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\begin{document}

\keywords{Ramanujan zeta-function, Ramanujan's tau-function, entire Ramanujan zeta-function, Ramanujan $\Xi$-function, Riemann-Hardy conjecture}

\subjclass[2020]{Primary:11-04; Secondary: 11Y35; 11Y55}

\title[Riemann-Hardy Conjecture]{On the Riemann-Hardy conjecture for the Ramanujan zeta-function}

\author{Xiao-Jun Yang}

\email{dyangxiaojun@163.com; xjyang@cumt.edu.cn}

\address{School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China}


\begin{abstract}
In this article we propose the integral, series and product representations for the Ramanujan zeta-function.
We suggest a variant for the Conrey-Ghosh product for the entire Ramanujan zeta-function.
We present some variants for the product for the Ramanujan $\Xi$-function. We prove that all of its zeros are real.
Along the way we obtain the truth of the Riemann-Hardy conjecture.
\end{abstract}

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\section{Introduction} {\label{sec:1}}
In his remarkable paper~\cite{1}, Ramanujan in 1916 introduced the zeta-function
$L_\tau \left( s \right)$, that in his honor is now called the Ramanujan
zeta-function, which is defined as~\cite{2}
\begin{equation}
\label{eq1}
L_\tau \left( s \right)=\sum\limits_{k=1}^\infty {\frac{\tau \left( k
\right)}{k^s}} \mbox{ }\left( {Re\left( s \right)>13/2} \right),
\end{equation}
and conjectured the arithmetical function $\tau \left( n \right)$ that in
his honor is now called the Ramanujan's tau-function, which is expressed by
(for example, see~\cite{3}, p.156)
\begin{equation}
\label{eq2}
G\left( x \right)=\sum\limits_{n=0}^\infty {\tau \left( n \right)x^n}
=x\prod\limits_{n=1}^\infty {\left( {1-x^n} \right)^{24}} \mbox{ }\left(
{\left| x \right|<1} \right),
\end{equation}
which was proved by Mordell in 1917~\cite{4,5}, where $s\in \mathds{C}$, $x\in \mathds{
R}$ and $k\in \mathds{N}$, if we denote the sets of the complex numbers, natural
numbers and real numbers by $\mathds{C}$ , $\mathds{N}$ and $\mathds{C}$, and the
real and imaginary parts of $s\in \mathds{C}$ by $Re\left( s \right)$ and
$Im\left( s \right)$, respectively.
Before the work of Deligne~\cite{6}, Ramanujan conjectured an Euler product for
Eq.~(\ref{eq1}), which can be expressed as~\cite{2,7,8}
\begin{equation}
\label{eq3}
L_\tau \left( s \right)=\prod\limits_p {\left( {1-\tau \left( n
\right)p^{-s}+p^{11-2s}} \right)^{-1}} =\prod\limits_p^\infty {\left(
{1-\alpha _1 p^{-s}} \right)^{-1}\left( {1-\overline {\alpha _2 } p^{-s}}
\right)^{-1}} ,
\end{equation}
where $p$ is prime and $Re\left( s \right)>13/2$.
The second term of Eq.~(\ref{eq3}) (see~\cite{3}, p.164) was conjectured by Ramanujan~\cite{1}
and proved by Mordell in 1917~\cite{4}, and the third term of Eq. (\ref{eq3}) was proved
by Deligne~\cite{6}.
Rankin in 1939~\cite{9} suggested the entire Ramanujan zeta-function $\xi
_\tau \left( s \right)$ by
\begin{equation}
\label{eq4}
\xi _\tau \left( s \right)=\left( {2\pi } \right)^{-s}\Gamma \left( s
\right)L_\tau \left( s \right),
\end{equation}
which leads to the functional equation
\begin{equation}
\label{eq5}
\xi _\tau \left( s \right)=\xi _\tau \left( {12-s} \right),
\end{equation}
proved by Wilton~\cite{10} and denoted by Conrey and Ghosh~\cite{2}.
By the result of the Weierstrass and Hadamard work on the entire functions~\cite{11},
Conrey and Ghosh pointed out that the entire Ramanujan zeta-function of order $\alpha =1$ can be rewritten
in the form~\cite{2}
\begin{equation}
\label{eq6}
\xi _\tau \left( s \right)=\xi _\tau \left( 0 \right)e^{\gamma
s}\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{s}{\rho _\vartheta }}
\right)e^{s/\rho _\vartheta }} ,
\end{equation}
where $\rho _\vartheta $ run the nontrivial zeros of $\xi _\tau \left( s
\right)$, $\vartheta \in \mathds{N}$ and $\gamma $ is a constant.
For $\varpi \ne 0$ and $i=\sqrt {-1} $, Rankin in 1939 showed~\cite{9}
\begin{equation}
\label{eq7}
\Delta _\tau \left( \varpi \right)=\frac{1}{\varpi ^{12}}\Delta _\tau \left(
{-\frac{1}{\varpi }} \right),
\end{equation}
which implies that~\cite{12}
\begin{equation}
\label{eq8}
\xi _\tau \left( s \right)=\int\limits_0^\infty {\varpi ^{s-1}\Delta _\tau
\left( {i\varpi } \right)d\varpi } ,
\end{equation}
and (see~\cite{11,12}, which is the special case of Wilton~\cite{10})
\begin{equation}
\label{eq9}
\Xi _\tau \left( t \right)=\xi _\tau \left( {6+it}
\right)=\int\limits_0^\infty {\varpi ^{5+it}\Delta _\tau \left( {i\varpi }
\right)d\varpi } =\int\limits_{-\infty }^\infty {F\left( l \right)e^{ilt}dl}
,
\end{equation}
where
\begin{equation}
\label{eq10}
F\left( l \right)=\Delta _\tau \left( {ie^l} \right)e^{6l}
\end{equation}
and~\cite{13}
\begin{equation}
\label{eq11}
\Delta _\tau \left( \varpi \right)=\eta \left( \varpi \right)^{24}=e^\varpi
\prod\limits_{k=1}^\infty {\left( {1-e^{k\varpi }} \right)^{24}}
=\sum\limits_{k=1}^\infty {\tau \left( k \right)e^{2\pi ik\varpi }} ,
\end{equation}
with
\begin{equation}
\label{eq12}
\eta \left( {-\frac{1}{\varpi }} \right)=\sqrt {\frac{\varpi }{i}} \eta
\left( \varpi \right),
\end{equation}
which satisfies the Dedekind eta function~\cite{14}
\begin{equation}
\label{eq13}
\eta \left( \varpi \right)=e^{\varpi /24}\prod\limits_{k=1}^\infty {\left(
{1-e^{k\varpi }} \right)} .
\end{equation}
Following the idea of Chirre and Casta\~{n}\'{o}n~\cite{14a}, $\Xi _\tau \left( t \right)$ is called the Ramanujan $\Xi$-function.
Based on the above results mentioned, the interesting topics for the Ramanujan
zeta-function have been conjectured by Hardy as follows (see~\cite{3}, p.174):
\begin{itemize}
  \item
 (Critical Strip) \\
 The critical strip for the Ramanujan zeta-function
is $11/2<Re\left( s \right)<13/2$;
  \item

 (Trivial Zeros)\\
 The trivial zeros for the Ramanujan zeta-function
is $s=-w$ with $w\in \mathds{N}\cup \left\{ \mbox{0} \right\}_{ }$;
\item
(Riemann-Hardy Conjecture)
\begin{conjecture}
\label{C1}
The nontrivial zeros for the Ramanujan zeta-function lie on the critical line $Re\left( s \right)=6$.
\end{conjecture}
\end{itemize}
Hardy in 1940 (see~\cite{3}, p.174) proposed the Riemann-Hardy conjecture for the Ramanujan zeta-function associated with
the Ramanujan's tau-function, that in his honor is as an analogue of Riemann conjecture for the
Riemann zeta-function. Up to now, it remains an important unsolved problem in the analytic
number theory.
Moreover, a great many of the interesting problems similar to the interesting topics for the Riemann
zeta-function have been reported by Hardy~\cite{3}. For example,
an analogue of the theorem of Hardy for the entire Ramanujan zeta-function was proposed by Hardy (see~\cite{3}, p.174) and proved by Lekkerkerker~\cite{15}.
An analogue of the von Mangoldt-like formula for the entire Ramanujan
zeta-function was conjectured by Hardy (see~\cite{3}, p.174) and proved by Ki~\cite{17}.
Ferguson and coauthors~\cite{18} reported 18 nontrivial zeros
for the Ramanujan zeta-function. An analogue of the Rieman-Siegel-like formula for the Ramanujan zeta-function (\ref{eq1}) was
conjectured by Hardy (see~\cite{3}, p.174) and proved by Keiper~\cite{19}.


Let $\varphi_\vartheta$ run the positive zeros for $\Xi _\tau \left( t \right)$.
In 1983, Hafner~\cite{13} proposed the equivalent form of \textit{Conjecture \ref{C1}}, which states that
\begin{equation}
\label{eq14}
N_0 \left( T \right)=N\left( T \right),
\end{equation}
where
\begin{equation}
\label{eq15}
N_0 \left( T \right)=\# \left\{ {s=6+it : 0\le
t \le T, \Xi _\tau \left( {s} \right)=0}
\right\}
\end{equation}
and
\begin{equation}
\label{eq16}
N\left( T \right)=\# \left\{ {s=\sigma +it
: 0\le t \le T,11/2<\sigma <13/2, \xi _\tau \left( {s} \right)=0} \right\}.
\end{equation}
Moreover, other statements equivalent to \textit{Conjecture \ref{C1}} were
reported in Moreno~\cite{7}. The main targets of our article are to proceed to prove
\textit{Conjecture \ref{C1}}, and to propose the series and product
formulas to give the structure of the product formulas to obtain this
conjecture. The structure of this article is given as follows. In Section \ref{sec:2}
we propose the integral and series representations for (\ref{eq1}) and the Lekkerkerker theorem.
In Section \ref{sec:3} we suggest the product formulas for
(\ref{eq1}) and (\ref{eq4}).
In Section \ref{sec:4} we prove that all zeros of the Ramanujan $\Xi$-function (\ref{eq9}) are real.
In Section \ref{sec:5} we also prove the truth of \textit{Conjecture \ref{C1}}.
Finally, we propose some equivalent theorems in Section \ref{sec:6}.

\section{The integral and series representations}{\label{sec:2}}

In this section we consider the integral and series representations for the Ramanujan
zeta-function.

Now we consider the remark on the work of Wilton to consider the integral
representations of them.

\begin{remark}
\label{R1}
In 1929, Wilton (see~\cite{10}, formula (5.2)) proved that
\[
\xi _\tau \left( s \right)=\int\limits_1^\infty {\left( {\varpi
^{s-6}+\varpi ^{6-s}} \right)\varpi ^6G\left( {e^{-2\pi \varpi }}
\right)\frac{d\varpi }{\varpi }} ,
\]
which leads to
\[
\begin{array}{l}
 \xi _\tau \left( s \right) \\
 =\int\limits_1^\infty {\left( {\varpi ^{s-6}\mbox{+}\varpi ^{6-s}}
\right)\varpi ^6G\left( {e^{-2\pi \varpi }} \right)\frac{d\varpi }{\varpi }}
\\
 =2\int\limits_1^\infty {\left( {\frac{e^{\left( {s-6} \right)\log \varpi
}\mbox{+}e^{-\left( {s-6} \right)}}{2}} \right)\varpi ^6G\left( {e^{-2\pi
\varpi }} \right)\frac{d\varpi }{\varpi }} \\
 =2\int\limits_1^\infty {\cosh \left( {\left( {s-6} \right)\log \varpi }
\right)\varpi ^5G\left( {e^{-2\pi \varpi }} \right)d\varpi } . \\
 \end{array}
\]
Thus, we have
\[
\xi _\tau \left( s \right)=2\int\limits_1^\infty {\cosh \left( {\left( {s-6}
\right)\log \varpi } \right){\mathds{X}}\left( \varpi \right)d\varpi } ,
\]
where
\begin{equation}
\label{eq17}
{\mathds{X}}\left( \varpi \right)=\varpi ^5G\left( {e^{-2\pi \varpi }} \right).
\end{equation}
\end{remark}

\begin{theorem}
\label{T1}
Let $\Delta _\tau \left( \varpi \right)=\sum\limits_{k=1}^\infty {\tau
\left( k \right)e^{k\varpi }} $. If $\xi _\tau \left( s
\right)$ is defined in (\ref{eq4}), then we have for $s\in \mathds{C}$,
\begin{equation}
\label{eq18}
\xi _\tau \left( s \right)=2\int\limits_1^\infty {\cosh \left( {\left( {s-6}
\right)\log \varpi } \right){\mathds{X}}\left( \varpi \right)d\varpi } ,
\end{equation}
where
\[
{\mathds{X}}\left( \varpi \right)=\varpi ^5G\left( {e^{-2\pi \varpi }}
\right)=\varpi ^5\Delta _\tau \left( {i\varpi } \right).
\]
\end{theorem}

\begin{proof}
By the definition of the entire Ramanujan zeta-function and (\ref{eq7}), we have
\begin{equation}
\label{eq19}
\begin{array}{l}
 \xi _\tau \left( s \right)\\
 =\int\limits_0^\infty {\varpi ^{s-1}\Delta _\tau
\left( {i\varpi } \right)d\varpi } \\
 =\int\limits_1^\infty {\varpi ^s\Delta _\tau \left( {i\varpi }
\right)\frac{d\varpi }{\varpi }} +\int\limits_0^1 {\varpi ^s\Delta _\tau
\left( {i\varpi } \right)\frac{d\varpi }{\varpi }} \\
 =\int\limits_1^\infty {\varpi ^s\Delta _\tau \left( {i\varpi }
\right)\frac{d\varpi }{\varpi }} -\int\limits_1^\infty {\varpi ^{12-s}\Delta
_\tau \left( {i\varpi } \right)\frac{d\varpi }{\varpi }} \\
 =\int\limits_1^\infty {\left( {\varpi ^s-\varpi ^{12-s}} \right)\Delta
_\tau \left( {i\varpi } \right)\frac{d\varpi }{\varpi }} \\
 =\int\limits_1^\infty {\left( {\varpi ^{s-6}-\varpi ^{6-s}} \right)\varpi
^6\Delta _\tau \left( {i\varpi } \right)\frac{d\varpi }{\varpi }} \\
 \end{array}
\end{equation}
and
\begin{equation}
\label{eq20}
\int\limits_0^1 {\varpi ^s\Delta _\tau \left( {i\varpi }
\right)\frac{d\varpi }{\varpi }} =-\int\limits_1^\infty {\varpi
^{12-s}\Delta _\tau \left( {i\varpi } \right)\frac{d\varpi }{\varpi }} ,
\end{equation}
where
\[
\Delta _\tau \left( {i\varpi } \right)=\frac{\Delta _\tau \left( {-1/i\varpi
} \right)}{\varpi ^{12}}.
\]
By (\ref{eq2}) and (\ref{eq11}), we arrive at the identity
\begin{equation}
\label{eq21}
G\left( {e^{-2\pi \varpi }} \right)=\Delta _\tau \left( {i\varpi } \right),
\end{equation}
which leads to
\[
{\mathds{X}}\left( \varpi \right)=\varpi ^5G\left( {e^{-2\pi \varpi }}
\right)=\varpi ^5\Delta _\tau \left( {i\varpi } \right).
\]
It follows that
\begin{equation}
\label{eq22}
\begin{array}{l}
 \xi _\tau \left( s \right)\\
 =\int\limits_1^\infty {\left( {\varpi
^{s-6}-\varpi ^{6-s}} \right)\varpi ^6\Delta _\tau \left( {i\varpi }
\right)\frac{d\varpi }{\varpi }} \\
 =\int\limits_1^\infty {\left( {e^{\left( {s-6} \right)\log \varpi
}-e^{-\left( {s-6} \right)}} \right)\varpi ^5\Delta _\tau \left( {i\varpi }
\right)d\varpi } \\
=2\int\limits_1^\infty {\left( {\frac{e^{\left( {s-6}
\right)\log \varpi }-e^{-\left( {s-6} \right)}}{2}} \right)\varpi ^5\Delta
_\tau \left( {i\varpi } \right)d\varpi } \\
 =2\int\limits_1^\infty {\cosh \left( {\left( {s-6} \right)\log \varpi }
\right)\varpi ^5\Delta _\tau \left( {i\varpi } \right)d\varpi }\\
=2\int\limits_1^\infty {\cosh \left( {\left( {s-6} \right)\log \varpi }
\right){\mathds{X}}\left( \varpi \right)d\varpi }.\\
 \end{array}
\end{equation}
Then, the desired result follows.

Hence, we finish the proof of \textit{Theorem \ref{T1}}.
\end{proof}

\begin{theorem}
\label{T2}
Let $\Delta _\tau \left( \varpi \right)=\sum\limits_{k=1}^\infty {\tau
\left( k \right)e^{k\varpi }} $.
If $\xi _\tau \left( s \right)$ is defined in (\ref{eq4}), then we have
for $m\in \mathds{N}\cup \left\{ 0 \right\}$ and $s\in \mathds{C}$,
\begin{equation}
\label{eq23}
\xi _\tau \left( s \right)=\sum\limits_{m=0}^\infty {\beta _m \left( {s-6}
\right)^{2m}} ,
\end{equation}
where
\begin{equation}
\label{eq24}
\beta _m =\frac{2}{\Gamma \left( {2m+1} \right)}\int\limits_1^\infty {\left(
{\log \varpi } \right)^{2m}{\mathds{X}}\left( \varpi \right)d\varpi } .
\end{equation}
\end{theorem}

\begin{proof}
By \textit{Theorem \ref{T1}}, we now show that
\begin{equation}
\label{eq25}
\begin{array}{l}
 \xi _\tau \left( s \right) \\
 =2\int\limits_1^\infty {\cosh \left( {\left( {s-6} \right)\log \varpi }
\right){\mathds{X}}\left( \varpi \right)d\varpi } \\
 =2\int\limits_1^\infty {\sum\limits_{m=0}^\infty {\frac{\left( {\left(
{s-6} \right)\log \varpi } \right)^{2m}}{\Gamma \left( {2m+1} \right)}} {\mathds{
X}}\left( \varpi \right)d\varpi } \\
 =\sum\limits_{m=0}^\infty {\left( {2\int\limits_1^\infty {\frac{\left(
{\log \varpi } \right)^{2m}}{\Gamma \left( {2m+1} \right)}{\mathds{X}}\left(
\varpi \right)d\varpi } } \right)\left( {s-6} \right)^{2m}} \\
 =\sum\limits_{m=0}^\infty {\left( {\frac{2}{\Gamma \left( {2m+1}
\right)}\int\limits_1^\infty {\left( {\log \varpi } \right)^{2m}{\mathds{
X}}\left( \varpi \right)d\varpi } } \right)\left( {s-6} \right)^{2m}} \\
 =\sum\limits_{m=0}^\infty {\beta _m \left( {s-6} \right)^{2m}} , \\
 \end{array}
\end{equation}
where
\begin{equation}
\label{eq26}
\beta _m =\frac{2}{\Gamma \left( {2m+1} \right)}\int\limits_1^\infty {\left(
{\log \varpi } \right)^{2m}{\mathds{X}}\left( \varpi \right)d\varpi } .
\end{equation}
Thus, the result follows.
\end{proof}

\begin{theorem}
\label{T3}
Let $\Delta _\tau \left( \varpi \right)=\sum\limits_{k=1}^\infty {\tau
\left( k \right)e^{k\varpi }} $. If $\Xi _\tau \left( t
\right)$ is defined in (\ref{eq9}), then we have for $t\in \mathds{R}$,
\begin{equation}
\label{eq27}
\Xi _\tau \left( t \right)=2\int\limits_1^\infty {\cos \left( {t\log \varpi
} \right){\mathds{X}}\left( \varpi \right)d\varpi } .
\end{equation}
\end{theorem}

\begin{proof}
Making use of (\ref{eq9}) and \textit{Theorem \ref{T1}}, we give
\begin{equation}
\label{eq28}
\begin{array}{l}
 \Xi _\tau \left( t \right)=\xi _\tau \left( {6+it} \right) \\
 =2\int\limits_1^\infty {\cosh \left\{ {\left[ {\left( {6+it} \right)-6}
\right]\log \varpi } \right\}{\mathds{X}}\left( \varpi \right)d\varpi } \\
 =2\int\limits_1^\infty {\cosh \left( {it\log \varpi } \right){\mathds{X}}\left(
\varpi \right)d\varpi } \\
 =2\int\limits_1^\infty {\cos \left( {t\log \varpi } \right){\mathds{X}}\left(
\varpi \right)d\varpi } , \\
 \end{array}
\end{equation}
and the result follows.
\end{proof}

\begin{remark}
In 1929, Wilton (see~\cite{10}, formulas (5.31) and (5.32)) proved that
\[
\Xi _\tau \left( t \right)=\int\limits_{-\infty }^\infty {\Lambda \left( l
\right)e^{ilt}dl} ,
\]
where
\[
\Lambda \left( l \right)=e^{6l}G\left( {e^{-2\pi e^l}} \right).
\]
It has been proved by Ferguson and coauthors~\cite{18} that
\[
\Xi _\tau \left( t \right)=2\int\limits_0^\infty {e^{6x}\Delta _\tau \left(
{ie^x} \right)\cos \left( {tx} \right)dx} ,
\]
which can be rewritten as (\ref{eq27}) when $x=\log \varpi $.

Thus, we find that
\[
\begin{array}{l}
 \Xi _\tau \left( t \right) \\
 =\int\limits_{-\infty }^\infty {e^{6l}G\left( {e^{-2\pi e^l}}
\right)e^{ilt}dl} \\
=2\int\limits_0^\infty {e^{6x}\Delta _\tau \left( {ie^x}
\right)\cos \left( {tx} \right)dx} \\
 =2\int\limits_1^\infty {\cos \left( {t\log \varpi } \right){\mathds{X}}\left(
\varpi \right)d\varpi } . \\
 \end{array}
\]
\end{remark}

\begin{theorem}
\label{T4}
Let $\Delta _\tau \left( \varpi \right)=\sum\limits_{k=1}^\infty {\tau
\left( k \right)e^{k\varpi }} $.
If $\Xi _\tau \left( t \right)$ is defined in (\ref{eq9}), then we have
for $m\in \mathds{N}\cup \left\{ 0 \right\}$,
\begin{equation}
\label{eq29}
\Xi _\tau \left( t \right)=\sum\limits_{m=0}^\infty {\left( {-1}
\right)^m\beta _m t^{2m}} ,
\end{equation}
where
\begin{equation}
\label{eq30}
\beta _m =\frac{2}{\Gamma \left( {2m+1} \right)}\int\limits_1^\infty {\left(
{\log \varpi } \right)^{2m}{\mathds{X}}\left( \varpi \right)d\varpi } .
\end{equation}
\end{theorem}

\begin{proof}
By \textit{Theorem \ref{T2}} and (\ref{eq9}), one obtains
\begin{equation}
\label{eq31}
\Xi _\tau \left( t \right)=\xi _\tau \left( {6+it}
\right)=\sum\limits_{m=0}^\infty {\beta _m \left[ {\left( {6+it} \right)-6}
\right]^{2m}} =\sum\limits_{m=0}^\infty {\left( {-1} \right)^m\beta _m
t^{2m}} ,
\end{equation}
where
\begin{equation}
\label{eq32}
\beta _m =\frac{2}{\Gamma \left( {2m+1} \right)}\int\limits_1^\infty {\left(
{\log \varpi } \right)^{2m}{\mathds{X}}\left( \varpi \right)d\varpi } .
\end{equation}
Thus, we finish the proof.
\end{proof}

\begin{remark}
It is clearly seen that \textit{Theorem \ref{T3}} also reduces to the desired
result.

By (\ref{eq27}) and (\ref{eq29}), we find that
\begin{equation}
\label{eq33}
\Xi _\tau \left( {-t} \right)=\Xi _\tau \left( t \right),
\end{equation}
which is in agreement with the result of Conrey and Ghosh~\cite{12}.

Moreover, there exist
\begin{equation}
\label{eq34}
\beta _m =\left[ {\frac{d^{2m}\Xi _\tau \left( t \right)}{dt^{2m}}}
\right]_{t=0} =\Xi _\tau ^{\left( {2m} \right)} \left( 0 \right)
\end{equation}
such that
\begin{equation}
\label{eq35}
\Xi _\tau \left( t \right)=\sum\limits_{m=0}^\infty {\left( {-1}
\right)^m\Xi _\tau ^{\left( {2m} \right)} \left( 0 \right)t^{2m}} .
\end{equation}
In view of (\ref{eq34}) and (\ref{eq30}), we arrive at
\begin{equation}
\label{eq36}
\beta _m =\left[ {\frac{d^{2m}\Xi _\tau \left( t \right)}{dt^{2m}}}
\right]_{t=0} =\Xi _\tau ^{\left( {2m} \right)} \left( 0
\right)=\frac{2}{\Gamma \left( {2m+1} \right)}\int\limits_1^\infty {\left(
{\log \varpi } \right)^{2m}{\mathds{X}}\left( \varpi \right)d\varpi } .
\end{equation}
Similarly, we also present
\begin{equation}
\label{eq37}
\beta _m =\left( {-1} \right)^m\left[ {\frac{d^{2m}\xi _\tau \left( s
\right)}{ds^{2m}}} \right]_{s=6} =\left( {-1} \right)^m\xi _\tau ^{\left(
{2m} \right)} \left( 6 \right).
\end{equation}
\end{remark}

By using the above remark, one has the following corollaries:

\begin{corollary}
\label{Co1}
If $\Xi _\tau \left( t \right)$
is defined in (\ref{eq9}), then we have for $m\in \mathds{N}\cup \left\{ 0 \right\}$,
\begin{equation}
\label{eq38}
\Xi _\tau \left( t \right)=\sum\limits_{m=0}^\infty {\left( {-1}
\right)^m\beta _m t^{2m}} ,
\end{equation}
where
\begin{equation}
\label{eq39}
\beta _m =\Xi _\tau ^{\left( {2m} \right)} \left( 0 \right).
\end{equation}
\end{corollary}

\begin{corollary}
\label{Co2}
If $\xi _\tau \left( s \right)$
is defined in (\ref{eq4}), then we have for $m\in \mathds{N}\cup \left\{ 0 \right\}$,
\begin{equation}
\label{eq40}
\xi _\tau \left( s \right)=\sum\limits_{m=0}^\infty {\beta _m \left( {s-6}
\right)^{2m}} ,
\end{equation}
where
\begin{equation}
\label{eq41}
\beta _m =\left( {-1} \right)^m\xi _\tau ^{\left( {2m} \right)} \left( 6
\right).
\end{equation}
\end{corollary}

In 1955, Lekkerkerker~\cite{15} showed that $\Xi _\tau \left( t \right)$ has
infinitely many zeros by Lekkerkerker theorem.

\begin{lemma}
(Lekkerkerker theorem)
\label{L2}
If $\Xi _\tau \left( t \right)$ is defined in (\ref{eq9}) and
\[
N_0 \left( T
\right)=\# \left\{ {s=6+it :0\le t \le T,\xi _\tau \left( {s} \right)=0} \right\},
\]
then we have
\begin{equation}
\label{eq43}
N_0 \left( T \right)>AT
\end{equation}
such that $\Xi _\tau \left( t \right)$ has infinitely many zeros, where $A$
is a positive constant.
\end{lemma}

\begin{proof}
See~\cite{15}.
\end{proof}

\section{The product formulas}{\label{sec:3}}

In this section we propose
a variant for the Conrey-Ghosh product for the entire Ramanujan zeta-function
and some product for Ramanujan zeta-function.

Now, we introduce the Conrey-Ghosh product
by applying the result of the Weierstrass and Hadamard~\cite{11}.

\begin{lemma}
(Conrey-Ghosh product)
\label{L3}

Let $\rho _\vartheta \in \mathds{C}$, $s\in \mathds{C}$ and $\vartheta \in {\rm
N}$. Suppose that $\rho _\vartheta $ run the nontrivial zeros of $\xi _\tau
\left( s \right)$ and $\gamma $ is a constant. If $\xi _\tau \left( s
\right)$ is defined in (\ref{eq4}), then we have
\begin{equation}
\label{eq44}
\xi _\tau \left( s \right)=\xi _\tau \left( 0 \right)e^{\gamma
s}\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{s}{\rho _\vartheta }}
\right)e^{s/\rho _\vartheta }} .
\end{equation}
\end{lemma}

\begin{proof}
For the proof of the Conrey-Ghosh product, see~\cite{2}.
\end{proof}

\begin{theorem}
\label{T5}

Let $\rho _\vartheta \in \mathds{C}$, $s\in \mathds{C}$ and $\vartheta \in {\rm
N}$. Suppose that $\rho _\vartheta $ run the nontrivial zeros of $\xi _\tau
\left( s \right)$ and $\gamma $ is a constant defined in (\ref{eq6}). If $\xi _\tau
\left( s \right)$ is defined in (\ref{eq4}) and $\xi _\tau \left( 0 \right)\ne 0$,
then we have
\begin{equation}
\label{eq45}
\xi _\tau \left( 6 \right)\ne 0
\end{equation}
and
\begin{equation}
\label{eq46}
\xi _\tau \left( {12} \right)\ne 0.
\end{equation}
\end{theorem}

\begin{proof}
By (\ref{eq5}) and \textit{Lemma \ref{L3}}, one has
\begin{equation}
\label{eq47}
\xi _\tau \left( s \right)=\xi _\tau \left( {12-s} \right)
\end{equation}
such that
\begin{equation}
\label{eq48}
\xi _\tau \left( {12} \right)=\xi _\tau \left( 0 \right)\ne 0
\end{equation}
and
\begin{equation}
\label{eq49}
\xi _\tau \left( 6 \right)=\xi _\tau \left( 0 \right)e^{6\gamma
}\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{6}{\rho _\vartheta }}
\right)e^{6/\rho _\vartheta }} \ne 0
\end{equation}
since $\rho _\vartheta \ne 6\in \mathds{R}$.

Thus, we complete the proof of \textit{Theorem \ref{T5}}.
\end{proof}

\begin{theorem}
\label{T6}
Let $\rho _\vartheta \in \mathds{C}$ and $\vartheta \in \mathds{N}$. Suppose
that $\rho _\vartheta $ run the nontrivial zeros of $\xi _\tau \left( s
\right)$ and $\gamma $ is a constant. If $\xi _\tau \left( s \right)$ is
defined in (\ref{eq4}) and $\Xi _\tau \left( t \right)$ is defined in (\ref{eq9}), then we
have
\begin{equation}
\label{eq50}
\xi _\tau \left( s \right)=\xi _\tau \left( 6 \right)e^{\gamma \left( {s-6}
\right)}\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{s-6}{\rho
_\vartheta -6}} \right)e^{\left( {s-6} \right)/\rho _\vartheta }}
\end{equation}
where $s\in \mathds{C}$, and
\begin{equation}
\label{eq51}
\Xi _\tau \left( t \right)=\xi _\tau \left( 6 \right)e^{i\gamma
t}\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{it}{\rho _\vartheta
-6}} \right)e^{it/\rho _\vartheta }} ,
\end{equation}
where $t\in \mathds{R}$.
\end{theorem}

\begin{proof}
With \textit{Lemma \ref{L3}} and (\ref{eq49}) one gives
\begin{equation}
\label{eq52}
\begin{array}{l}
 \xi _\tau \left( s \right)\\
 =\xi _\tau \left( 0 \right)e^{\gamma
s}\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{s}{\rho _\vartheta }}
\right)e^{s/\rho _\vartheta }} \\
=\xi _\tau \left( 0 \right)e^{\gamma
s}\prod\limits_{\vartheta =1}^\infty {\frac{\rho _\vartheta -s}{\rho
_\vartheta }e^{s/\rho _\vartheta }} \\
 =\xi _\tau \left( 0 \right)e^{\gamma s}\prod\limits_{\vartheta =1}^\infty
{\left( {\frac{\rho _\vartheta -6}{\rho _\vartheta -6}\cdot \frac{\rho
_\vartheta -s}{\rho _\vartheta }} \right)e^{s/\rho _\vartheta }} \\
=\xi _\tau
\left( 0 \right)e^{\gamma s}\prod\limits_{\vartheta =1}^\infty {\left(
{\frac{\rho _\vartheta -6}{\rho _\vartheta }\cdot \frac{\rho _\vartheta
-s}{\rho _\vartheta -6}} \right)e^{s/\rho _\vartheta }} \\
 =\xi _\tau \left( 0 \right)e^{\gamma s}\prod\limits_{\vartheta =1}^\infty
{\left( {\frac{\rho _\vartheta -6}{\rho _\vartheta }\cdot \frac{\rho
_\vartheta -s}{\rho _\vartheta -6}} \right)e^{s/\rho _\vartheta }} \\
=\xi
_\tau \left( 0 \right)e^{\gamma s}\prod\limits_{\vartheta =1}^\infty
{\frac{\rho _\vartheta -6}{\rho _\vartheta }} \prod\limits_{\vartheta
=1}^\infty {\frac{\rho _\vartheta -s}{\rho _\vartheta -6}e^{s/\rho
_\vartheta }} \\
 =\xi _\tau \left( 0 \right)e^{\gamma s}\prod\limits_{\vartheta =1}^\infty
{\left( {1-\frac{6}{\rho _\vartheta }} \right)} \prod\limits_{\vartheta
=1}^\infty {\frac{\rho _\vartheta -6-\left( {s-6} \right)}{\rho _\vartheta
-6}e^{s/\rho _\vartheta }} \\
 =\xi _\tau \left( 0 \right)e^{\gamma s}\prod\limits_{\vartheta =1}^\infty
{\left( {1-\frac{6}{\rho _\vartheta }} \right)} \prod\limits_{\vartheta
=1}^\infty {\left( {1-\frac{s-6}{\rho _\vartheta -6}} \right)e^{s/\rho
_\vartheta }} \\
=\xi _\tau \left( 6 \right)e^{\gamma \left( {s-6}
\right)}\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{s-6}{\rho
_\vartheta -6}} \right)e^{\left( {s-6} \right)/\rho _\vartheta }} , \\
 \end{array}
\end{equation}
which completes the proof of (\ref{eq50}).

Putting $s=6+it$ into (\ref{eq50}), one obtains (\ref{eq51}).

Thus, we finish the proof of \textit{Theorem \ref{T6}}.
\end{proof}

\begin{theorem}
\label{T7}
Let $\rho _\vartheta \in \mathds{C}$ and $\vartheta \in \mathds{N}$. Suppose
that $\rho _\vartheta $ run the nontrivial zeros of $\xi _\tau \left( s
\right)$ and $\gamma $ is a constant. Then we have for $s\in \mathds{C}$,
\begin{equation}
\label{eq53}
L_\tau \left( s \right)=\frac{\xi _\tau \left( 0 \right)\left( {2\pi }
\right)^se^{\gamma s}}{\Gamma \left( s \right)}\prod\limits_{\vartheta
=1}^\infty {\left( {1-\frac{s}{\rho _\vartheta }} \right)e^{s/\rho
_\vartheta }} ,
\end{equation}
and
\begin{equation}
\label{eq54}
L_\tau \left( s \right)=\frac{\xi _\tau \left( 6 \right)\left( {2\pi }
\right)^se^{\gamma \left( {s-6} \right)}}{\Gamma \left( s
\right)}\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{s-6}{\rho
_\vartheta -6}} \right)e^{\left( {s-6} \right)/\rho _\vartheta }}.
\end{equation}
\end{theorem}

\begin{proof}
By (\ref{eq4}) and \textit{Lemma \ref{L3}}, one obtains
\begin{equation}
\label{eq55}
L_\tau \left( s \right)=\frac{\xi _\tau \left( 0 \right)\left( {2\pi }
\right)^se^{\gamma s}}{\Gamma \left( s \right)}\prod\limits_{\vartheta
=1}^\infty {\left( {1-\frac{s}{\rho _\vartheta }} \right)e^{s/\rho
_\vartheta }} ,
\end{equation}
and by (\ref{eq4}) and \textit{Theorem \ref{T6}}, one gives
\begin{equation}
\label{eq56}
\zeta _\tau \left( s \right)=\frac{\xi _\tau \left( 6 \right)\left( {2\pi }
\right)^se^{\gamma \left( {s-6} \right)}}{\Gamma \left( s
\right)}\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{s-6}{\rho
_\vartheta -6}} \right)e^{\left( {s-6} \right)/\rho _\vartheta }} .
\end{equation}
Thus, we finish the proof of \textit{Theorem \ref{T7}}.
\end{proof}

\begin{remark}
By \textit{Theorem \ref{T7}}, it is clearly shown that $L_\tau \left( s \right)$ has
the trivial zeros $s=-w$ with $w\in \mathds{N}\cup \left\{ \mbox{0} \right\}$,
controlled by poles of $\Gamma \left( s \right)$, and that
$\rho _\vartheta $ run the nontrivial zeros of $L_\tau \left( s \right)$
(for more details, see~\cite{3}, p.174).
Hardy proved in 1940 (see~\cite{3}, p.174) that $L_\tau \left( {11/2+it}
\right)\ne 0$ and that $L_\tau \left( {13/2+it} \right)\ne 0$ for $t\in \mathds{
R}$.
\end{remark}

\section{Real zeros} {\label{sec:4}}

In this section we will prove that all of the zeros for the Ramanujan $\Xi$-function $\Xi _\tau \left( t \right)$ are real by
the Lekkerkerker theorem and \textit{Theorem \ref{T6}}.

\begin{corollary}
\label{Co3}

Let $t\in \mathds{R}$. If $\Xi _\tau \left( t \right)$ is defined in (\ref{eq9}),
then all of the zeros of $\Xi _\tau \left( t \right)$ are real.
\end{corollary}

\begin{proof}
Applying \textit{Theorem \ref{T6}}, we have
\begin{equation}
\label{eq1a}
\Xi _\tau \left( t \right)=\Xi _\tau \left( 0 \right)e^{i\gamma
t}\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{it}{\rho _\vartheta
-6}} \right)e^{it/\rho _\vartheta }} ,
\end{equation}
where $t\in \mathds{R}$, $\vartheta \in \mathds{N}$, and the product runs over
zeroes $\rho _\vartheta $ of the entire Ramanujan zeta-function $\xi_\tau \left( s \right)$.

In order to investigate the zeros of $\Xi _\tau \left( t \right)$, we now
rewrite (\ref{eq1a}) as
\begin{equation}
\label{eq2a}
\Xi _\tau \left( t \right)=\Xi _\tau \left( 0 \right)e^{i\gamma
t}\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{it}{\psi _\vartheta }}
\right)e^{it/\rho _\vartheta }} ,
\end{equation}
where $t\in \mathds{R}$, $\vartheta \in \mathds{N}$, and the product runs over the
zeroes $\psi _\vartheta $ of $\Xi _\tau \left( t \right)$ and the zeroes
$\rho _\vartheta $ of the entire Ramanujan zeta-function $\xi_\tau \left( s \right)$.

Now, let us consider the first hypothesis that all zeros of $\Xi _\tau
\left( t \right)$ are given by
\begin{equation}
\label{eq3a}
\psi _\vartheta =\widetilde{\sigma }_\vartheta +i\widetilde{\tau }_\vartheta
,
\end{equation}
where $\widetilde{\sigma }_\vartheta \in \mathds{R}\backslash \left\{ 0
\right\}$ and $\widetilde{\tau }_\vartheta \in \mathds{R}\backslash \left\{ 0
\right\}$, and the second hypothesis that all zeros of $\Xi _\tau \left( t
\right)$ are
\begin{equation}
\label{eq4a}
\psi _\vartheta =\ell _\vartheta \quad ,
\end{equation}
where $\ell _\vartheta \in \mathds{R}\backslash \left\{ 0 \right\}$. Our idea
is that if two cases are false, then we have that all zeros of $\Xi
_\tau \left( t \right)$ are real.

We now start to prove that they are false.

\textit{Hypothesis 1.}

Now, we consider the first hypothesis that the complex zeros for $\Xi
_\tau \left( t \right)$ can be expressed in the form
\begin{equation}
\label{eq5a}
\psi _\vartheta =\widetilde{\sigma }_\vartheta +i\widetilde{\tau }_\vartheta
\end{equation}
where $\widetilde{\sigma }_\vartheta \in \mathds{R}\backslash \left\{ 0
\right\}$ and $\widetilde{\tau }_\vartheta \in \mathds{R}\backslash \left\{ 0
\right\}$.

By (\ref{eq2a}) and (\ref{eq5a}), we now consider
\begin{equation}
\label{eq6a}
\begin{array}{l}
 \Xi _\tau \left( t \right) \\
 =\Xi _\tau \left( 0 \right)e^{i\gamma t}\prod\limits_{\vartheta =1}^\infty
{\left( {1-\frac{it}{\widetilde{\sigma }_\vartheta +i\widetilde{\tau
}_\vartheta }} \right)e^{it/\rho _\vartheta }} \\
 =\Xi _\tau \left( 0 \right)e^{i\gamma t}\prod\limits_{\vartheta =1}^\infty
{\left( {1-\frac{it}{\psi _\vartheta }} \right)e^{it/\rho _\vartheta }} . \\
 \end{array}
\end{equation}
Putting $t=\left( {s-6} \right)/i$ into (\ref{eq6}), we now give
\begin{equation}
\label{eq7a}
\begin{array}{l}
 \xi _\tau \left( s \right)=\Xi _\tau \left( {\left( {s-6} \right)/i}
\right) \\
 =\xi _\tau \left( 6 \right)e^{\gamma \left( {s-6}
\right)}\prod\limits_{\vartheta =1}^\infty {\left(
{1-\frac{s-6}{\widetilde{\sigma }_\vartheta +i\widetilde{\tau }_\vartheta }}
\right)e^{\left( {s-6} \right)/\rho _\vartheta }} \\
 =\xi _\tau \left( 6 \right)e^{\gamma \left( {s-6}
\right)}\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{s-6}{\psi
_\vartheta }} \right)e^{\left( {s-6} \right)/\rho _\vartheta }} \\
 \end{array}
\end{equation}
since $\Xi _\tau \left( 0 \right)=\xi _\tau \left( 6 \right)$.

Let $\rho _\vartheta =Re\left( {\rho _\vartheta } \right)+i\psi _\vartheta $
run all roots of $\xi _\tau \left( s \right)=0$, where $Re\left( {\rho
_\vartheta } \right)\in \mathds{R}\backslash \left\{ 0 \right\}$ and $\psi
_\vartheta \in \mathds{R}\backslash \left\{ 0 \right\}$. Then, we have from
(\ref{eq7a}) that
\begin{equation}
\label{eq8a}
\xi _\tau \left( {\rho _\vartheta } \right)=\xi _\tau \left( 6
\right)e^{\gamma \left( {\rho _\vartheta -6}
\right)}\prod\limits_{n=1}^\infty {\left( {1-\frac{\rho _\vartheta
-6}{\widetilde{\sigma }_\vartheta +i\widetilde{\tau }_\vartheta }}
\right)e^{\left( {\rho _\vartheta -6} \right)/\rho _\vartheta }} =0,
\end{equation}
where $\xi _\tau \left( 6 \right)\ne 0$.

From (\ref{eq8a}) we show
\begin{equation}
\label{eq9a}
\widetilde{\sigma }_\vartheta +i\widetilde{\tau }_\vartheta -\left( {\rho
_\vartheta -6} \right)=0
\end{equation}
since $\xi _\tau \left( 6 \right)\ne 0$ and $e^{\gamma \left( {\rho
_\vartheta -6} \right)}\prod\limits_{n=1}^\infty {e^{\left( {\rho _\vartheta
-6} \right)/\rho _\vartheta }} \ne 0$.

By (\ref{eq9a}), we present
\begin{equation}
\label{eq10a}
s_\vartheta =Re\left( {\rho _\vartheta } \right)+i\psi _\vartheta
=\widetilde{\sigma }_\vartheta +6+i\widetilde{\tau }_\vartheta ,
\end{equation}
where $\widetilde{\sigma }_\vartheta \in \mathds{R}\backslash \left\{ 0
\right\}$ and $\widetilde{\tau }_\vartheta \in \mathds{R}\backslash \left\{ 0
\right\}$.

By $\widetilde{\sigma }_\vartheta \in \mathds{R}\backslash \left\{ 0 \right\}$
and (\ref{eq10a}), that the critical line is expressed as
\begin{equation}
\label{eq11a}
Re\left( {s_\vartheta } \right)=\widetilde{\sigma }_\vartheta +6,
\end{equation}
which is in contradiction with \textit{Lemma \ref{L2}} (Lekkerkerker theorem), which states
infinitely many zeros for $\xi _\tau \left( s \right)$ lie on the critical line $Re\left( s \right)=6$.

\textit{Hypothesis 2.}

Now, we give the hypothesis that $\Xi _\tau \left( t \right)$ has zeros
\begin{equation}
\label{eq12a}
\psi _\vartheta =\ell _\vartheta ,
\end{equation}
where $\ell _\vartheta \in \mathds{R}\backslash \left\{ 0 \right\}$.

By inserting (\ref{eq12a}) into (\ref{eq2a}), we suggest that
\begin{equation}
\label{eq13a}
\Xi _\tau \left( t \right)=\Xi _\tau \left( 0 \right)e^{i\gamma
t}\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{it}{\ell _\vartheta }}
\right)e^{it/\rho _\vartheta }} .
\end{equation}
Substituting $t=\left( {s-6} \right)/i$ and (\ref{eq13a}), we give
\begin{equation}
\label{eq14a}
\begin{array}{l}
 \xi _\tau \left( s \right) \\
 =\Xi _\tau \left( {\left( {s-1/2} \right)/i} \right) \\
 =\Xi _\tau \left( 0 \right)e^{i\gamma t}\prod\limits_{\vartheta =1}^\infty
{\left[ {1-\frac{i\left[ {\left( {s-6} \right)/i} \right]}{\ell _\vartheta
}} \right]e^{it/\rho _\vartheta }} \\
 =\xi _\tau \left( 6 \right)e^{\gamma \left( {s-6}
\right)}\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{s-6}{\ell
_\vartheta }} \right)e^{\left( {s-6} \right)/\rho _\vartheta }} \\
 \end{array}
\end{equation}
since $\Xi _\tau \left( 0 \right)=\xi _\tau \left( 6 \right)$.

Let $\rho _\vartheta =Re\left( {\rho _\vartheta } \right)+i\psi _\vartheta $
such that $\xi _\tau \left( {s_\vartheta } \right)=0$, where $Re\left( {\rho
_\vartheta } \right)\in \mathds{R}\backslash \left\{ 0 \right\}$ and $\psi
_\vartheta \in \mathds{R}\backslash \left\{ 0 \right\}$.

From (\ref{eq14a}), we show that
\begin{equation}
\label{eq15a}
\xi _\tau \left( {\rho _\vartheta } \right)=\xi _\tau \left( 6
\right)e^{\gamma \left( {\rho _\vartheta -6} \right)}\prod\limits_{\vartheta
=1}^\infty {\left( {1-\frac{\rho _\vartheta -6}{\ell _\vartheta }}
\right)e^{\left( {\rho _\vartheta -6} \right)/\rho _\vartheta }} =0.
\end{equation}
From (\ref{eq15a}) we have that
\begin{equation}
\label{eq16a}
\rho _\vartheta -6-\ell _\vartheta =0.
\end{equation}
Since $\ell _\vartheta \in \mathds{R}\backslash \left\{ 0 \right\}$, (\ref{eq16a})
implies
\begin{equation}
\label{eq17a}
\rho _\vartheta =6\mbox{+}\ell _\vartheta \in \mathds{R},
\end{equation}
which is in contradiction with the well-known fact
\begin{equation}
\label{eq18a}
\rho _\vartheta =Re\left( {\rho _\vartheta } \right)+i\psi _\vartheta \in
\mathds{C},
\end{equation}
where $Re\left( {\rho _\vartheta } \right)\in \mathds{R}\backslash \left\{ 0
\right\}$ and $\psi _\vartheta \in \mathds{R}\backslash \left\{ 0 \right\}$.

Moreover, (\ref{eq17a}) and \textit{Lemma \ref{L2}} (Lekkerkerker theorem) conflict with each other.

In sum, all zeros of $\Xi _\tau \left( t \right)$ are real since two
cases are false.

Hence, we finish the proof.
\end{proof}

\begin{remark}
We have to point out that \textit{Corollary \ref{Co3}} is a sufficient condition for (\ref{eq9}) to have
all infinitely many real zeros.
\end{remark}

\section{The nontrivial zeros} {\label{sec:5}}

In this section we present the detailed account of the proof of
\textit{Conjecture \ref{C1}}.

To begin with, we consider the product of the Ramanujan $\Xi$-function
with the aid of \textit{Corollary \ref{Co3}}.

By \textit{Theorems \ref{T4}} and \textit{\ref{T6}}, we now structure the entire
Ramanujan $\Xi$-function by
\begin{equation}
\label{eq57}
\Xi _\tau \left( t \right)=\xi _\tau \left( 6 \right)e^{i\gamma
t}\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{it}{\rho _\vartheta
-6}} \right)e^{it/\rho _\vartheta }} =\sum\limits_{m=0}^\infty {\left( {-1}
\right)^m\beta _m t^{2m}} ,
\end{equation}
where $\beta _m$ are the coefficients and $\gamma $ is the constant.

In view of \textit{Corollary \ref{Co3}}, the second term of (\ref{eq57}) gives us to show
that all of the zeros of $\Xi _\tau \left( t \right)$ are real.

By (\ref{eq58}), we have $\psi _\vartheta \in \mathds{C}$ such that
\begin{equation}
\label{eq58}
\Xi _\tau \left( {\psi _\vartheta } \right)=0,
\end{equation}
which leads to
\begin{equation}
\label{eq59}
\xi _\tau \left( 6 \right)e^{i\gamma \psi _\vartheta
}\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{i\psi _\vartheta }{\rho
_\vartheta -6}} \right)e^{it/\rho _\vartheta }} =0,
\end{equation}
or, alternatively,
\begin{equation}
\label{eq60}
\sum\limits_{m=0}^\infty {\left( {-1} \right)^m\beta _m \left( {\psi
_\vartheta } \right)^{2m}} =\sum\limits_{m=0}^\infty {\left( {-1}
\right)^m\beta _m \psi _\vartheta ^{2m} } =0.
\end{equation}
With (\ref{eq59}) and \textit{Theorem \ref{T5}} we arrive at
\begin{equation}
\label{eq61}
1-\frac{i\psi _\vartheta }{\rho _\vartheta -6}=0,
\end{equation}
where $\xi _\tau \left( 6 \right)\ne 0$ and $\vartheta \in \mathds{N}$.

From (\ref{eq61}) one gets
\begin{equation}
\label{eq62}
\rho _\vartheta =6+i\psi _\vartheta
\end{equation}
for $\vartheta \in \mathds{N}$.

Substituting (\ref{eq62}) into (\ref{eq51}) and (\ref{eq54}), we have
\begin{equation}
\label{eq63}
\begin{array}{l}
 \Xi _\tau \left( t \right)\\
 =\xi _\tau \left( 6 \right)e^{i\gamma
t}\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{it}{\rho _\vartheta
-6}} \right)e^{it/\rho _\vartheta }} \\
 =\xi _\tau \left( 6 \right)e^{i\gamma t}\prod\limits_{\vartheta =1}^\infty
{\left[ {1-\frac{it}{\left( {6+i\psi _\vartheta } \right)-6}}
\right]e^{it/\left( {6+i\psi _\vartheta } \right)}} \\
 =\xi _\tau \left( 6 \right)e^{i\gamma t}\prod\limits_{\vartheta =1}^\infty
{\left( {1-\frac{it}{i\psi _\vartheta }} \right)e^{it/\left( {6+i\psi
_\vartheta } \right)}} \\
 =\xi _\tau \left( 6 \right)e^{i\gamma t}\prod\limits_{\vartheta =1}^\infty
{\left( {1-\frac{t}{\psi _\vartheta }} \right)e^{it/\left( {6+i\psi
_\vartheta } \right)}} \\
 \end{array}
\end{equation}
and
\begin{equation}
\label{eq64}
\begin{array}{l}
 \xi _\tau \left( s \right)\\
 =\xi _\tau \left( 6 \right)e^{\gamma \left( {s-6}
\right)}\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{s-6}{\rho
_\vartheta -6}} \right)e^{\left( {s-6} \right)/\rho _\vartheta }} \\
 =\xi _\tau \left( 6 \right)e^{\gamma \left( {s-6}
\right)}\prod\limits_{\vartheta =1}^\infty {\left[ {1-\frac{s-6}{\left(
{6+i\psi _\vartheta } \right)-6}} \right]e^{\left( {s-6} \right)/\left(
{6+i\psi _\vartheta } \right)}} \\
 =\xi _\tau \left( 6 \right)e^{\gamma \left( {s-6}
\right)}\prod\limits_{\vartheta =1}^\infty {\left[ {1-\frac{s-6}{i\psi
_\vartheta }} \right]e^{\left( {s-6} \right)/\left( {6+i\psi _\vartheta }
\right)}} . \\
 \end{array}
\end{equation}
Similarly, by (\ref{eq62}), (\ref{eq54}) can be rewritten in the form
\begin{equation}
\label{eq65}
\zeta _\tau \left( s \right)=\frac{\xi _\tau \left( 6 \right)\left( {2\pi }
\right)^se^{\gamma \left( {s-6} \right)}}{\Gamma \left( s
\right)}\prod\limits_{\vartheta =1}^\infty {\left[ {1-\frac{s-6}{i\psi
_\vartheta }} \right]e^{\left( {s-6} \right)/\left( {6+i\psi _\vartheta }
\right)}} .
\end{equation}
From (\ref{eq64}) and (\ref{eq65}), one also has (\ref{eq61}) and (\ref{eq62}) once again.

Thus, we finish the proof of \textit{Conjecture \ref{C1}}.

\section{The equivalent theorems} {\label{sec:6}}

In this section we propose some equivalent theorems for the Ramanujan
zeta-function based on the true of \textit{Conjecture \ref{C1}}.

From (\ref{eq62}) we find that $\rho _\vartheta =6+i\psi _\vartheta $, $\overline
{\rho _\vartheta } =6-i\psi _\vartheta $, $12-\rho _\vartheta =6-i\psi
_\vartheta $, and $1-\overline {\rho _\vartheta } =6+i\psi _\vartheta $ are the
zeros for the entire Ramanujan zeta-function and the nontrivial zeros for
the Ramanujan zeta-function.

By (\ref{eq63}) and $\varphi _\vartheta =\left| {\psi _\vartheta } \right|$, we
have
\begin{equation}
\label{eq66}
\begin{array}{l}
 \Xi _\tau \left( t \right)\\
 =\xi _\tau \left( 6 \right)e^{i\gamma
t}\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{t}{\psi _\vartheta }}
\right)e^{it/\left( {6+i\psi _\vartheta } \right)}} \\
 =\xi _\tau \left( 6 \right)e^{i\gamma t}\prod\limits_{\vartheta =1}^\infty
{\left( {1-\frac{t}{\varphi _\vartheta }} \right)e^{it/\left( {6+i\varphi
_\vartheta } \right)}} \prod\limits_{\vartheta =1}^\infty {\left(
{1+\frac{t}{\varphi _\vartheta }} \right)e^{it/\left( {6-i\varphi _\vartheta
} \right)}} \\
 =\xi _\tau \left( 6 \right)e^{i\gamma t}\prod\limits_{\vartheta =1}^\infty
{\left[ {\left( {1-\frac{t}{\varphi _\vartheta }} \right)\left(
{1+\frac{t}{\varphi _\vartheta }} \right)} \right]e^{i12t/\left( {6+i\varphi
_\vartheta } \right)\left( {6-i\varphi _\vartheta } \right)}} \\
 =\xi _\tau \left( 6 \right)e^{i\gamma t}\prod\limits_{\vartheta =1}^\infty
{\left( {1-\frac{t^2}{\varphi _\vartheta ^2 }} \right)e^{i12t/\left(
{36+\varphi _\vartheta ^2 } \right)}} . \\
 \end{array}
\end{equation}
Because (\ref{eq66}) is valid for $t\in \mathds{R}$ and $\Xi _\tau \left( t \right)$
is the even function, one has
\begin{equation}
\label{eq67}
\Xi _\tau \left( {-t} \right)=\xi _\tau \left( 6 \right)e^{-i\gamma
t}\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{t^2}{\varphi
_\vartheta ^2 }} \right)e^{-i12t/\left( {36+\varphi _\vartheta ^2 }
\right)}}
\end{equation}
and
\begin{equation}
\label{eq68}
\Xi _\tau \left( t \right)=\xi _\tau \left( 6 \right)e^{i\gamma
t}\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{t^2}{\varphi
_\vartheta ^2 }} \right)e^{i12t/\left( {36+\varphi _\vartheta ^2 } \right)}}
\end{equation}
such that
\begin{equation}
\label{eq69}
\xi _\tau \left( 6 \right)e^{-i\gamma t}\prod\limits_{\vartheta =1}^\infty
{\left( {1-\frac{t^2}{\varphi _\vartheta ^2 }} \right)e^{-i12t/\left(
{36+\varphi _\vartheta ^2 } \right)}} =\xi _\tau \left( 6 \right)e^{i\gamma
t}\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{t^2}{\varphi
_\vartheta ^2 }} \right)e^{i12t/\left( {36+\varphi _\vartheta ^2 } \right)}}
.
\end{equation}
Thus, by (\ref{eq69}), one gets
\begin{equation}
\label{eq70}
e^{2i\gamma t}\prod\limits_{\vartheta =1}^\infty {e^{i24t/\left( {36+\varphi
_\vartheta ^2 } \right)}} =1,
\end{equation}
which leads to
\begin{equation}
\label{eq71}
2i\gamma t+i24t\sum\limits_{\vartheta =1}^\infty {\frac{1}{36+\varphi
_\vartheta ^2 }=0} .
\end{equation}
Thus,
\begin{equation}
\label{eq72}
\sum\limits_{\vartheta =1}^\infty {\frac{1}{36+\varphi _\vartheta
^2 }}=-\frac{\gamma}{12}
\end{equation}
and
\begin{equation}
\label{eq73}
\Xi _\tau \left( t \right)=\xi _\tau \left( 6 \right)\prod\limits_{\vartheta
=1}^\infty {\left( {1-\frac{t^2}{\varphi _\vartheta ^2 }} \right)} .
\end{equation}
Moreover, (\ref{eq68}) reduces to
\begin{equation}
\label{eq74}
\Xi _\tau \left( 0 \right)=\xi _\tau \left( 6 \right)\ne 0.
\end{equation}

\begin{corollary}
\label{Co4}
Suppose that $\Xi _\tau \left( t \right)$ is defined in (\ref{eq9}), where $t\in
\mathds{R}$. Let $\vartheta \in \mathds{N}$ such that $\psi _\vartheta $ run the
zeros for $\Xi _\tau \left( t \right)$ and $\varphi _\vartheta =\left| {\psi
_\vartheta } \right|$. Then we have
\begin{equation}
\label{eq75}
\Xi _\tau \left( t \right)=\Xi _\tau \left( 0 \right)\prod\limits_{\vartheta
=1}^\infty {\left( {1-\frac{t^2}{\varphi _\vartheta ^2 }} \right)}
\end{equation}
and
\begin{equation}
\label{eq76}
\Xi _\tau \left( t \right)=\Xi _\tau \left( 0 \right)\prod\limits_{\vartheta
=1}^\infty {\left( {1-\frac{t}{\psi _\vartheta }} \right)}.
\end{equation}
\end{corollary}


\begin{proof}
Combining (\ref{eq73}) and (\ref{eq68}), we carry out (\ref{eq75}).

In view of (\ref{eq75}), we have
\begin{equation}
\label{eq77}
\begin{array}{l}
 \Xi _\tau \left( t \right)\\
=\Xi _\tau \left( 0
\right)\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{t^2}{\varphi
_\vartheta ^2 }} \right)} \\
 =\Xi _\tau \left( 0 \right)\prod\limits_{\vartheta =1}^\infty {\left[
{\left( {1+\frac{t}{\varphi _\vartheta }} \right)\left( {1-\frac{t}{\varphi
_\vartheta }} \right)} \right]} \\
=\Xi _\tau \left( 0
\right)\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{t}{\psi
_\vartheta }} \right)} . \\
 \end{array}
\end{equation}
Thus, we finish the proof of \textit{Corollary \ref{Co4}}.
\end{proof}

\begin{theorem}
\label{T8}
Let $\vartheta \in \mathds{N}$ and $t\in \mathds{R}$. The following
representations for $\Xi _\tau \left( t \right)$ are equivalent:
\begin{description}
  \item[(1A)]
\begin{equation}
\label{eq78}
\Xi _\tau \left( t \right)=\int\limits_0^\infty {\varpi ^{5+it}\Delta _\tau
\left( {i\varpi } \right)d\varpi } ,
\end{equation}
where $\Delta _\tau \left( \varpi \right)$ is defined in (\ref{eq11});

  \item[(2A)]
\begin{equation}
\label{eq79}
\Xi _\tau \left( t \right)=2\int\limits_1^\infty {\cos \left( t
\right)\varpi ^5\Delta _\tau \left( {i\varpi } \right)d\varpi } ,
\end{equation}
where $\Delta _\tau \left( \varpi \right)$ is defined in (\ref{eq11});

  \item[(3A)]
\begin{equation}
\label{eq80}
\Xi _\tau \left( t \right)=\sum\limits_{m=0}^\infty {\left( {-1}
\right)^m\beta _m t^{2m}} ,
\end{equation}
where $\beta _m $ is defined in (\ref{eq36});

  \item[(4A)]
\begin{equation}
\label{eq81}
\Xi _\tau \left( t \right)=\xi _\tau \left( 6 \right)e^{i\gamma
t}\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{t}{\psi _\vartheta }}
\right)e^{it/\left( {6+i\psi _\vartheta } \right)}} ,
\end{equation}
where $\psi _\vartheta $ run the zeros for $\Xi _\tau \left( t \right)$ and
$\gamma $ is defined in (\ref{eq72});

  \item[(5A)]
\begin{equation}
\label{eq82}
\Xi _\tau \left( t \right)=\Xi _\tau \left( 0 \right)\prod\limits_{\vartheta
=1}^\infty {\left( {1-\frac{t^2}{\varphi _\vartheta ^2 }} \right)} ,
\end{equation}
where $\varphi _\vartheta =\left| {\psi _\vartheta } \right|$;

  \item[(6A)]
\begin{equation}
\label{eq83}
\Xi _\tau \left( t \right)=\Xi _\tau \left( 0 \right)\prod\limits_{\vartheta
=1}^\infty {\left( {1-\frac{t}{\psi _\vartheta }} \right)} ,
\end{equation}
where $\psi _\vartheta $ run the zeros for $\Xi _\tau \left( t \right)$.
\end{description}
\end{theorem}

\begin{proof}
Eq.~(\ref{eq78}) (see (\ref{eq9}) in Section \ref{sec:1}) was proved by Conrey and Ghosh
~\cite{12}. (\ref{eq79}) is shown in \textit{Theorem \ref{T3}}, and (\ref{eq80}) was presented in
\textit{Theorem \ref{T4}}. (\ref{eq81}) was given by \textit{Theorem \ref{T6}} and
\textit{Conjecture \ref{C1}}. (\ref{eq82}) and (\ref{eq83}) were presented in \textit{Corollary
\ref{Co4}}. Thus, we finish the proof of \textit{Theorem \ref{T8}}. These are what we
wanted to show.
\end{proof}


\begin{theorem}
\label{T9}
Let $\vartheta \in \mathds{N}$ and $s\in \mathds{C}$. The following
representations for $\xi _\tau \left( s \right)$ are equivalent:
\begin{description}
  \item[(1B)]

\begin{equation}
\label{eq84a}
\xi _\tau \left( s \right)=\int\limits_0^\infty {\varpi ^{s-1}\Delta _\tau
\left( {i\varpi } \right)d\varpi },
\end{equation}

where $\Delta _\tau \left( \varpi \right)$ is defined in (\ref{eq11});

  \item[(2B)]
\begin{equation}
\label{eq84}
\xi _\tau \left( s \right)=2\int\limits_1^\infty {\cosh \left( t \right)\mathds{
X}\left( \varpi \right)d\varpi } ,
\end{equation}
where $\Delta _\tau \left( \varpi \right)$ is defined in (\ref{eq11});

  \item[(3B)]
\begin{equation}
\label{eq85}
\xi _\tau \left( s \right)=\sum\limits_{m=0}^\infty {\beta _m t^{2m}} ,
\end{equation}
where $\beta _m $ is defined in (\ref{eq36});

  \item[(4B)]
\begin{equation}
\label{eq86}
\xi _\tau \left( s \right)=\xi _\tau \left( 6 \right)e^{\gamma \left( {s-6}
\right)}\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{s-6}{i\psi
_\vartheta }} \right)e^{\left( {s-6} \right)/\left( {6+i\psi _\vartheta }
\right)}} ,
\end{equation}
where $\psi _\vartheta $ run the zeros for $\Xi _\tau \left( t \right)$ and
$\gamma $ is defined in (\ref{eq72});

  \item[(5B)]
\begin{equation}
\label{eq87}
\xi _\tau \left( s \right)=\xi _\tau \left( 6 \right)\prod\limits_{\vartheta
=1}^\infty {\left( {1+\frac{\left( {s-6} \right)^2}{\varphi _\vartheta ^2 }}
\right)} ,
\end{equation}
where $\varphi _\vartheta =\left| {\psi _\vartheta } \right|$;

  \item[(6B)]
\begin{equation}
\label{eq88}
\xi _\tau \left( s \right)=\xi _\tau \left( 6 \right)\prod\limits_{\vartheta
=1}^\infty {\left( {1-\frac{s-6}{i\psi _\vartheta }} \right)} ,
\end{equation}
where $\psi _\vartheta $ run the zeros for $\Xi _\tau \left( t \right)$.
\end{description}
\end{theorem}

\begin{proof}
Eq.~(\ref{eq84a}) (see (\ref{eq8}) in Section \ref{sec:1}) was proved in~\cite{12}. (\ref{eq84}) was
presented in \textit{Theorem \ref{T1}}, and (\ref{eq85}) was presented in \textit{Theorem
\ref{T2}}. (\ref{eq86}) was proved by \textit{Theorem \ref{T6}} and \textit{Conjecture \ref{C1}}.
Putting $t=-i\left( {s-6} \right)$ into (\ref{eq82}) and (\ref{eq83}), one obtains
\begin{equation}
\label{eq89}
\xi _\tau \left( s \right)=\Xi _\tau \left( 0 \right)\prod\limits_{\vartheta
=1}^\infty {\left( {1+\frac{\left( {s-6} \right)^2}{\varphi _\vartheta ^2 }}
\right)} =\xi _\tau \left( 6 \right)\prod\limits_{\vartheta =1}^\infty
{\left( {1+\frac{\left( {s-6} \right)^2}{\varphi _\vartheta ^2 }} \right)}
,
\end{equation}
and
\begin{equation}
\label{eq90}
\xi _\tau \left( s \right)=\Xi _\tau \left( 0 \right)\prod\limits_{\vartheta
=1}^\infty {\left( {1-\frac{s-6}{i\psi _\vartheta }} \right)} =\xi _\tau
\left( 6 \right)\prod\limits_{\vartheta =1}^\infty {\left(
{1-\frac{s-6}{i\psi _\vartheta }} \right)} ,
\end{equation}
since (\ref{eq74}) is valid.

Hence we complete of the proof.
\end{proof}

\begin{theorem}
\label{T10}
Let $\vartheta \in \mathds{N}$ and $s\in \mathds{C}$. The following
representations for $L_\tau \left( s \right)$ are equivalent:
\begin{description}
  \item[(1C)]
\begin{equation}
\label{eq91}
L_\tau \left( s \right)=\frac{\left( {2\pi } \right)^s}{\Gamma \left( s
\right)}\int\limits_0^\infty {\varpi ^{s-1}\Delta _\tau \left( {i\varpi }
\right)d\varpi } ,
\end{equation}
where $\Delta _\tau \left( \varpi \right)$ is defined in (\ref{eq11});

  \item[(2C)]
\begin{equation}
\label{eq92}
L_\tau \left( s \right)=\frac{2^{1+s}\pi ^s}{\Gamma \left( s
\right)}\int\limits_1^\infty {\cosh \left( t \right){\mathds{X}}\left( \varpi
\right)d\varpi } ,
\end{equation}
where $\Delta _\tau \left( \varpi \right)$ is defined in (\ref{eq11});

  \item[(3C)]
\begin{equation}
\label{eq93}
L_\tau \left( s \right)=\frac{\left( {2\pi } \right)^s}{\Gamma \left( s
\right)}\sum\limits_{m=0}^\infty {\beta _m t^{2m}} ,
\end{equation}
where $\beta _m $ is defined in (\ref{eq36});

  \item[(4C)]
\begin{equation}
\label{eq94}
L_\tau \left( s \right)=\frac{\xi _\tau \left( 6 \right)\left( {2\pi }
\right)^s}{\Gamma \left( s \right)}e^{\gamma \left( {s-6}
\right)}\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{s-6}{i\psi
_\vartheta }} \right)e^{\left( {s-6} \right)/\left( {6+i\psi _\vartheta }
\right)}} ,
\end{equation}
where $\psi _\vartheta $ run the zeros for $\Xi _\tau \left( t \right)$ and
$\gamma $ is defined in (\ref{eq72});

  \item[(5C)]
\begin{equation}
\label{eq95}
L_\tau \left( s \right)=\frac{\xi _\tau \left( 6 \right)\left( {2\pi }
\right)^s}{\Gamma \left( s \right)}\prod\limits_{\vartheta =1}^\infty
{\left( {1+\frac{\left( {s-6} \right)^2}{\varphi _\vartheta ^2 }} \right)}
,
\end{equation}
where $\varphi _\vartheta =\left| {\psi _\vartheta } \right|$;

  \item[(6C)]
\begin{equation}
\label{eq96}
L_\tau \left( s \right)=\frac{\xi _\tau \left( 6 \right)\left( {2\pi }
\right)^s}{\Gamma \left( s \right)}\prod\limits_{\vartheta =1}^\infty
{\left( {1-\frac{s-6}{i\psi _\vartheta }} \right)} ,
\end{equation}
where $\psi _\vartheta $ run the zeros for $\Xi _\tau \left( t \right)$.
\end{description}
\end{theorem}

\begin{proof}
By (\ref{eq4}), we have
\begin{equation}
\label{eq97}
L_\tau \left( s \right)=\frac{\left( {2\pi } \right)^s}{\Gamma \left( s
\right)}\xi _\tau \left( s \right),
\end{equation}
and by (\ref{eq97}) and \textit{Theorem \ref{T9}}, we obtain the required results.

Thus, the proof of \textit{Theorem \ref{T10}} is finished.
\end{proof}

We now define
\begin{equation}
\label{eq98}
{\mathds{M}}_\tau \left( t \right)=L_\tau \left( {6+it} \right)=\frac{\left(
{2\pi } \right)^{6+it}}{\Gamma \left( {6+it} \right)}\xi _\tau \left( {6+it}
\right)=\frac{\left( {2\pi } \right)^{6+it}}{\Gamma \left( {6+it}
\right)}\Xi _\tau \left( t \right),
\end{equation}
where $\left( {2\pi } \right)^{6+it}/\Gamma \left( {6+it} \right)\ne 0$ for
$t\in \mathds{R}$.

\begin{theorem}
\label{T11}
Let $\vartheta \in \mathds{N}$ and $t\in \mathds{R}$. The following
representations for ${\mathds{M}}_\tau \left( t \right)$ are equivalent:
\begin{description}
  \item[(1D)]
\begin{equation}
\label{eq99}
{\mathds{M}}_\tau \left( t \right)=\frac{\left( {2\pi } \right)^{6+it}}{\Gamma
\left( {6+it} \right)}\int\limits_0^\infty {\varpi ^{5+it}\Delta _\tau
\left( {i\varpi } \right)d\varpi } ,
\end{equation}
where $\Delta _\tau \left( \varpi \right)$ is defined in (\ref{eq11});

  \item[(2D)]
\begin{equation}
\label{eq100}
{\mathds{M}}_\tau \left( t \right)=\frac{2^{7+it}\pi ^{6+it}}{\Gamma \left(
{6+it} \right)}\int\limits_1^\infty {\cos \left( t \right){\mathds{X}}\left(
\varpi \right)d\varpi } ,
\end{equation}
where $\Delta _\tau \left( \varpi \right)$ is defined in (\ref{eq11});

  \item[(3D)]
\begin{equation}
\label{eq101}
{\mathds{M}}_\tau \left( t \right)=\frac{\left( {2\pi } \right)^{6+it}}{\Gamma
\left( {6+it} \right)}\sum\limits_{m=0}^\infty {\left( {-1} \right)^m\beta
_m t^{2m}} ,
\end{equation}
where $\beta _m $ is defined in (\ref{eq36});

  \item[(4D)]
\begin{equation}
\label{eq102}
{\mathds{M}}_\tau \left( t \right)=\frac{\left( {2\pi } \right)^{6+it}\xi _\tau
\left( 6 \right)e^{i\gamma t}}{\Gamma \left( {6+it}
\right)}\prod\limits_{\vartheta =1}^\infty {\left( {1-\frac{t}{\psi
_\vartheta }} \right)e^{it/\left( {6+i\psi _\vartheta } \right)}} ,
\end{equation}
where $\psi _\vartheta $ run the zeros for $\Xi _\tau \left( t \right)$ and
$\gamma $ is defined in (\ref{eq72});

  \item[(5D)]
\begin{equation}
\label{eq103}
{\mathds{M}}_\tau \left( t \right)=\frac{\Xi _\tau \left( 0 \right)\left( {2\pi }
\right)^{6+it}}{\Gamma \left( {6+it} \right)}\prod\limits_{\vartheta
=1}^\infty {\left( {1-\frac{t^2}{\varphi _\vartheta ^2 }} \right)} ,
\end{equation}
where $\varphi _\vartheta =\left| {\psi _\vartheta } \right|$;

  \item[(6D)]
\begin{equation}
\label{eq104}
{\mathds{M}}_\tau \left( t \right)=\frac{\Xi _\tau \left( 0 \right)\left( {2\pi }
\right)^{6+it}}{\Gamma \left( {6+it} \right)}\prod\limits_{\vartheta
=1}^\infty {\left( {1-\frac{t}{\psi _\vartheta }} \right)} ,
\end{equation}
where $\psi _\vartheta $ run the zeros for $\Xi _\tau \left( t \right)$.
\end{description}
\end{theorem}

\begin{proof}
Combining (\ref{eq98}) and \textit{Theorem \ref{T10}}, we arrive at the desired results.
\end{proof}

Now, we consider that the Hafner theorem was conjectured by Hafner in 1983, as a statement
equivalent to \textit{Conjecture \ref{C1}}~\cite{13}.

\begin{theorem}
(Hafner theorem)
\label{T12}

Let
\begin{equation}
\label{eq105}
N_0 \left( T \right)=\# \left\{ {s =6+it : 0\le
t \le T, \xi _\tau \left( {s} \right)=0}
\right\}
\end{equation}
and
\begin{equation}
\label{eq106}
N\left( T \right)=\# \left\{ {s =\sigma +it
: 0\le t \le T,11/2<\sigma <13/2, \xi _\tau \left( {s} \right)=0} \right\}.
\end{equation}
Then we have
\begin{equation}
\label{eq107}
N_0 \left( T \right)=N\left( T \right).
\end{equation}
\end{theorem}

\begin{proof}
Let $0\le t \le T$ and $11/2<\sigma <13/2$.
With the aid of (\ref{eq62}), one has
\begin{equation}
\label{eq108}
\rho _\vartheta =6+i\varphi_\vartheta
\end{equation}
such that
\begin{equation}
\label{eq109}
\Xi _\tau \left( {\rho _\vartheta } \right)=0.
\end{equation}
Thus, the result follows.
\end{proof}

\begin{remark}
By the above results, we show the following comments.
\begin{itemize}
  \item
By theory of the entire functions~\cite{20}, it is observed that
$\Xi _\tau \left( t \right)$ is an even entire function of order $\alpha =1$
with the exponent of convergence $\lambda =1$ and genus $\beta =0$, and of
growth $\left( {1,0} \right)$. $\xi _\tau \left( s \right)$ is an entire
function of order $\alpha =1$ with the exponent of convergence
$\lambda =1$ and genus $\beta =0$, and is of growth $\left( {1,0} \right)$.

  \item
It was proposed in 1940 by Hardy~\cite{3} that the nontrivial zeros $\rho _\vartheta $
for $L_\tau \left( s \right)$  lie on the critical line $Re\left(s \right)=6$ and in the critical
strip $11/2<Re\left( s \right)<13/2$, and that the trivial zeros for $L_\tau \left(s \right)$ are
$s=-w$ with $w\in \mathds{N}\cup \left\{ \mbox{0} \right\}$.
The trivial and nontrivial zeros,
critical line and critical trip for the Ramanujan zeta-function
in the entire complex plane are shown in Fig.~\ref{fig1}.
\begin{figure}
  \centering
  \includegraphics[width=9cm]{fig1.eps}
  \caption{The blue points are represented as
  the nontrivial zeros for $L_\tau \left( s \right)$.
  The amaranth points are represented as
  the trivial zeros for $L_\tau \left( s \right)$.
  All nontrivial zeros
for $L_\tau \left( s \right)$  lie on the critical line $Re\left(s \right)=6$ and in the critical
strip $11/2<Re\left( s \right)<13/2$.}\label{fig1}
\end{figure}
Thus, it is clear to see that \textit{Conjecture \ref{C1}} is true.
\end{itemize}
\end{remark}

\begin{acknowledgements}
This work is supported by the Yue-Qi Scholar of the China University of Mining and
Technology (No. 102504180004).
\end{acknowledgements}

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\end{document}
