum $\spec(D)$ of the operator $D$
is discrete and for every $\varepsilon\in(0,\frac{\pi}{2})$ there exist $R>0$ such that $\spec(D)$
is contained in the set $B(0,R)\cup L_{[-\varepsilon,\varepsilon]}\subset \mathbb{C}$.
\end{lem}
\begin{proof}
 The discreteness of the spectrum follows from \cite[Theorem 8.4]{Sh}. For the second statement see \cite[Theorem 9.3]{Sh}.
\end{proof}
Let $\lambda_{k}$ be an eigenvalue of $D$ and
$V_{\lambda_{k}}$ be the corresponding eigenspace.
This is a finite dimensional subspace of $C^{\infty}(X,E)$ invariant under $D$.
We have that for every $k\in\mathbb{N}$, there exist $N_{k}\in\mathbb{N}$ such that
\begin{align*}
 &(D-\lambda_{k}\Id)^{N_{k}}V_{\lambda_{k}}=0\\
&\lim_{k\rightarrow \infty}\lvert \lambda_{k}\rvert=\infty.
\end{align*}
\begin{defi}
We call algebraic multiplicity $m(\lambda_{k})$ of the eigenvalue $\lambda_{k}$ the dimension of the corresponding
eigenspace $V_{\lambda_{k}}$.
\end{defi}
By Lemma 5.6, we get Lemma 5.8, which describes the spectrum of the square root $(D^{\sharp}_{\chi}(\sigma))^{2}$ of the twisted Dirac operator.
\begin{lem}
Let $\varepsilon\in(0,\frac{\pi}{2})$ be an angle such that the principal symbol $\sigma_{(D^{\sharp}_{\chi}(\sigma))^{2}}(x,\xi)$ of $(D^{\sharp}_{\chi}(\sigma))^{2}$,
for $\xi\in T_{x}^{*}X,\xi\neq 0$ does not take values in $ L_{[-\varepsilon,\varepsilon]}$.
Then, the spectrum $\spec((D^{\sharp}_{\chi}(\sigma))^{2})$ of the twisted Dirac operator $(D^{\sharp}_{\chi}(\sigma))^{2}$
is discrete and for every $\varepsilon$ there exists $R>0$ such that $\spec((D^{\sharp}_{\chi}(\sigma))^{2})$
is contained in the set $B(-1,R)\cup L_{[-\varepsilon,\varepsilon]}\subset \mathbb{C}$.
\end{lem}
\begin{proof}
As in the proof of Lemma 5.6.
\end{proof}
Let $\theta$ be an Agmon angle for the operator $(D^{\sharp}_{\chi}(\sigma))^{2}$. Then, by definition of the Agmon angle and Lemma 5.8, there exists $\varepsilon>0$
such that
\begin{equation*}
 \spec((D^{\sharp}_{\chi}(\sigma))^{2})\cap L_{[\theta-\varepsilon,\theta+\varepsilon]}=\emptyset.
\end{equation*}
 Since $({D^{\sharp}}_{\chi}(\sigma))^{2}$ has discrete spectrum, there exists also an $r_{0}>0$ such that
\begin{equation*}
\spec((D^{\sharp}_{\chi}(\sigma))^{2})\cap\{z\in\mathbb{C}:\lvert z+1\rvert\leq 2r_{0}\}=\emptyset.
\end{equation*}
 We define a contour $\Gamma_{\theta,r_{0}}$ as follows.
\begin{equation*}
 \Gamma_{\theta,r _{0}}=\Gamma_{1}\cup\ \Gamma_{2}\cup\Gamma_{3},
\end{equation*}
where $\Gamma_{1}=\{-1+re^ {i\theta}\colon\infty>r\geq r_{0}\}$, $\Gamma_{2}=\{-1+r_{0}e^ {ia}\colon\theta\leq a\leq \theta+2\pi\}$,
$\Gamma_{3}=\{-1+re^{i(\theta+2\pi)}\colon r_{0}\leq r< \infty\}$.
On $\Gamma_{1}$, $r$ runs from $\infty$ to $r_{0}$, $\Gamma_{2}$ is oriented counterclockwise, and on $\Gamma_{3}$,
$r$ runs from $r_{0}$ to $\infty$.
We put
\begin{align}
e^{-t(D^{\sharp}_{\chi}(\sigma))^{2}}=&\frac{i}{2\pi}\int_{\Gamma_{\theta,r_{0}}}e^{-t\lambda}\big((D^{\sharp}_{\chi}(\sigma))^{2}-\lambda\Id\big)^{-1} d\lambda\\
D^{\sharp}_{\chi}(\sigma)e^{-t(({D^{\sharp}}_{\chi}(\sigma))^{2}}=&\frac{i}{2\pi}\int_{\Gamma_{\theta,r_{0}}}\lambda^{1/2}e^{-t\lambda}\big((D^{\sharp}_{\chi}(\sigma))^{2}-\lambda\Id\big)^{-1} d\lambda
\end{align}
We have $\lvert e^{-t\lambda}\rvert\leq e^{-t\text{\Re}(\lambda)}$. Furthermore, by \cite[Corollary 9.2]{Sh},
there exist a positive constant $c>0$ such that $\lVert((D^{\sharp}_{\chi}(\sigma))^{2}-\lambda\Id)^{-1}\rVert\leq c \lvert \lambda\rvert^{-1}$.
Hence, the integrals in (5.2) and (5.3) are  well defined.

By \cite[Lemma 2.4]{M1}, $D^{\sharp}_{\chi}(\sigma)e^{-t{(D^{\sharp}_{\chi}(\sigma)})^{2}}$
is an integral operator. Let $ K_{t}^{\tau_{s}(\sigma),\chi}$ be its kernel function.
Let $F$ be a fundamental domain of $\Gamma$. We consider the space $L^{2}(\widetilde{X},\widetilde{E}_{\tau_{s}(s)}\otimes \widetilde{E}_{\chi})^{\Gamma}$
of sections $f$ of $\widetilde{E}_{\tau_{s}(s)}\otimes \widetilde{E}_{\chi}$ such that $f(\gamma \widetilde{x})=\gamma f(\widetilde{x})$, $\forall\gamma\in\Gamma, \widetilde{x}\in\widetilde{X}$.\\
For $f\in L^{2}(X,E_{\tau_{s}(s)}\otimes E_{\chi})\cong L^{2}(\widetilde{X},\widetilde{E}_{\tau_{s}(s)}\otimes \widetilde{E}_{\chi})^{\Gamma}$, we have
\begin{align}
D^{\sharp}_{\chi}(\sigma)e^{-t{(D^{\sharp}_{\chi}(\sigma)})^{2}}f(x)\notag&=\int_{X}K_{t}^{\tau_{s}(\sigma),\chi}(x,y)f(y)dy\\\notag
&=\int_{\widetilde{X}}(K_{t}^{\tau_{s}(\sigma)}(\widetilde{x},\widetilde{y})\otimes \Id_{V_{\chi}})f(\widetilde{y})d\widetilde{y}\\
&=\sum_{\gamma \in \Gamma}\int_{F}
(K_{t}^{\tau_{s}(\sigma)}(\widetilde{x},\gamma\widetilde{y})\otimes \chi(\gamma)\Id_{V_{\chi}})f(\widetilde{y})d\widetilde{y},
\end{align}
where $x,y\in X$ and $\widetilde{x},\widetilde{y}\in\widetilde{X}$ are lifts of $x,y$ to $\widetilde{X}$, respectively.
The kernel function $K_{t}^{\tau_{s}(\sigma)}$ is the kernel
associated with the operator $\widetilde{D}(\sigma)e^{-t(\widetilde{D}(\sigma))^{2}}$. It belongs to the Harish-Chandra $L^{q}$-Schwartz space$
(\mathcal{C}^{q}(G)\otimes \End(V_{\tau_{s}(\sigma)}))^{K\times K}$,
as it is defined in \cite[p. 161-162]{BM}.
Hence, we can interchange summation and integration in the right hand side of (5.4) and get
\begin{equation*}
 K_{t}^{\tau_{s}(\sigma),\chi}(x,x')=\sum_{\gamma \in \Gamma}K_{t}^{\tau_{s}(\sigma)}(g^{-1}\gamma g')\otimes \chi(\gamma),
\end{equation*}
where $x=\Gamma g, x'=\Gamma g',g,g'\in G$.\\
By \cite[Proposition 2.5]{M1}, $D^{\sharp}_{\chi}(\sigma)e^{-t{(D^{\sharp}_{\chi}(\sigma)})^{2}}$ is a trace class operator, and its trace is given by
\begin{equation*}
\Tr(D^{\sharp}_{\chi}(\sigma)e^{-t(D^{\sharp}_{\chi}(\sigma))^{2}})=\sum_{\gamma \in \Gamma}\tr \chi(\gamma)\int_{\Gamma\backslash G}\tr K_{t}^{\tau_{s}(\sigma)}(g^{-1}\gamma g)dg.
\end{equation*}
We put
\begin{equation}
 k_{t}^{\tau_{s}(\sigma)}(g)=\tr K_{t}^{\tau_{s}(\sigma)}(g).
\end{equation}
We use the trace formula in \cite[Proposition 6.1]{M1} for non-unitary twists.
 \begin{align*}
  \Tr(D^{\sharp}_{\chi}(\sigma)e^{-t(D^{\sharp}_{\chi}(\sigma))^{2}})=&\dim(V_{\chi})\Vol(X)(k^{\tau_{s}(\sigma)}_{t})(e)\\
  &+\frac{1}{2\pi}\sum_{[\gamma]\neq e} \frac{l(\gamma)\tr(\chi(\gamma))}{n_{\Gamma}(\gamma)D(\gamma)}\sum_{\sigma\in\widehat{M}}
  \overline{\tr\sigma(m_{\gamma})}\int_{\mathbb{R}}\Theta_{\sigma,\lambda}(k_{t}^{\tau_{s}(\sigma)})e^{-il(\gamma)\lambda}d\lambda.
\end{align*}
We continue analyzing the trace formula above in terms of characters.
We want to compute the Fourier transform $\Theta_{\sigma,\lambda}( k_{t}^{\tau_{s}(\sigma)})$ of $ k_{t}^{\tau_{s}(\sigma)}$.
Following \cite{MS}, we let $(\pi,\mathcal{H}_{\pi})$ be an unitary admissible representation of $G$ in a Hilbert space $\mathcal{H}_{\pi}$. We let $\mathcal{H}^{\infty}_{\pi}$ be the subspace of of smooth
vectors of $\mathcal{H}_{\pi}$.
We set
\begin{equation}
 \pi(K_{t}^{\tau_{s}(\sigma)}):=\int_{G}\pi(g)\otimes K_{t}^{\tau_{s}(\sigma)}(g)dg.
\end{equation}
This defines a bounded trace class operator on $\mathcal{H}_{\pi}\otimes V_{\tau_{s}(\sigma)}$.
By \cite[p.160-161]{BM}, relative to the splitting
\begin{equation*}
 \mathcal{H}_{\pi}\otimes V_{\tau_{s}(\sigma)}=(\mathcal{H}_{\pi}\otimes V_{\tau_{s}(\sigma)})^K\oplus [(\mathcal{H}_{\pi}\otimes V_{\tau_{s}(\sigma)})^K]^{\perp},
\end{equation*}
$ \widetilde{\pi}(K_{t}^{\tau_{s}(\sigma)})$ has the form
\begin{equation}
  \widetilde{\pi}(K_{t}^{\tau_{s}(\sigma)})= \begin{pmatrix}
\pi(K_{t}^{\tau_{s}(\sigma)}) & 0 \\
 0 & 0
\end{pmatrix},
\end{equation}
with $\pi(K_{t}^{\tau_{s}(\sigma)})$ acting on $(\mathcal{H}_{\pi}\otimes V_{\tau_{s}(\sigma)})^K$.
We consider orthonormal bases $(\xi_{n}), n\in \mathbb{N}, (e_{j}), j=1,\cdots,k$ of the vector spaces $\mathcal{H}_{\pi}, V_{\tau_{s}}(\sigma)$, respectively, where $k:=\dim(V_{\tau_{s}(\sigma)})$.
By (5.7), we have
\begin{equation}
 \Tr(\pi(K_{t}^{\tau_{s}(\sigma)}))=\Tr(\widetilde{\pi}(K_{t}^{\tau_{s}(\sigma)})).
\end{equation}
Hence,
\begin{align}
 \Tr(\widetilde{\pi}(K_{t}^{\tau_{s}(\sigma)}))=\notag&\sum_{n}\sum_{j}\langle \widetilde{\pi}(K_{t}^{\tau_{s}(\sigma)})(\xi_n\otimes e_j),(\xi_n\otimes e_j)\rangle\\\notag
 &=\sum_{n}\sum_{j}\int_{G}\langle \pi(g)\xi_n,\xi_n\rangle\langle K_{t}^{\tau_{s}(\sigma)}(g)e_j,e_j\rangle dg\\\notag
 &=\sum_{n}\int_{G}\langle\pi(g)\xi_n,\xi_n\rangle k_{t}^{\tau_{s}(\sigma)}(g)dg\\\notag
 &=\sum_{n}\langle\pi(k_{t}^{\tau_{s}(\sigma)})\xi_n,\xi_n\rangle\\
 &=\Tr\pi(k_{t}^{\tau_{s}(\sigma)}).
 \end{align}
Let $(X_{i})_{i=1}^{d}$ be an orthonormal basis of $\mathfrak{p}$. We consider the operator acting on $(\mathcal{H}^{\infty}_{\pi}\otimes V_{\tau_{s}(\sigma)})^{K}$, defined by
\begin{equation}
 \widetilde{D}_{\tau_{s}(\sigma)}(\pi):=\sum_{i=1}^{d}X_{i}\cdot(\pi(X_{i})\otimes\Id).
\end{equation}
In \cite[p.77]{Pf}, it is proved that $ \widetilde{D}_{\tau_{s}(\sigma)}(\pi)$ maps $(\mathcal{H}^{\infty}_{\pi}\otimes V_{\tau_{s}(\sigma)})^{K}$ to $(\mathcal{H}^{\infty}_{\pi}\otimes V_{\tau_{s}(\sigma)})^{K}$.
By (5.6) we get
\begin{equation}
 \widetilde{\pi}(K_{t}^{\tau_{s}(\sigma)})=e^{-tc(\sigma)} \widetilde{D}_{\tau_{s}(\sigma)}(\pi)\circ\widetilde{\pi}(H_{t}^{\tau_{s}(\sigma)}),
\end{equation}
where
\begin{equation*}
\widetilde{\pi}(H_{t}^{\tau_{s}(\sigma)})=\int_{G}\pi(g)\otimes H_{t}^{\tau_{s}(\sigma)}dg.
\end{equation*}
The kernel function $H_{t}^{\tau_{s}(\sigma)}$ corresponds to the integral operator
\begin{equation*}
 e^{-t(\widetilde{D}(\sigma))^{2}}f(g)=e^{-tc(\sigma)}\int_{G}H_{t}^{\tau_{s}(\sigma)}(g^{-1}g')f(g')dg',
\end{equation*}
 where $(\widetilde{D}(\sigma))^{2}$ as in (4.7). $H_{t}^{\tau_{s}(\sigma)}$ belongs to the Harish-Chandra $L^{q}$-Schwartz space
 $(\mathcal{C}^{q}(G)\otimes \End(V_{\tau_{s}(\sigma)}))^{K\times K}$.\\
As above (see equations (5.7), (5.8), (5.9)), the operator $\widetilde{\pi}(H_{t}^{\tau_{s}(\sigma)})$ relative to the splitting,
\begin{equation*}
 \mathcal{H}_{\pi}\otimes V_{\tau_{s}(\sigma)}=(\mathcal{H}_{\pi}\otimes V_{\tau_{s}(\sigma)})^K\oplus [(\mathcal{H}_{\pi}\otimes V_{\tau_{s}(\sigma)})^K]^{\perp},
\end{equation*}
takes the form
\begin{equation}
  \widetilde{\pi}(H_{t}^{\tau_{s}(\sigma)})= \begin{pmatrix}
\pi(H_{t}^{\tau_{s}(\sigma)}) & 0 \\
 0 & 0
\end{pmatrix},
\end{equation}
with $\pi(H_{t}^{\tau_{s}(\sigma)})$ acting on $(\mathcal{H}_{\pi}\otimes V_{\tau_{s}(\sigma)})^K$.
Then, it follows that
\begin{equation}
 e^{t\pi(\Omega)}\Id=\pi(H_{t}^{\tau_{s}(\sigma)}),
\end{equation}
where $\Id$ denotes the identity on the space $(\mathcal{H}_{\pi}^{\infty}\otimes V_{\tau_{s}(\sigma)})^K$ (\cite[Corollary 2.2]{BM}).
We have $(\mathcal{H}_{\pi}\otimes V_{\tau_{s}(\sigma)})^K=(\mathcal{H}_{\pi}^{\infty}\otimes V_{\tau_{s}(\sigma)})^K$ and
\begin{equation}
 \Tr(\pi(k_{t}^{\tau_{s}(\sigma)}))=e^{(\pi(\Omega)-c(\sigma))t} \Tr(\widetilde{D}_{\tau_{s}(\sigma)}(\pi)|_{(\mathcal{H}_{\pi}^{\infty}\otimes V_{\tau_{s}(\sigma)})^K}).
\end{equation}
We recall that the representation space of $\tau_{s}(\sigma)$ is given by
\begin{equation*}
 V_{\tau_{s}(\sigma)}= V_{\tau(\sigma)}\otimes S.
\end{equation*}
Let $\pi$ be the unitary principal series representation $\pi_{\sigma,\lambda}$ defined as in \cite[p. 7-8]{Spil}.
By \cite[Proposition 3.6]{MS}, we have for $(\sigma',V_{\sigma'})\in\widehat{M}$,
\begin{equation}
 \Tr(\widetilde{D}_{\tau_{s}(\sigma)}(\pi_{\sigma',\lambda}))=\lambda\big(\dim(V_{\sigma'}\otimes V_{\tau(\sigma)}\otimes S^{+})^{M}-\dim(V_{\sigma'}\otimes V_{\tau(\sigma)}\otimes S^{-})^{M}\big).
\end{equation}
Following \cite[Corollary 7.6]{Pf}, we let $\check{\sigma}'$ be the contragredient representation of $\sigma'$.
Since $\check{\sigma}'\cong\sigma'$, we observe by equation (4.1) in Proposition 4.1 that for $\nu_{n}>0$,
\begin{equation}
 [\dim(V_{\sigma'}\otimes V_{\tau(\sigma)}\otimes S^{+})^{M}-\dim(V_{\sigma'}\otimes V_{\tau(\sigma)}\otimes S^{-})^{M}]=[\sigma-w\sigma:\sigma'].
\end{equation}
Since $\sigma'\in\widehat{M}$ we have that $\sigma'\in\{\sigma,w\sigma\}$, otherwise the right hand side of (5.16) vanishes.
In \cite[p. 28-29] {Spil}, the character $\Theta_{\sigma,\lambda}(q_{t}^{\sigma})$ for the case (a), is computed.
Recall that
\begin{equation*}
 \Theta_{\sigma,\lambda}(q_{t}^{\sigma})=
 \sum_{\tau \in \widehat{K}}m_{\tau}(\sigma) \Theta_{\sigma,\lambda}(q_{t}^{\tau}),
\end{equation*}
where
\begin{align*}
q_{t}^{\sigma}=&\sum_{\tau \in \widehat{K}}m_{\tau}(\sigma) q_{t}^{\tau},
\end{align*}
\begin{equation*}
 q_{t}^{\tau}=\tr Q_{t}^{\tau}(g),
\end{equation*}
and $Q^{\tau}_{t}\in (\mathcal{C}^{q}(G)\otimes \End(V_{\tau}))^{K\times K}$ is the kernel associated to the operator
$e^{-t\widetilde{A}_{\tau}}$.
By \cite[equation (5.35)]{Spil}, we have
\begin{equation*}
 \Theta_{\sigma,\lambda}(q_{t}^{\sigma})=\sum_{\tau \in \widehat{K}}m_{\tau}(\sigma)
 e^{t\pi_{\sigma,\lambda}(\Omega)}[\tau\rvert_{M}:\sigma],
\end{equation*}
where
\begin{equation*}
 \pi_{\sigma,\lambda}(\Omega)=-\lambda^{2}+c(\sigma).
\end{equation*}
This is proved in \cite[p.48]{Ar}.
We study here the case (b).
This means that by (4.5), for $\sigma'\in\widehat{M}$,
\begin{align}
& \Theta_{\sigma',\lambda}(q_{t}^{\sigma})=e^{tc(\sigma)}e^{-t\lambda^{2}},\quad\text{if}\quad \sigma'\in \{\sigma, w \sigma\},\\\notag
& \Theta_{\sigma',\lambda}(q_{t}^{\sigma})=0,\quad\text{if}\quad \sigma'\notin\{\sigma, w \sigma\}.
\end{align}
If we put together (5.14)---(5.17), we obtain
\begin{align}
 &\Theta_{\sigma',\lambda}( k_{t}^{\tau_{s}(\sigma)})=\lambda e^{-t\lambda^{2}} , \quad \text{if}\quad \sigma'=\sigma\\
 &\Theta_{\sigma',\lambda}( k_{t}^{\tau_{s}(\sigma)})=-\lambda e^{-t\lambda^{2}} , \quad \text{if}\quad \sigma'= w \sigma\\
 &\Theta_{\sigma',\lambda}( k_{t}^{\tau_{s}(\sigma)})=0, \quad \text{if} \quad \sigma'\notin\{\sigma, w \sigma\}.
\end{align}
For the identity contribution we use the fact that when $s$ is restricted to $M$ it decomposes as $s^{+}+s^{-}$.
If $ \nu_{\sigma}=(\nu_{1},\ldots,\nu_{n-1},\nu_{n})$ is the highest weight of $\sigma$,
then the highest weight of $w\sigma$ is given by $ \nu_{w\sigma}=(\nu_{1},\ldots,\nu_{n-1},-\nu_{n})$.
Specifically, for the half spin representations $s^{\pm}$ we have
\begin{equation*}
  \nu_{ws^{\pm}}=(\frac{1}{2},\ldots,\mp\frac{1}{2}).
\end{equation*}
Hence,
\begin{equation*}
 ws^{\pm}=s^{\mp}.
\end{equation*}
The Plancherel polynomial is an even polynomial of $\lambda$ and also $P_{s^{+}}(i\lambda)=P_{ws^{+}}(-i\lambda)=P_{s^{-}}(i\lambda)$.
Hence,
\begin{align}
 k_{t}^{\tau_{s}(\sigma)}(e)\notag&=\sum_{\sigma\in\widehat{M}}\int_{\mathbb{R}}\Theta_{\sigma,\lambda}( k_{t}^{\tau_{s}(\sigma)})P_{\sigma}(i\lambda)d\lambda\\
&= \int_{\mathbb{R}}\lambda e^{-t\lambda^{2}}P_{s^{+}}(i\lambda)d\lambda
+\int_{\mathbb{R}}-\lambda e^{-t\lambda^{2}}P_{s^{-}}(i\lambda)d\lambda=0.
\end{align}
For the hyperbolic contribution we use (5.18)---(5.20).
\begin{equation*}
 \Tr(D^{\sharp}_{\chi}(\sigma)e^{-t(D^{\sharp}_{\chi}(\sigma))^{2}})=
 \frac{1}{2\pi}\sum_{[\gamma]\neq e} \frac{l(\gamma)\tr(\chi(\gamma)\otimes(\sigma(m_{\gamma})-w\sigma(m_{\gamma})))}{D(\gamma) n_{\Gamma}(\gamma)}
 \int_{\mathbb{R}}\lambda e^{-t\lambda^{2}}e^{-il(\gamma)\lambda}d\lambda.
\end{equation*}
Equivalently,
\begin{equation*}
 \Tr(D^{\sharp}_{\chi}(\sigma)e^{-t(D^{\sharp}_{\chi}(\sigma))^{2}})=
  \sum_{[\gamma]\neq e}\frac{-2\pi i}{(4\pi t)^{3/2}}\frac{l^{2}(\gamma)\tr(\chi(\gamma)\otimes(\sigma(m_{\gamma})-w\sigma(m_{\gamma})))}
  {n_{\Gamma}(\gamma)D(\gamma)}e^{-l^{2}(\gamma)/4t}.
\end{equation*}
All in all, we have proved the following theorems.
\begin{thm}
 For every $\sigma \in \widehat{M}$ we have for case (b)
 \begin{equation}\label{f:trace formula 3}
 \Tr(D^{\sharp}_{\chi}(\sigma)e^{-t(D^{\sharp}_{\chi}(\sigma))^{2}})=
  \sum_{[\gamma]\neq e}\frac{-2\pi i}{(4\pi t)^{3/2}}\frac{l^{2}(\gamma)\tr(\chi(\gamma)\otimes(\sigma(m_{\gamma})-w\sigma(m_{\gamma})))}
  {n_{\Gamma}(\gamma)D(\gamma)}e^{-l^{2}(\gamma)/4t}.
 \end{equation}
\end{thm}

\begin{thm}
 For every $\sigma \in \widehat{M}$ we have for case (b)
   \begin{align}\label{f:trace formula 2}
  \Tr(e^{-tA_{\chi}^{\sharp}(\sigma)})=\notag&2\dim(V_{\chi})\Vol(X)\int_{\mathbb{R}}e^{-t\lambda^{2}}P_{\sigma}(i\lambda)d\lambda\\
  &+\sum_{[\gamma]\neq e}\frac{l(\gamma)}{n_{\Gamma}(\gamma)}L_{sym}(\gamma;\sigma+w\sigma)
  \frac{e^{-l(\gamma)^{{2}}/4t}}{(4\pi t)^{1/2}},
 \end{align}
where
\begin{equation*}
 L_{sym}(\gamma;\sigma)= \frac{\tr(\sigma(m_{\gamma})\otimes\chi(\gamma))e^{-|\rho|l(\gamma)}}{\det(\Id-\Ad(m_{\gamma}a_{\gamma})_{\overline{\mathfrak{n}})}}.
\end{equation*}
\end{thm}

\begin{center}
\section{\textnormal{Meromorphic continuation of the super zeta function}}
\end{center}

Let $N\in\mathbb{N}$. Let $s_{i},i=1,\ldots,N$ be complex numbers such that $s_{i}\in\mathbb{C}-\spec(-{D^{\sharp}_{\chi}(\sigma)}^{2})$.
We consider the resolvent operator
\begin{equation*}
 R(s_{i}^{2})=({D^{\sharp}_{\chi}(\sigma)}^{2}+s_{i}^{2})^{-1}.
\end{equation*}
We want to obtain the trace class property of the operators
\begin{align*}
&\prod_{i=1}^{N}R(s_{i}^{2})\\
D^{\sharp}_{\chi}(\sigma)&\prod_{i=1}^{N}R(s_{i}^{2}).
\end{align*}
In order to obtain this property, we take sufficient large $N\in\mathbb{N}$,
such that
\begin{itemize}
\item for $N>\frac{d}{2}$,
\begin{equation}
 \Tr(\prod_{i=1}^{N}R(s_{i}^{2}))<\infty.
\end{equation}
\item  for $N>\frac{d}{2}+1$,
\begin{equation}
 \Tr(D^{\sharp}_{\chi}(\sigma)\prod_{i=1}^{N}R(s_{i}^{2}))<\infty.
\end{equation}
\end{itemize}
We denote the space of pseudodifferential operators of order $k$ by $\psi DO^{k}$.
To prove the trace class property of the operators above, we
observe at first that $\prod_{i=1}^{N}R(s_{i}^{2})\in\psi DO^{-2N}$.

Let $\Delta$ be the Bochner-Laplace operator with respect
to some metric, acting on $C^{\infty}(X, E_{\tau_{s}(\sigma)}\otimes E_{\chi})$.
Then, $\Delta$ is a second-order elliptic differential operator,
which is formally self-adjoint and non-negative, i.e.,
$\Delta\geq 0$. Then, by Weyl's law, we have that
for $N>\frac{d}{2}$,
\begin{equation*}
 (\Delta+\Id)^{-N}
\end{equation*}
is a trace class operator.
Moreover,
\begin{equation*}
 B:=(\Delta+\Id)^{N}\prod_{i=1}^{N}R(s_{i}^{2})
\end{equation*}
is $\psi DO$ of order zero.
Hence, it defines a bounded operator in $L^{2}(X, E_{\tau_{s}(\sigma)}\otimes E_{\chi})$.
Thus,
\begin{equation*}
 \prod_{i=1}^{N}R(s_{i}^{2})=(\Delta+\Id)^{-N}B
\end{equation*}
is a trace class operator.

We recall here the following expressions of the resolvents. Let $s_{1},\ldots,s_{N}\in\mathbb{C}$ such that
Re$(s_{i}^{2})>-c$, for all $i=1,\ldots,N$, where $c$ is a real number such that $\spec\big(D_{\chi}^{\sharp}(\sigma)^{2}\big)\subset
\{z\in\mathbb{C}\colon \text{Re}(z)>c\}$.\\
Then,
\begin{align}
D^{\sharp}_{\chi}(\sigma){{(D^{\sharp}_{\chi}(\sigma)}^{2}+s_{i}^{2})}^{-1}&=\int_{0}^{\infty}e^{-ts_{i}^{2}}D^{\sharp}_{\chi}(\sigma)e^{-t{D^{\sharp}_{\chi}(\sigma)}^{2}}dt\\
(A_{\chi}^{\sharp}(\sigma)+s_{i}^{2})^{-1}&=\int_{0}^{\infty}e^{-ts_{i}^{2}}e^{-t{A_{\chi}^{\sharp}(\sigma)}}dt.
\end{align}
\begin{prop}
Let  $N\in\mathbb{N}$ with $N>d/2+1$.
Let $s_{1},\ldots,s_{N}\in\mathbb{C}$ with
 $s_{i}\neq s_{j}$ for all $i\neq j$ such that
\emph{Re}$(s_{i}^{2})>-c$, for all $i=1,\ldots,N$, where $c$ is a real number such that $\spec\big(D_{\chi}^{\sharp}(\sigma)^{2}\big)\subset
\{z\in\mathbb{C}\colon \emph{Re}(z)>c\}$.
Let $L^{s}(s):=\frac{d}{ds}\log(Z^{s}(s;\sigma,\chi))$ be the logarithmic derivative of the super zeta function. Then,
\begin{equation}
\label{eq super mer}
 \Tr(D^{\sharp}_{\chi}(\sigma)\prod_{i=1}^{N}({D^{\sharp}_{\chi}(\sigma)}^{2}+s_{i}^{2})^{-1})=-\frac{i}{2}\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)L^{s}(s_{i}).
\end{equation}
\end{prop}
\begin{proof}
By \cite[Lemma 6.1]{Spil} and formula (6.3), we have
\begin{equation*}
 D^{\sharp}_{\chi}(\sigma)\prod_{i=1}^{N}({D^{\sharp}_{\chi}(\sigma)}^{2}+s_{i}^{2})^{-1}=\int_{0}^{\infty}\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)e^{-ts_{i}^{2}}
   D^{\sharp}_{\chi}(\sigma)e^{-t{D^{\sharp}_{\chi}(\sigma)}^{2}}dt.
\end{equation*}
The operators $ D^{\sharp}_{\chi}(\sigma)\prod_{i=1}^{N}({D^{\sharp}_{\chi}(\sigma)}^{2}+s_{i}^{2})^{-1}$, and
 $D^{\sharp}_{\chi}(\sigma)e^{-t{D^{\sharp}_{\chi}}}$ are both of trace class.\\
Then,
\begin{align*}
 D^{\sharp}_{\chi}(\sigma)\prod_{i=1}^{N}({D^{\sharp}_{\chi}(\sigma)}^{2}+s_{i}^{2})^{-1}&=\int_{\epsilon}^{\infty}\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)e^{-ts_{i}^{2}}
   D^{\sharp}_{\chi}(\sigma)e^{-t{D^{\sharp}_{\chi}(\sigma)}^{2}}dt\\
&\xrightarrow[\epsilon \rightarrow 0]{R\rightarrow \infty} \int_{\epsilon}^{R}
\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)e^{-ts_{i}^{2}}
  D^{\sharp}_{\chi}(\sigma)e^{-t{D^{\sharp}_{\chi}(\sigma)}^{2}}dt,\\
\end{align*}
where the limit is taken with respect to the trace norm
$\lVert \mathcal{A}\rVert_{1}:=\Tr\lvert \mathcal{A}\rvert$, with
$\mathcal{A}= D^{\sharp}_{\chi}(\sigma)\prod_{i=1}^{N}({D^{\sharp}_{\chi}(\sigma)}^{2}+s_{i}^{2})^{-1}$, or $D^{\sharp}_{\chi}(\sigma)e^{-t{D^{\sharp}_{\chi}(\sigma)}^{2}}$.
We have
\begin{align*}
\Tr\Bigg(\int_{0}^{\infty}\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)&e^{-ts_{i}^{2}}
   D^{\sharp}_{\chi}(\sigma)e^{-t{D^{\sharp}_{\chi}(\sigma)}^{2}}dt\Bigg)=\\
&\Tr\Bigg(\lim_{\substack{\epsilon \rightarrow 0\\ R\rightarrow \infty}}\int_{\epsilon}^{R}\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)e^{-ts_{i}^{2}}
   D^{\sharp}_{\chi}(\sigma)e^{-t{D^{\sharp}_{\chi}(\sigma)}^{2}}dt\Bigg).
\end{align*}
But,
\begin{align*}
 \Tr\Bigg(\int_{\epsilon}^{R}\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)&e^{-ts_{i}^{2}}
   D^{\sharp}_{\chi}(\sigma)e^{-t{D^{\sharp}_{\chi}(\sigma)}^{2}}dt\Bigg)=\\
&\int_{\epsilon}^{R}\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)e^{-ts_{i}^{2}}
   \Tr(D^{\sharp}_{\chi}(\sigma)e^{-t{D^{\sharp}_{\chi}(\sigma)}^{2}})dt.
\end{align*}
Hence, it is sufficient to show that the limit
\begin{equation*}
 \lim_{\substack{\epsilon \rightarrow 0\\ R\rightarrow \infty}}\int_{\epsilon}^{R}\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)e^{-ts_{i}^{2}}
 \Tr(D^{\sharp}_{\chi}(\sigma)e^{-t{D^{\sharp}_{\chi}(\sigma)}^{2}})dt
\end{equation*}
exists. We study the behavior of the integral in the equation above as $\epsilon\rightarrow 0$.\\
By \cite[Lemma 6.3]{Spil}, we have that as $t\rightarrow 0^{+}$
\begin{equation*}
 \sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg) e^{-ts_{i}^{2}}=O(t^{N-1}).
\end{equation*}
Also, by \cite[Theorem 2.7, p.503-504]{GS}, there exists a short time asymptotic expansion of the kernel of the operator $D^{\sharp}_{\chi}(\sigma)e^{-t{D^{\sharp}_{\chi}(\sigma)}^{2}}$
\begin{equation*}
 \Tr(D^{\sharp}_{\chi}(\sigma)e^{-t{D^{\sharp}_{\chi}(\sigma)}^{2}})\sim_{t\rightarrow 0^{+}}t^{-d/2}.
\end{equation*}
We have that as $t\rightarrow 0^{+}$
\begin{equation*}
\Bigg\lvert\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)e^{-ts_{i}^{2}}
 \Tr(D^{\sharp}_{\chi}(\sigma)e^{-t{D^{\sharp}_{\chi}(\sigma)}^{2}}) \Bigg\lvert\leq Ct,
\end{equation*}
where $C$ is a positive constant.
All in all, we have proved
 \begin{equation*}
   \Tr(D^{\sharp}_{\chi}(\sigma)\prod_{i=1}^{N}({D^{\sharp}_{\chi}(\sigma)}^{2}+s_{i}^{2})^{-1})=\int_{0}^{\infty}\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)e^{-ts_{i}^{2}}
   \Tr(D^{\sharp}_{\chi}(\sigma)e^{-t{D^{\sharp}_{\chi}(\sigma)}^{2}})dt.
 \end{equation*}
We apply now the trace formula (5.22) for the operator $D^{\sharp}_{\chi}(\sigma)e^{-t{D^{\sharp}_{\chi}(\sigma)}^{2}}$. Then, we get
\begin{align}
 \Tr(D^{\sharp}_{\chi}(\sigma)\prod_{i=1}^{N}({D^{\sharp}_{\chi}(\sigma)}^{2}+s_{i}^{2})^{-1})\notag&=\int_{0}^{\infty}\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)e^{-ts_{i}^{2}}\\
  & \bigg\{\sum_{[\gamma]\neq e}\frac{-2\pi i}{(4\pi t)^{3/2}}\frac{l^{2}(\gamma)\tr(\chi(\gamma)\otimes(\sigma(m_{\gamma})-w\sigma(m_{\gamma})))}
  {n_{\Gamma}(\gamma)D(\gamma)}e^{-l^{2}(\gamma)/4t}\bigg\}.
\end{align}
If we use the formula (see \cite[p.146, (28)]{Er})
\begin{equation*}
 \int_{0}^{\infty}e^{-ts^{2}}\frac{1}{(4\pi t)^{3/2}}e^{-l^{2}(\gamma)/4t}dt=\frac{e^{-l(\gamma)s}}{4\pi l(\gamma)}
\end{equation*}
equation (6.6) becomes
\begin{align*}
\Tr(D^{\sharp}_{\chi}(\sigma)\prod_{i=1}^{N}({D^{\sharp}_{\chi}(\sigma)}^{2}+s_{i}^{2})^{-1})=&\frac{-i}{2}\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)\\
&\bigg\{\sum_{[\gamma]\neq e}\frac{l(\gamma)\tr(\chi(\gamma)\otimes(\sigma(m_{\gamma})-w\sigma(m_{\gamma})))}
{n_{\Gamma}(\gamma)D(\gamma)}e^{-l(\gamma)s_{i}}\bigg\}.
\end{align*}
Hence, by equation \eqref{f:log der super} we get
\begin{equation*}
 \Tr(D^{\sharp}_{\chi}(\sigma)\prod_{i=1}^{N}({D^{\sharp}_{\chi}(\sigma)}^{2}+s_{i}^{2})^{-1})=\frac{-i}{2}\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)L^{s}(s_{i}).
\end{equation*}
\end{proof}
The meromorphic continuation of the super zeta function follows from the Proposition (6.1) above.
\begin{thm}
The super zeta function $Z^{s}(s;\sigma,\chi)$ admits a meromorphic continuation to the whole complex plane $\mathbb{C}$. The singularities
are located at $\{s_{k}^{\pm}=\pm i\lambda_{k}:\lambda_{k}\in \spec(D^{\sharp}_{\chi}(\sigma)), k\in\mathbb{N}\}$
of order  $\pm m_{s}(\lambda_{k})$, where $m_{s}(\lambda_{k})=m(\lambda_{k})-m(-\lambda_{k})\in\mathbb{N}$ and
$m(\pm\lambda_{k})$ denotes the algebraic multiplicity of the eigenvalue $\pm\lambda_{k}$.
\end{thm}
\begin{proof}
We define the function $\Phi(s_{1},s_{2},\ldots,s_{N})$ of the complex variables $s_{1},s_{2},\ldots,s_{N}$ by
\begin{equation}
\Phi(s_{1},s_{2},\ldots,s_{N})=-\frac{i}{2}\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)L^{s}(s_{i}).
\end{equation}
By Lidskii's theorem and \cite[Lemma 6.1]{Spil}, (6.5) becomes
 \begin{equation}
\sum_{\lambda_{k}}\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg) m_{s}(\lambda_{k})\lambda_{k}\frac{1}{(\lambda_{k})^{2}+s_{i}^{2}}=\Phi(s_{1},s_{2},\ldots,s_{N}).
 \end{equation}
We fix the complex numbers $s_{i}, i=2,\ldots,N$ with $s_{i}\neq s_{j}$ for $i,j=2,\ldots, N$
and let the complex number $s=s_{1}$ vary.
Hence,
\begin{equation*}
\Phi(s,s_{2},\ldots,s_{N})=\Phi(s)
\end{equation*}
The term that contains the logarithmic derivative $L^{s}(s)$ in $\Phi(s)$ is of the form
\begin{equation}
 \frac{-i}{2}\bigg(\prod_{j=2}^{N}\frac{1}{s_{j}^{2}-s^{2}}\bigg)L^{s}(s).
\end{equation}
The term of
\begin{equation*}
\sum_{\lambda_{k}}\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg) m_{s}(\lambda_{k})\lambda_{k}\frac{1}{(\lambda_{k})^{2}+s_{i}^{2}}
\end{equation*}
which is singular at $s=\pm i\lambda_{k}$, $k\in \mathbb{N}$ is
\begin{equation*}
\bigg(\prod_{j=2}^{N}\frac{1}{s_{j}^{2}-s^{2}}\bigg) m_{s}(\lambda_{k})\lambda_{k}\frac{1}{(\lambda_{k})^{2}+s^{2}}.
\end{equation*}
If we multiply both sides of  (6.8) by
\begin{equation*}
 2i\prod_{j=2}^{N}(s_{j}^{2}-s^{2}),
\end{equation*}
we see that the residue of the logarithmic derivative $L^{s}(s)$ at $\pm i\lambda_{k}$ is $\pm m_{s}(\lambda_{k})$.

By (3.8), $L^{s}(s)$ decreases exponentially as Re$(s)\rightarrow \infty$. Hence, the integral
\begin{equation*}
 \int_{s}^{\infty}L^{s}(w)dw
\end{equation*}
over a path connecting $s$ and infinity
is well defined and
\begin{equation}
 \log Z^{s}(s;\sigma,\chi)=-\int_{s}^{\infty}L^{s}(w)dw.
\end{equation}
The integral above depends on the choice of the path, because $L^{s}(s)$
has singularities at $s_{k}^{\pm}$.
Nevertheless, since all the
residues of the singularities are integers, it follows that the exponential of the integral in
the right hand side of (6.10)
is independent of the choice of the path.
The meromorphic continuation of the super zeta function
$Z^{s}(s;\sigma,\chi)$ to the whole complex plane follows.

\end{proof}

\begin{center}
{\section{\textnormal{Meromorphic continuation of the symmetrized zeta function}}}
\end{center}

Let  $N\in\mathbb{N}$ with $N>d/2$.
We choose  $s_{1},\ldots,s_{N}\in\mathbb{C}$ with
 $s_{i}\neq s_{j}$ for all $i\neq j$ such that
 Re$(s_{i}^{2})>-r$, for all $i=1,\ldots,N$, where $r$ is a real number such that $\spec\big(A_{\chi}^{\sharp}(\sigma)\big)\subset
\{z\in\mathbb{C}\colon \text{Re}(z)>r\}$.\\
Then, by \cite[Lemma 6.1]{Spil} and equation (6.4), we have
\begin{equation*}
  \prod_{i=1}^{N}(A_{\chi}^{\sharp}(\sigma)+s_{i}^{2})^{-1}=\int_{0}^{\infty}\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)e^{-ts_{i}^{2}}
e^{-t{A_{\chi}^{\sharp}(\sigma)}}dt.
\end{equation*}
As in the proof of Proposition 6.1, we can consider the trace of the operators in the formula above and get
\begin{equation*}
  \Tr \prod_{i=1}^{N}(A_{\chi}^{\sharp}(\sigma)+s_{i}^{2})^{-1}=\int_{0}^{\infty}\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)e^{-ts_{i}^{2}}\Tr
e^{-t{A_{\chi}^{\sharp}(\sigma)}}dt.
\end{equation*}
We insert the trace formula (5.23) for the operator $e^{-tA_{\chi}^{\sharp}(\sigma)}$ and get
\begin{align}
\notag\Tr \prod_{i=1}^{N}(A_{\tau,\chi}^{\sharp}&(\sigma)+s_{i}^{2})^{-1}=\\\notag&\int_{0}^{\infty}\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)e^{-ts_{i}^{2}}
\bigg\{2\dim(V_{\chi})\Vol(X)\int_{\mathbb{R}}e^{-t\lambda^{2}}P_{\sigma}(i\lambda)d\lambda\\
&+\sum_{[\gamma]\neq e}\frac{l(\gamma)}{n_{\Gamma}(\gamma)}L_{sym}(\gamma;\sigma+w\sigma)
\frac{e^{-l(\gamma)^{{2}}/4t}}{(4\pi t)^{1/2}}\bigg\}dt.
\end{align}
The first sum in the right hand side of (7.1) includes the double integral
\begin{equation*}
 I=\int_{0}^{\infty}\int_{\mathbb{R}}\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)
e^{-ts_{i}^{2}}e^{-t\lambda^{2}}P_{\sigma}(i\lambda)d\lambda dt,
\end{equation*}
which has been computed in \cite[p. 33-34]{Spil}
\begin{align*}
I=\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)
\frac{\pi}{s_{i}}P_{\sigma}(s_{i}).
\end{align*}
Hence, equation (7.1) reads
\begin{align*}
 \Tr \prod_{i=1}^{N}(A_{\chi}^{\sharp}(\sigma)+s_{i}^{2})^{-1}&=\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)\frac{\pi}{s_{i}} 2\dim(V_{\chi})\Vol(X)P_{\sigma}(s_{i})\\\notag
&+\sum_{i=1}^{N}\frac{1}{2s_{i}}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)\sum_{[\gamma]\neq[e]} \frac{l(\gamma)}{n_{\Gamma}(\gamma)}L_{sym}(\gamma;\sigma+w\sigma)
e^{-s_{i}l(\gamma)}.
\end{align*}
By \eqref{f:log der symmetrized}, we can insert the logarithmic derivative $L_{S}(s)$ of the symmetrized zeta function.
Then, we get
\begin{align}
\label{eq mer sym}
 \Tr \prod_{i=1}^{N}(A_{\chi}^{\sharp}(\sigma)+s_{i}^{2})^{-1}\notag&=\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)\frac{\pi}{s_{i}} 2\dim(V_{\chi})\Vol(X)P_{\sigma}(s_{i})\\
&+\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)\frac{1}{2s_{i}}L_{S}(s_{i}).
\end{align}
Equation (7.2) will give the meromorphic continuation of the symmetrized zeta function.
\begin{thm}
The symmetrized zeta function $S(s;\sigma,\chi)$ admits a meromorphic continuation to the whole complex plane $\mathbb{C}$. The set of the singularities
equals $\{s_{k}^{\pm}=\pm i{\sqrt{\mu_{k}}}:\mu_{k}\in \spec(A^{\sharp}_{\chi}(\sigma)), k\in\mathbb{N}\}$.
The orders of the singularities are equal to $m(\mu_{k})$, where $m(\mu_{k})\in\mathbb{N}$ denotes the algebraic multiplicity of the eigenvalue $\mu_{k}$.
For $\mu_{0}=0$, the order of the singularity $s_{0}$ is equal to $2m(0)$.
\end{thm}
\begin{proof}
By \cite[Lemma 6.1]{Spil} and equation (7.2), we get
\begin{align}
\sum_{\mu_{k}}\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)\frac{m(\mu_{k})}{\mu_{k}+s_{i}^{2}}=\notag&\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)\frac{\pi}{s_{i}}2\dim(V_{\chi})\Vol(X)P_{\sigma}(s_{i})\\\notag
&+\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)\frac{1}{2s_{i}}L_{S}(s_{i}).
\end{align}
We multiply the last equation by $2s_{1}$.
\begin{align}
\sum_{\mu_{k}}\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)2s_{1}\frac{m(\mu_{k})}{\mu_{k}+s_{i}^{2}}=\notag&\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)\frac{4\pi s_{1}}{s_{i}}\dim(V_{\chi})\Vol(X)P_{\sigma}(s_{i})\\
&+\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)\frac{s_{1}}{s_{i}}L_{S}(s_{i}).
\end{align}
We define the function $\varXi(s_{1},\ldots,s_{N})$ of the complex variables $s_{1},\ldots,s_{N}$ by
\begin{equation*}
\varXi(s_{1},\ldots,s_{N}):=\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)\frac{s_{1}}{s_{i}}L_{S}(s_{i}).
\end{equation*}
We fix the complex numbers $s_{i}, i=2,\ldots,N$ with $s_{i}\neq s_{j}$ for $i,j=2,\ldots, N$
and let the complex number $s=s_{1}$ vary.
Then,
\begin{equation*}
 \varXi(s,\ldots,s_{N})=\varXi(s),
\end{equation*}
and equation (7.3) becomes
\begin{align}
\notag\sum_{\mu_{k}}\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)2s\frac{m(\mu_{k})}{\mu_{k}+s_{i}^{2}}
=\sum_{i=1}^{N}&\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)\frac{4\pi s}{s_{i}}\dim(V_{\chi})\Vol(X)P_{\sigma}(s_{i})\\
+&\varXi(s).
\end{align}
The term that contains the logarithmic derivative $L_{S}(s)$ in $\varXi(s)$ is of the form
\begin{equation}
 \bigg(\prod_{j=2}^{N}\frac{1}{s_{j}^{2}-s^{2}}\bigg)L_{S}(s).
\end{equation}
The term of
\begin{equation*}
 \sum_{\mu_{k}}\sum_{i=1}^{N}\bigg(\prod_{\substack{j=1\\ j\neq i}}^{N}\frac{1}{s_{j}^{2}-s_{i}^{2}}\bigg)2s\frac{m(\mu_{k})}{\mu_{k}+s_{i}^{2}},
\end{equation*}
which is singular at  $s=\pm i\sqrt{\mu_{k}}$, $k\in \mathbb{N}$ is
\begin{equation*}
\bigg(\prod_{j=2}^{N}\frac{1}{s_{j}^{2}-s^{2}}\bigg)2s\frac{m(\mu_{k})}{\mu_{k}+s^{2}}.
\end{equation*}

We multiply both sides of the
equality (7.4) by
\begin{equation*}
 \prod_{j=2}^N(s_j^2-s^2).
\end{equation*}
Then, the residues of $L_{S}(s)$ at the points $\pm i\sqrt{\mu_{k}}$
are $m(\mu_{k})$, for $k\neq0$ and $2m(0)$, for $k=0$.

By (3.7), $L_{S}(s)$ decreases exponentially as Re$(s)\rightarrow \infty$. Therefore, the integral
\begin{equation*}
 \int_{s}^{\infty}L_{S}(w)dw
\end{equation*}
over a path connecting $s$ and infinity is well defined and
\begin{equation}
 \log S(s;\sigma,\chi)=-\int_{s}^{\infty}L_{S}(w)dw.
\end{equation}
The integral above depends on the choice of the path, because $L_{S}(s)$
has singularities.
Since the residues of the singularities
are integers, we can use the same argument as in the proof of Theorem 6.2.
If we exponentiate the right hand side of (7.6), then this exponential
is independent of the choice of the path. The meromorphic continuation of the
symmetrized zeta function $S(s;\sigma,\chi)$ to the whole complex plane follows.

\end{proof}

\begin{center}
\section{\textnormal{Meromorphic continuation of the Selberg and Ruelle zeta function}}
\end{center}

\begin{thm}
The Selberg zeta function $Z(s;\sigma,\chi)$ admits a meromorphic continuation to the whole complex plane $\mathbb{C}$. The set of the singularities
equals to $\{s_{k}^{\pm}=\pm i\lambda_{k}:\lambda_{k}\in \spec(D^{\sharp}_{\chi}(\sigma)),k\in\mathbb{N}\}$.
The orders of the singularities are equal to $\frac{1}{2}(\pm m_{s}(\lambda_{k})+m(\lambda_{k}^{2})$.
For $\lambda_{0}=0$, the order of the singularity is equal to $m(0)$.
\end{thm}
\begin{proof}
 We observe at first that
\begin{equation*}
  Z(s;\sigma,\chi)=\sqrt{S(s;\sigma,\chi)Z^{s}(s;\sigma,\chi)}.
 \end{equation*}
Recall that by equation (4.8) we have $A^{\sharp}_{\chi}(\sigma)=(D^{\sharp}_{\chi}(\sigma))^{2}$. Hence, we can identify the eigenvalues $\mu_{k}$
of $A^{\sharp}_{\chi}(\sigma)$ with $\lambda_{k}^{2}$, where $\lambda_{k}\in \spec(D^{\sharp}_{\chi}(\sigma))$.
By Theorem 6.2 and Theorem 7.1, the product $S(s;\sigma,\chi)Z^{s}(s,\sigma,\chi)$ has its singularities at $s_{k}^{\pm}=\pm i\lambda_{k}$,
of order $\pm m_{s}(\lambda_{k})+m(\lambda_{k}^{2})$.
We need to prove that the order of the singularities of $Z(s;\sigma,\chi)$ is an even integer.
This follows from the definition of the algebraic multiplicities $m_{s}(\lambda_{k}),m(\lambda_{k}^{2})$
and the constuction of the locally homogenous vector bundles $E(\sigma),E_{\tau_{s}(\sigma)}$ associated to the representations $\tau(\equiv\tau_{\sigma})$ and
$\tau_{s}(\sigma)$ of $K$.
By \cite[ Proposition 5.4]{Spil} together with equation (4.5), and Proposition 4.1 together with equation (4.2),
we can choose representations $\tau$ of $K$, such that
$E(\sigma)=E_{\tau_{s}(\sigma)}$ up to a $\mathbb{Z}_{2}$-grading.
Hence, $
 m_{s}(\lambda_{k})\equiv m(\lambda_{k}^{2})\mod 2.
$
The assertion follows.
\end{proof}
\begin{thm}
For every $\sigma\in\widehat{M}$, the Ruelle zeta function $R(s;\sigma,\chi)$ admits a meromorphic continuation to the whole complex plane $\mathbb{C}$.
\end{thm}
\begin{proof}
The assertion follows from Theorem 8.1 together with \cite[Theorem 6.6]{Spil}.
\end{proof}

\bibliographystyle{amsalpha}

redpajama_set_name: RedPajamaArXiv


Follow: Send John to Together For The Gospel.
Hey y'all! Thanks for visiting my page.
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text:
Das Jagdschloss Gammelsbach, auch Jägerhaus Steingrund oder Jagdhaus Steingrund genannt, ist ein ehemaliges Jagdschloss in Gammelsbach (Steingrund 31), einem Ortsteil der Stadt Oberzent im Odenwaldkreis in Hessen.

Unter der Bauleitung des gräflichen Rentmeisters Klump ließ Ludwig II. Friedrich Graf zu Erbach-Fürstenau in den Jahren 1767 bis 1769 ein stattliches Jagdhaus errichten.

Baubeschreibung

Das ehemalige Jagdschloss ist ein stattliches, zweigeschossiges Bauwerk in Massivbauweise, verputzt und mit Ortskanten aus Haustein. Das Haus, Nord-Süd ausgerichtet, ist zu drei auf sechs Achsen geteilt mit segmentbogigen Fensteröffnungen und hat ein Krüppelwalmdach, dass im unteren Dachbereich mit fünf versetzten Bogendachgauben, im oberen Dachbereich mit drei Dreiecksgauben versehen ist. Das Anwesen hat die typisch barocke exponierte Lage im Blickpunkt eines kleinen, sich an dieser Stelle gabelnden Seitentals. Es steht mit zwei alten Linden auf einem hohen, von mächtigen, wehrmäßig angelegten Stützmauern terrassenartig umsäumten künstlich aufgeschütteten Plateau, das über zwei Rampen erschlossen ist. Nördlich davon steht ein kleines Gebäude als Querriegel vorgesetzt, ebenfalls mit einem Krüppelwalmdach versehen.

Heutige Nutzung
Das Anwesen ist restauriert, in Privatbesitz und kann nur von außen besichtigt werden.
Es ist mit den umliegenden Grundstücken ein hessisches Kulturdenkmal in seiner Sachgesamtheit.

Literatur
 Peter und Marion Sattler: Burgen und Schlösser im Odenwald. Verlag Edition Diesbach, 2004

Weblinks

 Schloss Gammelsbach auf der Seite wehrbauten.de

 Bild des ehemaligen Jagdschlosses

Einzelnachweise

Kulturdenkmal in Oberzent
Gammelsbach
Bauwerk in Oberzent
Barockbauwerk in Hessen
Bauwerk des Adelsgeschlechts Erbach
Gammelsbach
Gammelsbach

redpajama_set_name: RedPajamaWikipedia


text:
Tag Archives: President Muhammadu Buhari
Newswire : Nigeria to slash funds for essential food imports while currency crisis looms
Posted on August 21, 2019 by greenecodemocratcom
Nigeria currency
Aug. 12, 2019 (GIN) – Hard to imagine a steaming plate of Nigerian jollof rice without the rice. Or without fish. Or wheat. Rice, fish and wheat are Nigeria's top three food imports but foreign exchange for these staple food imports is about to end by order of President Muhammadu Buhari.
Presidential spokesman Garba Shehu said Tuesday the move to end imports of these essential foods is aimed at improving Nigeria's agricultural production and attaining food security.
"Don't give a cent to anybody to import food into the country," Buhari was quoted by his spokesman Shehu to say. "The foreign reserve will be conserved and utilized strictly for diversification of the economy, and not for encouraging more dependence on foreign food import bills," he added.
Imported milk and other dairy products will also be restricted from access to foreign exchange in an effort to boost local production and investment in ranches.
Some may have hoped these risky ideas would be forgotten in time. But a recently decided lawsuit in the UK may have prompted the government to advance the timetable.
The lawsuit, decided this month, gives a company called Process and Industrial Developments Ltd the right to pursue some $9 billion in assets from the Nigerian government over an aborted gas project.
Currently Nigeria spends US$22 billion on food imports annually. Rice, imported from Thailand and India, accounts for about US$1.65 billion which could make Nigeria the world's second largest importer of rice after China in 2019.
Many obstacles stand in the way of Nigeria becoming self-sufficient in food, some experts say. These include climate change, weeds, pests and diseases, farmers' limited access to credit, training, rudimentary and time consuming tools like hoes, slashers, sickles, axes and rakes.
Economic analyst Tokunbo Afikuyomi says making it harder for businesses to import food through official channels will push importers to find foreign exchange on the black market.
"Making it harder for businesses to import food through official channels is likely to lead to higher food prices as businesses use more expensive exchange rates or expensive domestic alternatives," Afikuyomi told CNN.
He said Nigeria's strategy should be to produce which foods it can grow cheaply and import others that are more expensive to make.
"Nigeria cannot produce all the food it eats — no country in the world is able to achieve this. Banning food imports to save foreign exchange is not the way to build a sustainable economy," he added.
Posted in Community, Crime, Education, history, Newswire | Tagged Nigeria to slash funds for essential food imports while currency crisis looms, President Muhammadu Buhari, Presidential spokesman Garba Shehu | Leave a comment
Newswire: U. S. Secretary of State fired, cutting short long awaited Africa visit
Posted on March 14, 2018 by greenecodemocratcom
Secretary of State Tillerson meets with Chadian foreign minister
Mar. 12, 2018 (GIN) – An extended visit to Africa this month by the U.S. Secretary of State to mend fences after the President's crude description of African and Caribbean countries was cut short this week by the dismissal of the embattled Secretary Rex Tillerson.
It was the first tour of the continent by the ex-Secretary who was the first high level U.S. representative to visit the continent since the President's vulgar remarks. Tillerson has been replaced by the CIA director, Mike Pompeo.
The President — who has long clashed will Tillerson— felt it was important to make the change now, as he prepares for talks with North Korean leader Kim Jong Un, as well as upcoming trade negotiations, three White House officials said.
"(The President's) statements shocked almost all Africans," Chadian Foreign Minister Mahamat Zene Cherif said at a press conference seated next to Tillerson, but added, "We made efforts on either side to move ahead and look at the future with optimism."
Quietly, however, Chadian President Idriss Déby made his anger known about his country being targeted in a Muslim travel ban despite close working relations between the two countries on anti-terrorism.
Other planned activities that were called off included the laying of a wreath at the memorial to victims of the 1998 U.S. embassy bombings in Nairobi and Dar es Salaam, and a working luncheon with Kenyan leaders.
A full day of meetings in Nigeria was reduced to a quick chat with President Muhammadu Buhari and his foreign minister before hopping a flight home.
Among the responses to the State Department's visit was a published piece by visiting professor of international relations at South Africa's University of the Witwatersrand, Joseph J. Stremlau, titled Three Reasons why Africa should treat the visit with scepticism.
The Secretary was scheduled to visit just five of Africa's 54 countries: Chad, Djibouti, Ethiopia, Kenya and Nigeria – the so-called "arc of instability," noted Stremlau..
"Evidently, counter terrorism is America's main Africa concern," he wrote. "If Trump and Tillerson were seriously interested in issues of trade, public health, and good governance, he would have at least included democratic South Africa on his agenda. Instead, Tillerson has prioritized repressive governments and ones under states of emergency."
Stremlau's second reason for skepticism is that the US hasn't shown much appetite for diplomatic engagement with Africa since Trump became president.
"By contrast, the US military is already deeply engaged in the struggle against counter terrorism."
"Neither Trump or Tillerson has announced an overarching Africa policy. No assistant secretary for Africa has been named, important embassies, including in South Africa, lack ambassadors. This is in sharp contrast to the US's active engagement on the military front," he pointed out.
"Finally, said Stremlau, "Tillerson has no credibility. He has been publicly criticized and even mocked by his commander-in-chief. Rumors persist that he will resign or be fired. And any claims that the government he represents means what it says is undermined by Trump's own false or misleading statements."
Stremlau congratulated African leaders for setting a dignified precedent when Trump compared African countries to dirty toilets.
Africans should also remind Tillerson of their appreciation of China's increasing importance as their leading development partner, Stremlau urged.
He added: "It's worth recalling that since the 1990s, Congress has consistently supported expanding economic and political partnerships with Africa.
"The reason for this is that congressmen have been pressed to do so by African-Americans as well as other sympathetic elements in America's diverse civil society, business, and philanthropic sectors.
"Networks such as these, as well as close ties at state and local government level that stretch throughout Africa may indeed be more important in the long run."
Posted in Community, Crime, Education, Newswire | Tagged Chadian foreign minister, Chadian Foreign Minister Mahamat Zene Cherif, Chadian President Idriss Déby, CIA director, cutting short long awaited Africa visit, Mike Pompeo, President Muhammadu Buhari, Secretary of State Tillerson, U. S. Secretary of State fired | Leave a comment
Newswire : Africa's floods largely go unnoticed despite higher death toll
Posted on September 7, 2017 by greenecodemocratcom
Women wades through flood waters in Niamey, Niger, west Africa
Sep. 5, 2017 (GIN) – "Floods in Africa in August killed 25 times more people than Hurricane Harvey did."
That was the headline of a recent story in Quartz online by Lagos-based writer Yomi Kazeem. "Like severe floods in southern Asia, the disasters in Africa have been largely under-reported compared to similar events in Houston where Hurricane Harvey, a once in a "500-year storm" has wreaked havoc," wrote Kazeem.
Across Texas, 50 people have been reported dead due to the tropical storm but across Africa, intense rains and mudslides killed at least 1,240 people in August, he pointed out.
Besides the mudslide in Sierra Leone, destructive floods have been reported in Niamey, Niger's capital city. Last week, thousands of people in Niamey were advised to evacuate their homes following severe flooding. Back in May, the United Nations had warned that more than 100,000 people were at risk of the flooding and so far this year, the death toll has topped 40 people. Hundreds of homes have been destroyed forcing stranded residents to take shelter in local schools.
In Ituri, DR Congo, over 200 people are believed to have died after a mudslide hit a fishing village in mid-August. At the time, Pacifique Keta, deputy governor of the northeastern province, said rescue operations were "complicated" given the mountainous terrain of the area and the continued adverse weather. Like in Freetown, residents in the area have mushroomed on steep hillsides over the years worsening the effects of floods.
Similarly, in Benue, a state in Nigeria's middle belt, more than 110,000 people have been displaced in after intense rains. In addition to thousands of homes, local markets and government offices were badly affected by the flood, according to the state emergency agency.
While President Muhammadu Buhari says he's received reports of the flood with "great concern" and has ordered relief efforts, very little has been forthcoming. Collins Uma, a Benue-based writer says most of the relief efforts made available so far are down to the "efforts of individuals and groups, not the government."
The floods in Benue are also not a new event. "This happens every year and will likely happen next year, but the government pays lip service to it," Benue-based writer Collins Uma tells Quartz. "If we had working drainage systems, this would not happen."
Climate scientists have been warning for several years that global warming is contributing the rise of the oceans and the intensification of major storms, like hurricanes, typhoons and other natural phenomenon.
Posted in Community, Crime, Education, General News, Newswire | Tagged a Benue-, Africa's floods largely go unnoticed despite higher death toll, Collins Uma, deputy governor, Pacifique Keta, President Muhammadu Buhari, Yomi Kazeem | Leave a comment
Newswire : Nigerian leader, ending 3 month hospital stay, returns with fiery speech
Nigerian President Buhari
Aug. 21, 2017 (GIN) – After a long absence due to ill health, President Muhammadu Buhari returned to Nigeria pledging to renew the fight against terror group Boko Haram and end a rash of ethnic violence which he blamed on "political mischief makers."
Buhari, 74, looking thin and frail, referred to a movement of Biafrans seeking an independent region in the southwest. The group had crossed a "national red line," he said. The separatists were "daring to question our collective existence as a nation. This is a step too far."
Fifty years ago, the people of "Igbo-land", waged a futile war of secession against the well-armed Nigerian government, citing the persecution of Igbos and control over oil production in the Niger Delta.
The Federal Military Government imposed a blockade which led to severe famine. Over the two and half years of war, between 500,000 and 2 million Biafran civilians perished from starvation.
While the Igbos comprise one of the three largest ethnic groups, they have fewer states than the Hausas in the north and the Yorubas in the south-west, resulting in a smaller budget allocation.
This, some feel, puts them behind the other regions. The south-east has not been at the forefront of Nigeria's development and none of its cities are major economic hubs.
Nnamdi Kanu, leader of the Indigenous People of Biafra, has called on his followers to boycott local and national elections scheduled for Nov. 18. Arrested in October 2015 for allegedly intending to levy war against Nigeria, he faces a trial on charges ranging from treasonable felony, terrorism and illegal possession of firearms.
Meanwhile, many Nigerians were disappointed that Buhari failed to comment on his prolonged health issues, but no reasons were given for his absence or the diagnosis.
Solomon Okoduwa told a reporter: "As a leader and the number one citizen, he should be able to tell his subjects the kind of headache, the kind of malaria, the kind of fever he went to treat in the UK that we cannot treat in Nigeria," he said.
Failure to disclose the nature of his health issues is widely seen as an indication that he has something to hide.
Nigerians will be watching whether he is going to make a third medical trip abroad this year. If that were to happen, calls for his resignation will surely be expressed even louder.
Posted in Community, Education, General News, Politics | Tagged Boko Haram, Federal Military Government, leader of the Indigenous People of Biafra, Nnamdi Kanu, President Muhammadu Buhari, Solomon Okoduwa | Leave a comment
82 kidnapped Nigerian schoolgirls are reunited with their families, after trade for 5 Boko Haram fighters
Posted on May 11, 2017 by greenecodemocratcom
By PADDY DINHAM FOR MAILONLINE
Kidnapped Nigerian schoolgirls Abducted by Boko Haram
82 schoolgirls who were kidnapped by Boko Haram have been released after more than three years in captivity.
They are the largest group yet to be released after years of tense negotiations between the government and the terrorist group, a Nigerian government spokesman said. Later government statements indicated that the girls were released in exchange for the government release of five Boko Haram fighters.
The girls were among about 220 students abducted from a secondary school in the northeastern town of Chibok in 2014, sparking a global campaign #bringbackourgirls supported by then-U.S. First Lady Michelle Obama, and a host of other celebrities.
· Enoch Mark, a Christian pastor whose two daughters were among those kidnapped, said he was told of the release by the Bring Back Our Girls pressure group and an official in Maiduguri. He added: 'This is good news to us. We have been waiting for this day. We hope the remaining girls will soon be released.'
Bring Back Our Girls said it was awaiting an official statement but added: 'Our hopes and expectations are high as we look forward to this news being true and confirmed.'
Boko Haram fighters stormed the Government Girls Secondary School in the remote town of Chibok on the evening of April 14, 2014 and kidnapped 276 girls. Fifty-seven managed to escape in the hours that followed but the remaining 219 were held by the group.
Boko Haram leader Abubakar Shekau claimed in a video message that they had converted to Islam. The kidnapping brought the insurgency to world attention, triggering global outrage that galvanized support across the globe.
21 Chibok girls were released in October in a deal brokered by Switzerland and the International Red Cross, while a handful of others have escaped or been rescued. However, a large number of the girls are still missing.
Last month President Muhammadu Buhari said in a statement that the government was 'in constant touch through negotiations, through local intelligence to secure the release of the remaining girls and other abducted persons unharmed'.
The girls were taken from a school in Chibok in the remote northeastern Borno state where Boko Haram has waged an insurgency aimed at creating an Islamic state that has killed thousands and displaced more than 2 million people.
Posted in Community, Crime, General News, history, Newswire, Politics, World News | Tagged 82 kidnapped Nigerian schoolgirls are reunited with their families, Abubakar Shekau, after trade for 5 Boko Haram fighters, President Muhammadu Buhari | Leave a comment
Nigerians put the brakes on superhighway cutting through rainforest
Posted on April 5, 2017 by greenecodemocratcom
Ekuri woman protesting superhighway
(TriceEdneyWire.com/GIN) – Bulldozers approaching the communities of the southern state of Cross River, with orders to raze up to a million homes and cut down an ancient tropical rainforest, were stopped in their tracks as an environmental impact statement for a proposed superhighway was rejected by officials.
As conceived, the roadway would link northern Nigeria to a proposed deep seaport in the south, covering 162 miles and displacing along the way some 180 indigenous communities, a national park and adjoining forest reserves that are home to some of the country's most endangered species.
But this week, at a public hearing with government ministers and stakeholders, the Minister of the Environment admitted the project could not go forward. "The EIA (Environmental Impact Assessment) is not of standard, it is too primary and does not qualify as a working document for such an international project," Minister Ibrahim Jibrin was reported to say.
The so-called Cross River Superhighway, the brainchild of the state's governor, Ben Ayade, has been on the drawing board for years. Two years ago it was announced that the "much anticipated construction" was "on course" and that President Muhammadu Buhari would be performing groundbreaking ceremonies Sept. 21, 2015.
The roadway would have cut through several protected areas such as the Cross River National Park, Ukpon River Forest Reserve, Cross River South Forest Reserve, Afi River Forest Reserve and Afi Mountain Wildlife Sanctuary – home to various threatened species, including Nigeria-Cameroon chimpanzees, drills, Preuss's red colobus monkeys, slender-snouted crocodiles and African gray parrots, among others.
Among the impacted communities is the Ekuri, whose conservation skills were recognized by the U.N. Development Program with a Equator Initiative Award for protecting biodiversity and reducing poverty.
Ekuri leaders say they supported the highway project at first, believing it would bring better transportation and greater economic opportunities to their people. But in a letter to the governor of Cross River State sent Feb. 7, the leaders withdrew their support, calling the project "a land grab in the guise of a Super Highway."
Ekuri communities manage some 83,000 acres of community-owned forest – one of the largest in West Africa. "We require schools, water, electricity but not the kind of road that will take our forest away," village leaders Stephen Oji and Abel Egbe told Premium Times news.
Posted in Community, Education, Newswire, Politics, School | Tagged Afi River Forest Reserve and Afi Mountain Wildlife Sanctuary, Ben Ayade, Cross River National Park, Cross River South Forest Reserve, EIA (Environmental Impact Assessment), Nigerians put the brakes on superhighway cutting through rainforest, President Muhammadu Buhari, state's governor, tropical rainforest, Ukpon River Forest Reserve | Leave a comment

redpajama_set_name: RedPajamaCommonCrawl


text:
Cochin Shipyard blast: Centre seeks report from CSL on explosion on under repair ONGC ship
Kochi | Published: February 14, 2018 4:29:24 PM
The Centre today sought a report from the Cochin Shipyard Limited (CSL) on the explosion and fire on an under repair ONGC drill ship, which claimed five lives.
KA police van outside Cochin Shipyard, where a blast occurred in an ONGC tanker that was under maintainance. (PTI)
The Centre today sought a report from the Cochin Shipyard Limited (CSL) on the explosion and fire on an under repair ONGC drill ship, which claimed five lives. Minister of State for Shipping Pon Radhakrishnan visited the shipyard this morning and took stock of the situation on board "SagarBhushan," a day after the ONGC ship was hit by the explosion while it was undergoing repair. "The Hon'ble Minister instructed CSL to conduct a thorough investigation and ascertain the root cause and report the details to Ministry of Shipping," a CSL statement said here. Talking to reporters, the Minister said there was an explosion but there was no sign of fire inside the ship. He said the CSL has begun a high level internal inquiry into the incident. Technical investigations are being carried out by DG Shipping and Factories and Boilers Department. "Police is also taking all steps.. after the enquiry only we can find out the reasons for the explosion," the Minister said, after visiting those who injured in a hospital here.
In a statement, the CSL had said five people died and seven were injured in the incident. The Minister appreciated the efforts taken by the CSL management in reaching out to the families of the deceased and also for promptly announcing the ex-gratia compensation to their kith and kin. He also enquired about the condition of those undergoing treatment and promised full support from the Ministry of Shipping in the company's endeavours to help them and their families, the CSL release said. During the interaction with the unions and associations, Radhakrishnan and the Chairman and Managing Director of the CSL, Madhu S Nair promised adequate support to all the affected families.
"The employees also appreciated the management for a positive and prompt response in the aftermath of the accident. The supervisory association expressed their appreciation for the remarkable and quick rescue operation and the presence of all senior executives on the spot without delay," the statement said. Principal Officer (I/C) Mercantile Marine Department under the DG Shipping, Ajithkumar Sukumaran said its investigation will cover all aspects including lapses that had occurred in conducting mandatory safety and security checks before commencing the repairing job. Union Minister for Road Transport and Highways, Shipping and Water Resources Nitin Gadkari had expressed shock over the loss of lives in the blast.
"Shocked by unfortunate blast at Cochin Shipyard… My heartfelt condolences to bereaved families. I have spoken to MD, Cochin Shipyard n asked him to provide all necessary medical support to victims n to initiate immed inquiry with help of concerned agencies (sic)," he had tweeted yesterday. CSL said the vessel had arrived at Cochin Shipyard on December 7, 2017 for dry dock repairs.
It was dry-docked — brought to land for repairs — on January 12, 2018 and it was to be undocked on February 28, 2018. The work completion was scheduled by April 7, 2018. "CSL places health, safety and environment (HSE) as one of its core value in operations," the company said in a statement yesterday.


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