\section{Introduction}
Since the epoch making paper of G.B.Riemann ~\cite{Ri59},
the study of the zeros of the Riemann zeta-function and the other zeta-functions are
one of the major fields of number theory.
He introduced the analytic method to the theory of prime distribution,
and clarified the relation between the primes and the zeros of the Riemann zeta-function.

After Riemann's paper,
the accumulation of studies about the zeros of zeta-functions are enormous ones
and many papers about them have been published every years.
However almost of them have dealt with the zeros of single zeta-function,
even if there exist some studies which deal with the zeros of a family of zeta-functions.
An example is the studies related with the GUE-conjecture ~\cite{KaSa99}.
It has given a new point of view to the study of the zeros of zeta-functions.
This example tells us that
studying the relation between the zeros of several different zeta-functions
would give a new insight to the theory of the zeros of zeta-functions.
Hence, in the present paper, we mainly concern to the relation between the zeros of different zeta-functions
rather than individual properties of the zeros of single zeta-function.
It is a continuation of the studies in ~\cite{Su04}.

In ~\cite{Su04} the author showed that there exists some relation
between the zeros of a $L$-function $L(s)$ belonging to the Selberg class
and the zeros of an associated $L$-function $L_\chi(s)$
twisted by a primitive Dirichlet character $\chi$
which is a generalization of Linnik's result in~\cite{Li47}.
Linnik's result is the asymptotic relation
\begin{equation}\label{101}
\sum_{{L(\rho,\chi)=0}\atop{0 < {\rm Re}(\rho) < 1}}
 \Gamma(\rho) x^{-\rho}
=\frac{1}{\tau(\overline{\chi})}
 \sum_{a=1}^q \overline{\chi}(a)
\sum_{{\zeta(\rho)=0}\atop{0 < {\rm Re}(\rho) < 1}}
 \Gamma(\rho) \left(x -2\pi i \frac{a}{q}\right)^{-\rho}
 +O(\log^2 x)
\end{equation}
as $x \to +0$, where $\zeta(s)$ is the Riemann zeta function and
$L(s,\chi)$ is the Dirichlet $L$-function attached to a primitive Dirichlet character $\chi$ mod $q$.
The relation $\eqref{101}$ suggests that
there exists a relation between the zeros of $\zeta(s)$ and $L(s,\chi)$.
In fact Sprindzuk showed that, under the original Riemann hypothesis (RH),
some properties of the zeros of $\zeta(s)$ are equivalent to
the RH for Dirichlet $L$-functions by using $\eqref{101}$
in ~\cite{Sp75}.
The author generalized the Sprindzuk type result to the case of the above pair  $L(s)$ and $L_\chi(s)$ in ~\cite{Su04}.
Further studies on Sprinduzuk's work in ~\cite{Sp75}, see Fujii ~\cite{Fu88,Fu92}.
The aim of this paper is to generalize the relation $\eqref{101}$ to the class of $L$-functions as wide as possible.
The author believes that such results will give a new view point to the theory of $L$-functions.

Our starting point is the observation about the zeros of principal automorphic $L$-functions.
We refer to ~\cite{RuSa96} for notations and properties of them.
Let $\pi=\otimes_p \pi_p$ be an irreducible cuspidal automorphic representation of $GL_N({\Bbb A}_{\Bbb Q})$
with unitary central character.
The associated $L$-function $L(s,\pi)$ is given by a product of local factors $L(s,\pi_p)$.
Except for a finite set of primes, $\pi_p$ is unramified.
The local factors $L(s,\pi_p)$ for unramified primes are given by
\begin{equation}\label{102}
L(s,\pi_p)=\prod_{j=1}^N (1-\alpha_{\pi}(p,j)p^{-s})^{-1}
\end{equation}
where $\alpha_{\pi}(p,j)$ are the eigen values of
the semi-simple conjugacy class $\{A_{\pi}(p)\} \in GL_N(\Bbb C)$ associated to $\pi_p$.
The generalized Ramanujan conjecture for cuspidal automorphic representation $\pi$
asserts that $|\alpha_{\pi}(p,j)|=1$ for unramified $p$.
\smallskip

\noindent
{\bf Observation 1.} Under the general Ramanujan conjecture,
the set of all zeros of $L(s,\pi_p)^{-1}$ for the unramified prime $p$
is a union of $n$-piece translations of the zeros of $\zeta_{p}^{-1}(s)=1-p^{-s}$.
\smallskip

The logarithmic derivative of $L(s,\pi)$ is written as
\begin{equation}\label{103}
-\frac{L^\prime}{L}(s,\pi)=\sum_{n=1}^\infty \frac{\Lambda_\pi(n)}{n^s}
\end{equation}
where $\Lambda_\pi(n)=\Lambda(n)a_\pi(n)$, $\Lambda(n)=\log p$ if $n=p^m$ and zero otherwise, and
\begin{equation}\label{104}
a_\pi(p^m)=\sum_{j=1}^N \alpha_{\pi}(p,j)^m.
\end{equation}

By an ``explicit formula''  we usually mean an equation that
represents the information of the Euler product and the functional equation
in terms of an explicit relation between the zeros of $L(s,\pi)$ and $\Lambda_\pi(p^m)$.
Let $h \in C_0^\infty({\Bbb R}_+)$ be a smooth compactly supported function,
and let $\widehat{h}(s)=\int_0^\infty h(u)u^{s-1} du$ be the Mellin transform of $h$.
Then
\begin{equation}\label{105}
\delta_\pi  \, \widehat{h}(0) -\sum_{L^\ast(\rho,\pi)=0}\widehat{h}(\rho) + \delta_\pi \, \widehat{h}(1)
=\sum_{p} W_\pi(h,p) + W_{\pi}(h,\infty),
\end{equation}
where $\delta_\pi=1$ if $\pi$ corresponds to $\zeta(s)$, and zero otherwise,
\begin{equation}\label{106}
W_\pi(h,p)=\sum_{m=1}^\infty (\Lambda_\pi(p^m) h(p^m) +\overline{\Lambda_\pi(p^m)} p^{-m}h(p^{-m})).
\end{equation}
On the other hand, Poisson's summation formula yields
\begin{equation}\label{107}
W_\pi(h,p)=\sum_{L(\rho,\pi_p)^{-1}=0} ( \widehat{h}(\rho) + \widehat{h}(1-\overline{\rho})).
\end{equation}

\noindent
{\bf Observation 2.}
Combining $\eqref{105}$ with $\eqref{107}$,
we obtain a relation between the zeros of $L(s,\pi)$ and the zeros of $L(s,\pi_p)^{-1}$.
(We deal with similar things more precisely in $\S 8.3$.)
\smallskip

From {\bf Observation 1} and {\bf Observation 2},
we can see a possibility of the generalization of $\eqref{101}$
to principal automorphic $L$-functions.
Further we notice a possibility of generalizing $\eqref{101}$ to $L$-functions
which have Euler products and functional equations, because, as explained above,
the explicit formula is mainly based on the existence of the Euler product and the functional equation.
However it is not so clear that how we generalize $\eqref{101}$.
To obtain a hint of the formulation, we recall the outline of the arguments in ~\cite{Su04}.

A special case of Theorem 1 in ~\cite{Su04} is stated as
\begin{equation}\label{108}
\sum_{{L(\rho,\chi)=0}\atop{0<{\rm Re}(\rho)<1}} \int_0^\infty h(xu) u^{\rho} \frac{du}{u} =
\sum_{{\zeta(\rho)=0}\atop{0<{\rm Re}(\rho)<1}} \int_0^\infty h(xu) \phi_\chi(u) u^{\rho} \frac{du}{u}
  + O(1)
\end{equation}
for sufficiently small $x>0$ and $h \in C_0^\infty({\Bbb R}_+)$ with $\int_0^\infty h(u)du=0$,
where the function $\phi_\chi$ is given by
\begin{equation}\label{109}
\phi_\chi(u) =\frac{1}{\tau(\bar{\chi})} \sum_{a=1}^q \overline{\chi}(a)e^{2\pi i au/q},
\end{equation}
and $\tau(\chi)=\sum_{a=1}^q \chi(a)e^{2\pi i a/q}$ is the ordinary Gauss sum.
This is a smooth version of the original result $\eqref{101}$.
From the well-known equation
\begin{equation}\label{110}
\chi(n) =\frac{1}{\tau(\bar{\chi})} \sum_{a=1}^q \overline{\chi}(a)e^{2\pi i an/q},
\end{equation}
we find that $\phi_\chi(n)=\chi(n)$ for $n \in {\Bbb Z}$.
That is, $\phi_\chi$ is an interpolation function of the Dirichlet coefficients $\chi(n)$ of $L(s,\chi)$.
The existence of such interpolation function plays a key roll in ~\cite{Su04}.
Now we explain it roughly.
To obtain $\eqref{108}$, we consider the sum $S(x)=\sum_{n=1}^\infty \Lambda(n)\chi(n)h(xn)$
for $h \in C_0^\infty({\Bbb R}_+)$ with $\int_0^\infty h(u)du=0$
and calculate the sum $S(x)$ in two ways.
By applying Weil's explicit formula for $L(s,\chi)$ and $u \mapsto h(xu)$,
we find that $S(x)$ is asymptotically equal to the left hand side of $\eqref{108}$.
Because $\phi_{\chi}(n)=\chi(n)$,
we can replace $\chi(n)$ by $\phi_\chi(n)$.
Denote by $\tilde{S}(x)$ the replaced sum.
Then, by applying Weil's explicit formula again for $\zeta(s)$ and $u \mapsto h(xu)\phi_{\chi}(u)$,
we find that $\tilde{S}(x)$ is asymptotically equal to the right hand side of $\eqref{108}$.
Since $S(x)=\widetilde{S}(x)$, we obtain $\eqref{108}$.
These arguments suggest that the existence of
suitable interpolation function of the Dirichlet coefficients
is very useful for our purpose.
Standing on the above consideration,
we adopt
Euler products,
functional equations and
interpolation functions of Dirichlet coefficients
as the axis of our formulation.

Here we comment on Euler products.
In ~\cite{Ku86},
Kurokawa showed that the properties of Euler products
are deeply related to the possibility of analytic continuation.
His results assert that for a wide class of Euler products
the unitary property of Euler products is equivalent to
the possibility of analytic continuation to the whole plane.
The Euler product also plays a very important role to establish our results.
Hence our results give a new reason for
the importance of Euler products in the theory of zeta-functions.
In our results, the Euler product works as a device which connects the zeros of different $L$-functions with each other.

This paper is organized as follows.
In $\S 2$ we prepare several notations and our settings.
In $\S 3$ we state our theorems.
In $\S 4$ we give several examples of our results.
In $\S 5$ we explain about Weil's explicit formula.
It is a main tool for the proof of our results.
In $\S 6$ we prove Theorem 1 and Theorem 2.
In $\S 7$ we prove Theorem 3 and Theorem 4.
In $\S 8$ we deal with some related topics.

\section{ Preliminary. }
In this section we explain our setting and prepare the notations.
Let $ C^\infty $ be the space of all smooth function on ${\Bbb R}_+ $, and
let $ C_0^\infty $ be the space of all smooth compactly supported function on ${\Bbb R}_+ $.
Let $C_D$ be the space of all smooth slowly increasing functions on ${\Bbb R}_+$, that is,
\begin{equation}\label{201}
C_D:=\{ \phi \in C^\infty  \, | \, |\phi(u)| \leq C u^k \text{ for some $k \geq 0$ and $C>0$ } \}.
\end{equation}
For $\phi \in C_D$
we define the Dirichlet series $L_\phi(s)$ as
\begin{equation}\label{202}
L_\phi(s) :=\sum_{n=1}^\infty \phi(n)n^{-s},
\end{equation}
for sufficiently large ${\rm Re}(s)$.
In the rest of this section,
we introduce the subclass $C_L$ of $C_D$ that
any elements in $C_L$ can be regarded
as an interpolation function of Dirichlet coefficients of some zeta-function.
To define $C_L$ we recall the concept of the {\it Selberg class}.
The Selberg class $\cal S$ is the class of all function $L(s)$ on ${\Bbb C}$
which satisfies the following five axioms;
\begin{enumerate}
\item[(S1)] $L(s)$ is expressed as an absolutely convergent Dirichlet series
$L(s)=\sum_{n=1}^\infty a(n)n^{-s}$ in the half plane ${\rm Re}(s)>1$.
\item[(S2)] There exists a non-negative integer $m$ such that $(s-1)^m L(s)$ is an entire function with finite order.
$($We denote by $m_L$ the smallest non-negative integer $m$ which satisfies this condition.$)$
\item[(S3)]
There exist some $Q>0$, $r \geq 1$, $\lambda_j>0$, ${\rm Re}(\mu_j) \geq 0$ $(1\leq j \leq r)$ and $|\omega|=1$,
such that {\it the complete $L$-function}
$$ L^\ast(s):=Q^s \prod_{j=1}^r \Gamma(\lambda_j s + \mu_j) L(s) $$
satisfies the functional equation $L^\ast(s)=\omega \overline{L^\ast(1-\bar{s})}$.
$($ The factor $\gamma(s):=Q^s \prod_{j=1}^r \Gamma(\lambda_j s + \mu_j) $ is called the {\it $\Gamma$-factor} of $L(s)$.$)$
\item[(S4)] For any positive $\varepsilon$, the coefficients $a(n)$ of $L(s)$ is estimated as $a(n) \ll n^{\varepsilon}$.
\item[(S5)] The logarithm of $L(s)$ is also expressed as
$$\log L(s)=\sum_{n=1}^\infty b(n)n^{-s},$$
where $b(n)$'s are zero unless $n=p^m$ $(m \geq 1)$.
Further, the estimate $b(n) \ll n^\theta$ holds for some $\theta < \frac 12$.
\end{enumerate}
From (S$5$) the logarithmic derivative of $L(s)$ also has the Dirichlet series expression
\begin{equation}\label{203}
-\frac{L^\prime}{L}(s)
=\sum_{n=1}^\infty \Lambda_L(n)n^{-s}, \quad
\end{equation}
where $\Lambda_L(n)=b(n)\log n$ is an analogue of
the von Mangoldt function $\Lambda(n)$ defined by
\begin{align}\label{204}
\Lambda(n)=
\begin{cases}
\log p & \text{if $n=p^m$ with $m \geq 1$,} \\
0      & \text{otherwise.}
\end{cases}
\end{align}
Futher, by the result of Conrey and Ghosh~\cite{CoGh93},
the Dirichlet coefficients $a(n)$ of $L(s) \in {\cal S}$
 are multiplicative. Moreover, the Euler product
\begin{equation}\label{205}
L(s)=\prod_p L_p(s), \quad \text{where} \quad L_p(s)=\sum_{m=0}^\infty a(p^m) p^{-ms},
\end{equation}
is absolutely convergent for ${\rm Re}(s) >1$ and
$L_p(s)$ is absolutely convergent for ${\rm Re}(s)>0$ for every $p$.
The factors $L_p(s)$ are called {\it Euler factors} of $L(s)$.
We remark that $b(p)=a(p)$ and
$\log L_p(s)=\sum_{m=1}^\infty b(p^m)p^{-ms}$ for every prime $p$.
Hence
\begin{equation}\label{206}
L_p(s) \not= 0 \quad \text{for ${\rm Re}(s)>\theta$ and every $p$,}
\end{equation}
since $b(n) \ll n^\theta$.
In additon, the $\Gamma$-factor has no zero and no pole
in the half-plane ${\rm Re}(s) >0$.
Therefore $L^\ast(s)$ has no zero
outside of the vertical strip $0 \leq {\rm Re}(s) \leq 1$.

We say that $L(s) \in {\cal S}$ has a {\it rank $N$ Euler product}, if $L_p(s)$ is expressed as
\begin{equation}\label{207}
L_p(s) = P_p(p^{-s})^{-1},
\end{equation}
where
\begin{equation}\label{208}
P_p(X)=1-c_1(p)X-c_2(p)X^2 - \cdots -c_N(p) X^N
\end{equation}
and $c_N(p) \not=0$ except for finitely many $p$.
When $L(s)$ has a rank $N$ Euler product,
we denote by $S_L$ the set of all primes for which $c_N(p)=0$.

Now we define the subclass $C_L(N)$ and $C_L$ of $C_D$ by
\begin{align}
C_L(N)  &:= \left\{
 \phi \in C_D \,
\left|
\aligned
\, & |\phi(u)| \leq C u^2 \quad \text{for some $C>0$,} \label{209} \\
\, & \text{$L_\phi(s)$ belongs to $\cal S$,} \\
\, & \text{$L_\phi(s)$ has a rank $N$ Euler product.}
\endaligned
\right.
\right\}, \\
C_L   &:= \cup_{N \geq 1} C_L(N)\label{210}.
\end{align}
From the definition, $\phi \in C_L$ satisfies
the Ramanujan-Deligne estimate $|\phi(n)| \ll_\varepsilon n^{\varepsilon}$
for any positive integer $n$,
even if $\phi(u)$ has rather high order as a function on ${\Bbb R}_+$.
\smallskip

\noindent
{\bf Remark.}
The condition $|\phi(u)| \leq C u^2$ is technical one to obtain a simple statement.
In the case we omit this condition, we can obtain similar results
although they are of a more complicated form.
See the proof of Lemma 6 in Section 6.

\section{Main Results}
\subsection{ Relations with $\zeta(s)$.}
\hfill

\begin{thm}
Let $\phi \in C_L(1)$.
Then we have
\begin{align} \label{301} \aligned
m_\phi \cdot &  \int_0^\infty h(xu) du
- \sum_{L_\phi^\ast(\rho)=0} \int_0^\infty h(xu) u^\rho \frac{du}{u} \\
=&
\, \int_0^\infty h(xu) \phi(u) du
 - \sum_{\zeta^\ast(\rho)=0} \int_0^\infty h(xu) \phi(u) u^\rho \frac{du}{u}
+O(1)
\endaligned \end{align}
as $x \to +0$ for any fixed $\varepsilon >0$,
where $m_\phi=m_{L_\phi}$ is the integer defined in $(${\rm S}$2)$.
The sum on the left hand side runs over all zeros of $L_\phi^\ast(s)$ counting with multiplicity.
And the sum on the right hand side runs over all zeros of $\zeta^\ast(s)$ counting with multiplicity.
\end{thm}

\noindent
{\bf Remark.}
Theorem 1 is a generalization of $\eqref{101}$. Compare with~\cite[Theorem 1]{Su04}.\\

\begin{thm}
Let $\phi \in C_L(N)$ with $N \geq 2$.
Then we have
\begin{align} \label{302} \aligned
m_\phi \cdot &  \int_0^\infty h(xu) du
- \sum_{L_\phi^\ast(\rho)=0} \int_0^\infty h(xu) u^\rho \frac{du}{u} \\
=&
\, \int_0^\infty h(xu) \phi(u) du
 - \sum_{\zeta^\ast(\rho)=0} \int_0^\infty h(xu) \phi(u) u^\rho \frac{du}{u}
+ J_{\phi}(x;h) + O(x^{-\frac 13 -\varepsilon})
\endaligned \end{align}
as $x \to +0$ for any fixed $\varepsilon >0$,
where $m_\phi=m_{L_\phi}$ is the integer defined in $(${\rm S}$2)$.
The sum on the left hand side runs over all zeros of $L_\phi^\ast(s)$ counting with multiplicity,
and the sum on the right hand side runs over all zeros of $\zeta^\ast(s)$ counting with multiplicity.
The function $J_{\phi}(x;h)$ is estimated as
\begin{equation}\label{303}
J_{\phi}(x;h) \ll_{\varepsilon, \phi, h} x^{-\frac{1}{2}-\varepsilon}
\end{equation}
as $x \to +0$ for any fixed $\varepsilon >0$.
Furthermore, if the estimate
\begin{equation}\label{304}
\sum_{p \leq T} c_{\phi,2}(p) \log p = A_\phi T^{\mu}  +O(T^{\nu +\varepsilon}).
\end{equation}
holds for the numbers $c_{\phi,2}(p)$ defined in $(2.8)$ with some constant $A_\phi$ and ${\rm Re}(\mu) > \nu \geq 0$,
then
\begin{equation}\label{305}
J_{\phi}(x;h) = \mu A_\phi \int_0^\infty h(xu^2) u^{\mu} \frac{du}{u}  + O(x^{-\frac {\nu}{2} -\varepsilon}).
\end{equation}
\end{thm}

\noindent
{\bf Remark.}
When $A_{\phi} \not= 0$, $J_{\phi}(x;h) \sim C_{\phi,h}x^{-\frac{\mu}{2}}$.
Hence ${\rm Re}(\mu)$ must be smaller than $1$ by $\eqref{303}$ in this case.
When $A_{\phi}=0$, $J_{\phi}(x;h)=O(x^{-\frac{\nu}{2} -\varepsilon})$.
Hence $\nu$ must be smaller than $1$ by $\eqref{303}$ in this case.

\begin{cor}
Let $\phi, \,\psi \in C_L(N)$. Suppose that $\phi(n)=\psi(n)$ for any positive integer $n$.
In the case $N \geq 2$,
we additionally suppose that $\phi$ $($and $\psi)$
satisfies the condition $(3.4)$ for the constants $A,\mu,\nu$.
Then we have
\begin{align}\label{306}
\aligned
\int_0^\infty & h(xu)\phi(u)du
- \sum_{\zeta^\ast(\rho)=0} \int_0^\infty h(xu) \phi(u) u^\rho \frac{du}{u} \\
=&
\int_0^\infty h(xu) \psi(u)du
- \sum_{\zeta^\ast(\rho)=0} \int_0^\infty h(xu) \psi(u) u^\rho \frac{du}{u}
+O(x^{-\frac {\nu}{2} -\varepsilon}) +O(x^{-\frac 13 -\varepsilon})
\endaligned
\end{align}
as $x \to 0$ for any fixed $\varepsilon >0$.
Sums on the both sides run over all zeros of $\zeta^\ast(s)$ counting with multiplicity.
\end{cor}

\subsection{ Rankin-Selberg type relations .}
Let $\phi \in C_L(M)$, $\psi \in C_L(N)$ with the Euler products
\begin{align*}
L_\phi(s)& =\prod_p P_{\phi,p}(p^{-s})^{-1}
      =\prod_{p} \prod_{i=1}^{{\rm deg}\,P_{\phi,p}} (1 -\alpha_{\phi}(p,i) p^{-s})^{-1}, \\
L_\psi(s)& =\prod_p P_{\psi,p}(p^{-s})^{-1}
      =\prod_{p} \prod_{j=1}^{{\rm deg}\,P_{\psi,p}} (1-\alpha_{\psi}(p,j) p^{-s})^{-1}.
\end{align*}
We use the notaion $S=S_{\phi \otimes \psi}=S_{\phi} \cup S_{\psi}$
and denote by $c_{\phi,l}(p)$ the $l$-th coefficient of polynomial $P_{\phi,p}(X)$ (cf. $\eqref{208}$).
Define the function $\widetilde{L}_{\psi \otimes \phi}(s)$ by
\begin{equation}\label{307}
\widetilde{L}_{\phi \otimes \psi}(s)
=\prod_{p \not\in S} \prod_{i=1}^M \prod_{j=1}^N (1-\alpha_{\phi}(p,i) \alpha_{\psi}(p,j)p^{-s})^{-1}
\end{equation}
for sufficiently large ${\rm Re}(s)$. Note that
\begin{equation}\label{308}
\widetilde{L}_{\phi \otimes \psi}(s)
= \prod_{p \not\in S} \prod_{i}(1-\alpha_{\phi}(p,j) \psi(p)p^{-s})^{-1}
= \prod_{p \not\in S} P_{\phi,p}( \psi(p)p^{-s} )^{-1},
\end{equation}
if $\psi \in C_L(1)$.
We say that the pair $(\phi,\psi) \in C_L(M) \times C_L(N)$ is the {\it Rankin-Selberg pair},
if there exist some polynomials $Q_p(X)$ with ${\rm deg}\,Q_p \leq MN$ for any $p \in S$
such that the function
\begin{equation}\label{309}
L_{\phi \otimes \psi}(s):=
\widetilde{L}_{\phi \otimes \psi}(s) \times
\prod_{p \in S} Q_p(p^{-s})^{-1}
\end{equation}
belongs to the Selberg class $S$.

\begin{thm}
Let $(\phi,\psi) \in C_L(N) \times C_L(1)$ be a Rankin-Selberg pair.
Then we have
\begin{equation}\label{310a}
\aligned
m_{\phi \otimes \psi} \cdot  & \int_0^\infty h(xu) du
- \sum_{L_{\phi \otimes \psi}^\ast(\rho)=0} \int_0^\infty h(xu) u^\rho \frac{du}{u} \\
=&
\,m_\phi \cdot \int_0^\infty h(xu) \psi(u) du
- \sum_{L_\phi^\ast(\rho)=0} \int_0^\infty h(xu) \psi(u) u^\rho \frac{du}{u}
+ O(x^{-\frac{1}{3}-\varepsilon})
\endaligned
\end{equation}
as $x \to 0$ for any fixed $\varepsilon$.
Additionally, we suppose that $|\phi(u)\psi(u)|\leq C u^2$ for some $C>0$.
Then
\begin{equation}\label{310b}
\aligned
m_{\phi \otimes \psi} \cdot  & \int_0^\infty h(xu) du
- \sum_{L_{\phi \otimes \psi}^\ast(\rho)=0} \int_0^\infty h(xu) u^\rho \frac{du}{u} \\
=&
\,m_\phi \cdot \int_0^\infty h(xu) \psi(u) du
- \sum_{L_\phi^\ast(\rho)=0} \int_0^\infty h(xu) \psi(u) u^\rho \frac{du}{u}
+ O(x^{-\frac{1}{3}-\varepsilon}) \\
=&
\, \int_0^\infty h(xu) \phi(u) \psi(u) du
- \sum_{\zeta^\ast(\rho)=0} \int_0^\infty h(xu) \phi(u) \psi(u) u^\rho \frac{du}{u}
+ J_{\phi,\psi}(x;h)  + O(x^{-\frac{1}{3}-\varepsilon})
\endaligned
\end{equation}
as $x \to 0$ for any fixed $\varepsilon$.
The function $J_{\phi,\psi}(x;h)$ is estimated as
\begin{equation}\label{311}
J_{\phi,\psi}(x;h) \ll_{\varepsilon, \phi,\psi, h} x^{-\frac{1}{2}-\varepsilon}
\end{equation}
as $x \to +0$ for any fixed $\varepsilon >0$.
Furthermore, if the estimate
\begin{equation}\label{312}
\sum_{p \leq T} c_{\phi,2}(p) \psi(p)^2 \log p = A_{\phi,\psi} T^{\mu}  +O(T^{\nu +\varepsilon}).
\end{equation}
holds with some constant $A_{\phi,\psi}$ and ${\rm Re}(\mu) > \nu \geq 0$,
\begin{equation}\label{313}
J_{\phi,\psi}(x;h) = \mu A_{\phi,\psi} \int_0^\infty h(xu^2) u^{\mu} \frac{du}{u}  + O(x^{-\frac {\nu}{2} -\varepsilon}).
\end{equation}
\end{thm}

\begin{thm}
Let $(\phi,\psi) \in C_L(M) \times C_L(N)$ be a Rankin-Selberg pair with $M,N \geq 2$.
Then
\begin{equation}\label{314a}
\aligned
m_{\phi \otimes \psi} & \int_0^\infty h(xu) du
- \sum_{L_{\phi \otimes \psi}^\ast(\rho)=0} \int_0^\infty h(xu) u^\rho \frac{du}{u} \\
=&
\,m_\phi \int_0^\infty h(xu) \mu(u) du
  - \sum_{L_\phi^\ast(\rho)=0} \int_0^\infty h(xu) \psi(u) u^\rho \frac{du}{u} \\
& +J_{\phi,\psi}^{(1)}(x;h) +J_{\phi,\psi}^{(2)}(x;h)  + O(x^{-\frac{1}{3}-\varepsilon })  \\
=&
\, \int_0^\infty h(xu) \phi(u) \psi(u) du
- \sum_{\zeta^\ast(\rho)=0} \int_0^\infty h(xu) \phi(u) \psi(u) u^\rho \frac{du}{u} \\
& +J_{\phi,\psi}^{(3)}(x;h) + J_{\phi,\psi}^{(4)}(x;h) + J_{\phi,\psi}^{(5)}(x;h) + O(x^{-\frac{1}{3}-\varepsilon })
\endaligned
\end{equation}
as $x \to +0$ for any fixed $\varepsilon >0$.
Additionally, we suppose that $|\phi(u)\psi(u)|\leq C u^2$ for some $C>0$.
Then
\begin{equation}\label{314b}
\aligned
m_{\phi \otimes \psi} & \int_0^\infty h(xu) du
- \sum_{L_{\phi \otimes \psi}^\ast(\rho)=0} \int_0^\infty h(xu) u^\rho \frac{du}{u} \\
=&
\,m_\phi \int_0^\infty h(xu) \mu(u) du
  - \sum_{L_\phi^\ast(\rho)=0} \int_0^\infty h(xu) \psi(u) u^\rho \frac{du}{u} \\
& +J_{\phi,\psi}^{(1)}(x;h) +J_{\phi,\psi}^{(2)}(x;h)  + O(x^{-\frac{1}{3}-\varepsilon })  \\
=&
\, \int_0^\infty h(xu) \phi(u) \psi(u) du
- \sum_{\zeta^\ast(\rho)=0} \int_0^\infty h(xu) \phi(u) \psi(u) u^\rho \frac{du}{u} \\
& +J_{\phi,\psi}^{(3)}(x;h) + J_{\phi,\psi}^{(4)}(x;h) + J_{\phi,\psi}^{(5)}(x;h) + O(x^{-\frac{1}{3}-\varepsilon })
\endaligned
\end{equation}
as $x \to +0$ for any fixed $\varepsilon >0$.
The functions $ J_{\phi,\psi}^{(k)}(x;h) $ are estimated as
\begin{equation}\label{315}
J_{\phi,\psi}^{(k)}(x;h) \ll x^{-\frac{1}{2}-\varepsilon } \quad (1 \leq k \leq 5)
\end{equation}
as $x \to +0$ for any fixed $\varepsilon >0$.
Moreover the asymptotic formulas
\begin{align}
J_{\phi,\psi}^{(1)}(x;h) = J_{\phi,\psi}^{(3)}(x;h)
&= \mu_1 A_1 \int_0^\infty h(xu^2) u^{\mu_1} \frac{du}{u}
 +O(x^{-\frac{\nu_1}{2}-\varepsilon}),  \label{316} \\
J_{\phi,\psi}^{(4)}(x;h)
&=  \mu_2 A_2 \int_0^\infty h(xu^2) u^{\mu_2} \frac{du}{u}
 +O(x^{-\frac{\nu_2}{2} -\varepsilon}),  \label{317} \\
\frac{1}{2}J_{\phi,\psi}^{(2)}(x;h) = \frac{1}{3}J_{\phi,\psi}^{(5)}(x;h)
&= \mu_3 A_3 \int_0^\infty h(xu^2) u^{\mu_3} \frac{du}{u}
 +O(x^{-\frac{\nu_3}{2} -\varepsilon}) \label{318}
\end{align}
hold, if the corresponding estimate
\begin{align}
\sum_{p \leq T}\Lambda_{\phi \otimes \phi}(p) c_{\psi,2}(p) = A_1 T^{\mu_1}+O(T^{ \nu_1+\varepsilon }), \label{thm4:1} \\
\sum_{p \leq T} \Lambda_{\psi \otimes \psi}(p) c_{\phi,2}(p) = A_2 T^{\mu_2}+O(T^{\nu_2+\varepsilon}),\label{thm4:2} \\
\sum_{p \leq T} c_{\phi,2}(p) c_{\psi,2}(p) \log p = A_3 T^{\mu_3}+O(T^{ \nu_3+\varepsilon }),\label{thm4:3}
\end{align}
hold respectively.
\end{thm}

\section{Examples}

In this section we give simple examples of Theorem 1 $\sim$ Theorem 4.

\subsection{ Dirichlet $L$-functions. }
Let $\chi$ mod $q$ be a primitive Dirichlet character.
Then the function $\phi_{\chi}$ defined in $\eqref{109}$ belongs to $C_L(1)$,
since $\phi_\chi$ is bounded on ${\Bbb R}_+$ and $L_{\phi_{\chi}}(s)=L(s,\chi)$.
Further
$$
\int_0^\infty h(xu) \phi_{\chi}(u)du
= \frac{-q}{2\pi i \,\tau(\overline{\chi})} \sum_{a=1}^q \frac{\overline{\chi}(a)}{a}
\int_0^\infty h^\prime(v)e^{2 \pi i \frac{av}{qx}}dv =O(1).
$$
Hence we re-obtain $\eqref{108}$ from Theorem 1.

\subsection{ Automorphic $L$-functions attached to cusp forms in $S_k(N)$. }
Let ${\frak h}=\{ z \in{\Bbb C};{\rm Im}(z)>0\}$ be the upper half plane
and let $\Gamma_0(N)$ be the Hecke subgroup of level $N$ of the full modular group.
Let $S_k(N)$ be the vector space of all holomorhic function $f$ on $\frak h$ such that
$f((az+b)/(cz+d))=(cz+d)^k f(z)$
for any $\bigl( \begin{smallmatrix} a&b\\c&d \end{smallmatrix} \bigr) \in \Gamma_0(N)$,
and $f(i\infty)=0$. It is well known that any $f \in S_k(N)$ has the Fourier expansion
\begin{equation}\label{401}
f(z)=\sum_{n=1}^\infty a_{f}(n) e^{ 2 \pi i n z}.
\end{equation}
By using the Fourier coefficients $\{a_{f}(n) \}$,
the automorpchic $L$-function $L(s,f)$ associated with $f$ is defined as
\begin{equation}\label{402}
L(s,f)=\sum_{n=1}^\infty a_{f}(n) n^{-s-\frac{k-1}{2}}.
\end{equation}
This series is absolutely convergent on the right-half plane ${\rm Re }(s)>1$
because of the estimate $\sum_{n \leq T} |a_{f}(n)|^2 \ll T^{k+1}$
obtained by the Rankin-Selberg method or
the more precise estimate $|a_{f}(n)| \ll_\varepsilon n^{\varepsilon}$ due to Deligne.
The automorphic $L$-function $L(s,f)$ can be extended to an entire function in $s$ and the function
$$L^\ast(s,f)=N^{\frac{2s+k-1}{4}}(2\pi)^{-s-\frac{k-1}{2}} \Gamma(s+\frac{k-1}{2}) L(s,f)$$
satisfies the functional equation
$$L^\ast(s,f)=\pm (-1)^{k/2} L^\ast(1-s,f)$$
where the sign $\pm$ is determined by the action of Fricke involution.
Moreover, if $f\in S_k(N)$ is a normalized Hecke-eigen newform (cf. ~\cite[chap.1.4]{Bu99}),
$L(s,f)$ has the Euler product
\begin{equation}\label{403}
L(s,f)=
\prod_{p|N} (1-a_{f}(p)p^{-\frac{k-1}{2}}p^{-s})^{-1}
\prod_{p \not\,|\, N} (1-a_{f}(p)p^{-\frac{k-1}{2}}p^{-s}+p^{-2s})^{-1}.
\end{equation}
Define the function $\phi_f:{\Bbb R}_+  \to {\Bbb C}$ by
\begin{equation}\label{404}
\phi_f(u)=u^{-\frac{k-1}{2}}\,\int_1^2 f(X+iu^{-1})e^{ -2\pi i u (X+iu^{-1}) }dX. \label{Def_1}
\end{equation}
From the definiton of $\phi_f$, $\phi_f(n)$ coincides with the (shifted) $n$-th Fourier coefficient $a_{f}(n)n^{-\frac{k-1}{2}}$,
and satisfies the estimate $|\phi_f(u)| \leq C\sqrt{u}$ for some $C>0$, because $Y^{k/2}|f(X+iY)|$ is bounded.
Also we can easily find that $c_2(p)=-1$ and $S_{\phi_f}=\{p\,;\, p|N \}$ for $\phi_f(u)$.
Therefore $\phi_f(u)$ belongs to $C_L(2)$ and
we find that
\begin{equation}\label{405}
\int_0^\infty h(xu)\phi_f(u)du=O(x^{N-\frac{3}{2}})
\end{equation}
for any fixed positive integer $N$ by using integration by parts suitable times.
Furthermore, if we assume that $\zeta(s)$ has no zero in ${\rm Re}(s) > \sigma$,
then
\begin{equation}\label{406}
\sum_{p\leq T, \,p \not\in S_{\phi} }c_2(p)\log p
= \sum_{p\leq T, \,p \not\in S_{\phi} } \log p = T +O(T^{\sigma+\varepsilon}).
\end{equation}
Hence we obtain the following result as a consequence of Theorem 2.

\begin{thm}
Let $f \in S_k(N)$ be a normalized Hecke-eigen cuspform.
Assume that $\zeta(s)$ has no zeros in ${\rm Re}(s) > \sigma$.
Then
\begin{equation}\label{407}
\aligned
\sum_{L^\ast(\rho,f)=0} & \int_0^\infty
h(xu)u^{\rho}\frac{du}{u} \\
= & \sum_{\zeta^\ast(\rho)=0}\int_0^\infty
h(xu) \phi_f(u) u^\rho \frac{du}{u}
-C(h) \, x^{-\frac 12}
+O(x^{-\frac{\sigma}{2}-\varepsilon }) +O(x^{-\frac 13 - \varepsilon })
\endaligned
\end{equation}
as $x \to +0$ for any fixed $\epsilon >0$ for any $h \in C_0^\infty$,
where $C(h)=2^{-1}\,\widehat{h}(1/2)$.
The sum on the right hand side runs over all zeros of $\zeta^\ast(s)$ counting with multiplicity.
And the sum on the left hand side runs over all zeros of $L^\ast(s,f)$ counting with multiplicity.
\end{thm}

\subsection{ Rankin-Selberg $L$-functions. }
Let $f,g \in S_k(1)$ be normalized Hecke-eigen cusp forms with Fourier expansions
\begin{equation}\label{408}
f(z)=\sum_{n=1}^\infty a_{f}(n)e^{ 2 \pi i n z},\qquad
g(z)=\sum_{n=1}^\infty a_{g}(n)e^{ 2 \pi i n z}.
\end{equation}
Define $\alpha_p$, $\beta_p$, $\gamma_p$ and $\delta_p$
by using the Euler product of $L(s,f)$, $L(s,g)$;
\begin{align*}
L(s,f)&=\prod_p (1 -a_f(p)p^{-\frac{k-1}{2}} p^{-s} +p^{-2s})^{-1}
      =\prod_p [(1 -\alpha_p p^{-s}) (1-\beta_p p^{-s})]^{-1}, \\
L(s,g)&=\prod_p (1 -a_g(p)p^{-\frac{k-1}{2}} p^{-s} +p^{-2s})^{-1}
      =\prod_p [(1-\gamma_p p^{-s})(1-\delta_p p^{-s})]^{-1}.
\end{align*}
The Rankin-Selberg $L$-function $L(s,f \otimes g)$ is defined by
\begin{equation*}
L(s,f \otimes g)=\prod_p
[(1-\alpha_p \gamma_p p^{-s})(1-\alpha_p \delta_p p^{-s})(1-\beta_p \gamma_p p^{-s})(1-\beta_p \delta_p p^{-s})]^{-1}.
\end{equation*}
Then $L(s,f \otimes g)=\zeta(2s)\sum_{n=1}^\infty a_{f}(n) a_{g}(n) n^{-s-k+1}$.
Moreover the completed $L$-function $L^\ast(s, f \otimes g)=(4\pi)^{-s-k+1}\Gamma(s+k-1)\Gamma(s)L(s, f \otimes g)$
satisfies the functional equation
$L^\ast(s, f \otimes g)=L^\ast(1-s, f \otimes g)$
(cf. ~\cite[chap.1.6]{Bu99}).
Further it is known that $s=1$ is a simple pole if $f=g$ and is a regular point otherwise.
Let
\begin{equation}\label{409}
\phi_{f \otimes g}(u):= \phi_f(u)\phi_g(u)
\end{equation}
where $\phi_f$, $\phi_g$ are defined in $\eqref{404}$.
Then $ |\phi_{f \otimes g}(u)| \leq C u $ for some $C>0$.
If $\zeta(s)$, $L(s,f \otimes f)$ and $L(s,g \otimes g)$ have no zeros in ${\rm Re}(s)>\sigma$,
we obtain
\begin{equation}\label{410}
\aligned
\sum_{p \leq T}\Lambda_{\phi_f \otimes \phi_f}(p) c_{\phi_g,2}(p)
= -\sum_{p \leq T}\Lambda_{f \otimes f}(p) = - T +O(T^{ \sigma+\varepsilon }),  \\
\sum_{p \leq T}\Lambda_{\phi_g \otimes \phi_g}(p) c_{\phi_g,2}(p)
= -\sum_{p \leq T} \Lambda_{g \otimes g}(p) = - T +O(T^{\sigma+\varepsilon}), \\
\sum_{p \leq T} c_{\phi_f,2}(p) c_{\phi_g,2}(p) \log p
=\sum_{p \leq T} \log p = T +O(T^{ \sigma+\varepsilon }).
\endaligned \end{equation}
Together with the above things, we obtain the following theorem as a consequence of Theorem 4.

\begin{thm}
Let $f,g \in S_k(1)$ be normalized Hecke-eigen cusp forms.
Assume that $\zeta(s)$, $L(s,f \otimes f)$ and $L(s,g \otimes g)$
have no zeros in ${\rm Re}(s) >\sigma$.
Then for any $h \in C_0^\infty$, the following formula holds:
\begin{equation}\label{411}
\aligned
\delta_{f=g} & \int_{0}^\infty h(xu)du
-\sum_{L^\ast(\rho,f \otimes g)=0}\int_0^\infty
h(xu) u^{\rho} \frac{du}{u} \\
\,
&= -\sum_{L^\ast(\rho,f)=0} \int_0^\infty
h(xu) \phi_g(u) u^\rho \frac{du}{u}
- C(h) \, x^{-\frac 12}
+O(x^{-\frac {\sigma}{2} - \varepsilon })
+O(x^{-\frac 13 - \varepsilon }) \\
\,
&= -\sum_{\zeta^\ast(\rho)=0}\int_0^\infty
h(xu) \phi_f(u) \phi_g(u) u^\rho \frac{du}{u}
- C(h) \, x^{-\frac 12}
+O(x^{-\frac {\sigma}{2} - \varepsilon })
+O(x^{-\frac 13 - \varepsilon } )
\endaligned
\end{equation}
as $x \to +0$ for any positive $\varepsilon$,
where $ C(h) =2^{-1} \, \widehat{h}(1/2)$, $\delta_{f=g}=1$ if $f=g$ and is zero otherwise.
\end{thm}

\section{Weil's Explicit Formula}
In this section we state a version of Weil's explicit formula.
It is one of the main tools for our proof of the results in this paper.
Define the involution $h \mapsto h^\ast$ on $C_0^\infty$ by
\begin{equation}\label{501}
h^\ast(u)= \frac 1u f(\frac 1u)
\end{equation}
and the Mellin transform of $h$ by
\begin{equation}\label{502}
\widehat{h}(s)=\int_0^\infty h(u)u^s \frac{du}{u}.
\end{equation}
Because $h$ has a compact support, the above integral is absolutely convergent for any $s \in {\Bbb C}$.
Further the Mellin inversion formula
\begin{equation}\label{503}
h(u)=\frac{1}{2\pi i}\int_{(\sigma)}\widehat{h}(s)u^{-s}ds
\end{equation}
is valid, where the path of integration is the vertical line ${\rm Re}(s)=\sigma$.

\begin{prop} {\bf [ Weil's Explicit Formula ]}
Let $L(s) \in {\cal S}$.
Then, for any $h \in C_0^\infty$,
\begin{equation}\label{504}
\aligned
m_L & \, \widehat{h}(0) -\sum_{L^\ast(\rho)=0}\widehat{h}(\rho) + m_L \, \widehat{h}(1) \\
=&\sum_{n=1}^\infty \{ \Lambda_L(n) h(n)+\overline{\Lambda_L(n)} h^\ast (n) \}
  +(2 \log Q+d\,C_E)\,h(1)+\sum_{j=1}^r W_{\lambda_j.\mu_j}(h),
\endaligned
\end{equation}
where $d=2\sum_{j=1}^r\lambda_j$ and $C_E$ is the Euler constant.
The functional $W_{\lambda,\mu}$ is given by
\begin{equation}\label{505}
W_{\lambda,\mu}(h) =
\int_1^\infty
\left[h_{\lambda,\mu}(u)+h_{\lambda,\mu}^\ast(u)-2h(1)u^{({\rm Re}(\mu)-1)/\lambda}\right]
\frac{u^{(1-{\rm Re}(\mu))/\lambda}}{u^{1/\lambda}-1}\frac{du}{u},
\end{equation}
\begin{equation}\label{506}
h_{\lambda,\mu}(u)  =h(u)u^{-i\,{\rm Im}(\mu)/\lambda}.
\end{equation}
The sum $\sum_{L^\ast(\rho)=0}$ runs over all zeros of $L^\ast(s)$ counting with multiplicity.
Sums and integrals contained in the both sides of $\eqref{504}$ are absolutely convergent,
because the Mellin transform $\widehat{h}$ decays very fast by the assumption on $h$.
\end{prop}

Proposition 1 is proved by a way similar to the proof of Weil's explicit formula in ~\cite{La70}.
There is no essential difference or difficulty in our case
because of conditions $({\rm S}1) \sim ({\rm S}5)$ for $L(s)$.
Hence we omit the proof of Proposition 1.

\section{Proof of Theorem 1 and Theorem 2}

\subsection{ Lemmas.}
In this part we prepare several lemmas which are necessary for our proof of Theorem 1 and Theorem2.
For $L(s) \in {\cal S}$, $h \in C_0^\infty$ and $x>0$,
we define the sum $S_L(x)$ by
\begin{equation}\label{601}
S_L(x) :=S_L(x;h) :=\sum_{n=1}^\infty \Lambda_L(n)h(xn).
\end{equation}

\begin{lem}
Let $L(s)=\sum_{n=1}^\infty c(n) n^{-s} \in {\cal S}$.
Then, for any $\varepsilon >0$,
\begin{equation}\label{602}
\sum_{p} \sum_{{l \leq m}\atop{p^m \leq T} } c(p^m) \log p
\ll_\varepsilon  T^{\frac 1l +\varepsilon } ,
\end{equation}
and
\begin{equation}\label{603}
\sum_{p} \sum_{{l \leq m}\atop{p^m \leq T} } \Lambda_L(p^m)
\ll_\varepsilon  T^{ \theta +\frac 1l +\varepsilon } ,
\end{equation}
where $\theta$ is the constant in axiom $({\rm S}5)$ of the Selberg class.
\end{lem}

\begin{pf}
We have
\begin{equation}\label{604}
\aligned
\sum_{p} &\sum_{{l \leq m}\atop{p^m \leq T} } c(p^m) \log p
  \ll \sum_{p^l \leq T} \sum_{ l \leq m \leq \frac{\log T}{\log p} } p^{m\varepsilon} \log p \\
& \leq T^\varepsilon \frac 1l \log T \sum_{p^l \leq T} \sum_{ 1 \leq m \leq \frac{\log T}{\log p} } 1
  \leq \frac {1}{\log2} T^{\varepsilon +\frac 1l} (\log T)^2
  \ll_{\varepsilon^\prime} T^{\frac 1l +\varepsilon^\prime}.
\endaligned
\end{equation}
This is assertion $\eqref{602}$.
Recall that $\Lambda_L(n) = b(n)\log n$ and $|b(n)| \ll n^\theta$.
Then we have
\begin{equation}\label{605}
\aligned
\sum_{p} & \sum_{{l \leq m}\atop{p^m \leq T} } \Lambda_L(p^m)
  = \sum_{p^l \leq T} \sum_{ l \leq m \leq \frac{\log T}{\log p} } b(p^m)\log p^m \\
& \ll \sum_{p^l \leq T} \sum_{ l \leq m \leq \frac{\log T}{\log p} } p^{m\theta} \log p^m
  \leq T^\theta \log T \sum_{p^l \leq T} \sum_{ 1 \leq m \leq \frac{\log T}{\log p} } 1 \\
& \leq T^{\theta + \frac 1l} \log T \frac {\log T}{\log 2} \sum_{p^l \leq T}
  \ll_\varepsilon T^{\theta +\frac 1l +\varepsilon}.
\endaligned
\end{equation}
This is our assertion $\eqref{603}$.
\end{pf}

\begin{lem}
Let $L(s) \in {\cal S}$. Then
\begin{equation}\label{606}
S_L(x)
=m_L \cdot \int_0^\infty h(xu) du
-\sum_{L^\ast(\rho)=0} \int_0^\infty h(xu) u^\rho \frac{du}{u}
+O(1).
\end{equation}
\end{lem}

\begin{pf}
Applying Proposition 1 to $L(s)$ and $u \mapsto h(xu)$ we have
\begin{equation}\label{607}
\aligned
S_L(x)=\,\,
& m_L \int_0^\infty h(xu) du
 -\sum_{L^\ast(\rho)=0} \int h(xu)u^\rho \frac{du}{u}
 + m_L \int_0^\infty h(xu) \frac{du}{u} \\
& -\sum_{n=1}^\infty  \overline{\Lambda_L(n)} n^{-1} h(xn^{-1}) \\
& -(2 \log Q + d\,C_E)\,h(x)
 -\sum_{j=1}^r W_{\lambda_j,\mu_j}(u \mapsto h(xu)).
\endaligned
\end{equation}

The third term on the right hand side is bounded, because it is equal to $\widehat{h}(0)$.
The fourth term and the fifth term on the right hand side is zero
for sufficiently small $x>0$ because the support of $h$ is compact.
Furthermore the sixth term on the right hand side is absorbed into the error term.
In fact
$$
W_{\lambda, \mu}(u \mapsto h(xu))
= \int_1^\infty h(xu)
\frac{u^{\frac{1-\mu}{\lambda}}}{u^{ \frac{1}{\lambda} } -1}\frac{du}{u}
= x^{\frac{\mu}{\lambda}}
\int_x^\infty h(v)
\frac{v^{ \frac{1-\mu}{\lambda} }}{ v^{ \frac{1}{\lambda} } -x^{\frac{1}{\lambda} }}\frac{dv}{v}
$$
for sufficiently small $x>0$ and the right hand side is bounded as $x \to +0$
since ${\rm Re}(\mu) \geq 0$ and $\lambda >0$.
\end{pf}

\begin{lem} Let $L(s) \in {\cal S}$. Then
\begin{equation}\label{608}
S_L(x)
=\sum_{p} \sum_{m=1}^{l-1} \Lambda_L(p^m) h( x p^m )
+O(x^{ -\theta -\frac 1l -\varepsilon})
\end{equation}
for any $\varepsilon >0$.
\end{lem}

\begin{pf}
It suffices to show that
\begin{equation}\label{609}
\sum_{p} \sum_{m=l}^{\infty} \Lambda_L(p^m) h( x p^m ) = O(x^{ -\theta -\frac 1l -\varepsilon}).
\end{equation}
By using partial summation (cf.~\cite[page 2]{SP94}) we obtain
$$
\sum_{p} \sum_{m=l}^{\infty} \Lambda_L(p^m) h( x p^m )
= - \int_l^\infty xu h^\prime (xu) \left( \sum_{p}\sum_{{l \leq m}\atop{p^m \leq u}} \Lambda_L(p^m) \right) \frac{du}{u}.
$$
From Lemma 1 the right hand side is estimated as
$$
O\left( \int_{l}^\infty
xu |h^\prime(xu)|u^{ \theta +\frac{1}{l} +\varepsilon }\frac{du}{u}
\right)
=O( x^{ -\theta -\frac{1}{l} -\varepsilon } ).
$$
This implies $\eqref{609}$.
\end{pf}

Here we describe the relation between the Dirichlet coefficients
of $L_\phi(s)$ and those of its logarithmic derivative $(L_\phi^\prime/L_\phi)(s)$.
Define the numbers $r_m(p)$ by
\begin{equation}\label{610}
X \frac{d}{dX} \log \, P_p(X)^{-1} = \sum_{m=1}^\infty r_m(p)X^m.
\end{equation}
By simple series calculations, we find that
\begin{equation}\label{611}
\phi(p^m)=
\begin{cases}
c_1(p)\phi(p^{m-1})+c_2(p)\phi(p^{m-2})+\cdots+c_m(p) & \text{if $m \leq n_p$} \\
c_1(p)\phi(p^{m-1})+c_2(p)\phi(p^{m-2})+\cdots+c_{n_p}(p)\phi(p^{m-n_p}) & \text{if $m > n_p$}
\end{cases}
\end{equation}
and
\begin{equation}\label{612}
r_m(p)= \left\{
\aligned
\phi( & p^m)  +c_2(p)\phi(p^{m-2}) +\cdots \\
& +(j-1)c_j(p)\phi(p^{m-j}) +\cdots +(m-1)c_m(p) \qquad \quad \quad \,\,\,\,\,\,\, \text{if $m \leq n_p$} \\
\phi( & p^m)  +c_2(p)\phi(p^{m-2}) +\cdots \\
& +(j-1)c_j(p)\phi(p^{m-j}) +\cdots +(n_p-1)c_{n_p}(p)\phi(p^{m-n_p}) \quad  \text{if $m > n_p$}
\endaligned
\right.
\end{equation}
where $n_p={\rm deg}\,P_p(X)$ and
the numbers $c_j(p)$ are the coefficients of the polynomial $P_p(X)$
defined in $\eqref{207}$, $\eqref{208}$.
Note that $\phi(p^0)=\phi(1)=1$
which can be seen from the form of the Euler product attached to $L_\phi(s)$.
Then the Dirichlet coefficient $\Lambda_\phi(n)$ of $(L_\phi^\prime/L_\phi)(s)$ is given by
\begin{equation}\label{613}
\Lambda_{\phi}(n) =
\begin{cases}
r_m(p) \log p & \text{ if $n=p^m$ with $m \geq 1$}, \\
0             & \text{otherwise.}
\end{cases}
\end{equation}
Additionally, it is useful for us to note the relation
\begin{equation}\label{614}
c_l(p) = (-1)^{l+1} \sum_{ {(i_1,\cdots,i_l)}\atop{1\leq i_1 \leq \cdots \leq i_l \leq M} }
\alpha_{\phi}(p,i_1) \cdots \alpha_{\phi}(p,i_l)
\end{equation}
where $\alpha_{\phi}(p,i) $ are the roots of
the polynomial $P_{\phi,p}(X)$ associated with $\phi \in C_L(M)$.

\begin{lem}
Let $\phi \in C_L(N)$.
Then
\begin{equation}\label{615}
\Lambda_{\phi}(n) \ll_\varepsilon n^{\varepsilon}.
\end{equation}
\end{lem}

\begin{pf}
From $\eqref{611}$ we have
\begin{equation}\label{616}
c_j(p) = \sum_{l=1}^j \sum_{{1\leq i_1,\cdots,i_l \leq j}\atop{i_1+\cdots+i_l=j}}
(-1)^{l+1}\phi(p^{i_1}) \cdots \phi(p^{i_l}).
\end{equation}
Because $|\phi(n)| \leq C_\varepsilon n^\varepsilon$ for any positive integer $n$,
$$
|c_j(p)| \leq
p^{j \varepsilon}
\sum_{l=1}^j \sum_{{1\leq i_1,\cdots,i_l \leq j}\atop{i_1+\cdots+i_l=j}}
C_\varepsilon^l =: p^{j \varepsilon}\, C_j,
$$
say. Therefore, by $\eqref{612}$, we obtain
$$
|r_m(p)| \leq
p^{m \varepsilon}(1 + \sum_{j=2}^{N} (j-1)C_j).
$$
Hence
$$
|\Lambda_\phi(p^m)|= |r_m(p)| \log p \leq C\,p^{m\varepsilon}\log p^m \leq C^\prime p^{m \varepsilon^\prime} .
$$
\end{pf}

From Lemma 4, we can take $\theta=\varepsilon$ for any fixed $\varepsilon >0$ in (S$5$)
for $L_\phi(s)$ with $\phi \in C_L$. Now we define $\widetilde{S}(x)$ by
\begin{equation}\label{617}
\widetilde{S}(x) :=\widetilde{S}_\phi(x;h)
:=\sum_{p} \sum_{m=1}^2 \Lambda_{\phi}(p^m) h( x p^m ).
\end{equation}

\begin{lem} Let $\phi \in C_L(N)$.
Then
\begin{equation}\label{618}
\aligned
\widetilde{S}(x)
& = \sum_{p} \log p \sum_{m=1}^2 \phi(p^m) h( x p^m )
  + \sum_{p \not\in S_\phi} C(p)h(xp^2) \log p
  + \sum_{p \in S_\phi} C(p)h(xp^2) \log p  \\
& =: \widetilde{S}_1(x) +\widetilde{S}_2(x) +\widetilde{S}_3(x)
\endaligned
\end{equation}
say, where
\begin{equation}\label{619}
C(p) =
\begin{cases}
c_2(p) & \text{if $n_p \geq 2$}, \\
0      & \text{if $n_p=1$}.
\end{cases}
\end{equation}
\end{lem}

\begin{pf}
This is a direct consequence of $\eqref{611}$ and $\eqref{612}$.
\end{pf}

\begin{lem}
Let $\phi \in C_L(N)$ and let $\widetilde{S}_1(x)$ be as above. Then
\begin{equation}\label{620}
\widetilde{S}_1(x)
=\int_0^\infty h(xu) \phi(u) du
-\sum_{\zeta^\ast(\rho)=0} \int_0^\infty h(xu) \phi(u) u^\rho \frac{du}{u}
+O(x^{-\frac{1}{3}-\varepsilon})
\end{equation}
for any $\varepsilon >0$.
\end{lem}

\begin{pf}
First we show that
\begin{equation}\label{621}
\widetilde{S}_1(x)
= \sum_{n=1}^\infty \Lambda(n)\phi(n)h(xn)
 +O( x^{-\frac {1}{3} -\varepsilon } )
\end{equation}
for any $\varepsilon >0$.
By a way similar to the proof of Lemma 3, we obtain
\begin{equation}\label{622}
\sum_{p} \sum_{m=3}^\infty \phi(p^m) h( x p^m )
= O( x^{-\frac {1}{3} -\varepsilon } )
\end{equation}
by using Lemma 1 and partial summation.
This leads to $\eqref{621}$ because
$$
\widetilde{S}_1(x)
= \sum_{n=1}^\infty \Lambda(n)\phi(n)h(xn)
-\sum_{p} \sum_{m=3}^\infty \phi(p^m) h( x p^m ).
$$
Applying Proposition 1 to $\zeta(s)$ and $u \mapsto h(xu) \phi(u)$
we have
\begin{equation}\label{623}
\aligned
\sum_{n=1}^\infty & \Lambda(n) h(xn) \phi(n) \\
=& \int_0^\infty h(xu) \phi(u)du
 -\sum_{\zeta^\ast(\rho)=0} \int h(xu) \phi(u) u^\rho \frac{du}{u}
 +\int_0^\infty h(xu) \phi(u) \frac{du}{u} \\
& -\sum_{n=1}^\infty  \overline{\Lambda(n)} n^{-1} h(xn^{-1})\phi(n^{-1}) \\
& -(\log \pi + \,C_E)\,h(x)\phi(1)
  -W_{1/2,0}(u \mapsto h(xu)\phi(u)).
\endaligned
\end{equation}
As for the third term and the sixth term on the right hand side, we have
\begin{equation}\label{624}
\aligned
\int_0^\infty & h(xu) \phi(u) \frac{du}{u}
\, - \, W_{1/2, 0}(u \mapsto h(xu)\phi(u)) \\
&= \int_0^\infty h(v) \phi(v/x) \frac{dv}{v}
-\int_x^\infty h(v)\phi(v/x)
\frac{v^2}{v^2-x^2}\frac{dv}{v} \\
&= x^2 \int_0^\infty h(v) \phi(v/x) \frac{1}{v^2-x^2} \frac{dv}{v}
-\int_0^x h(v)\phi(v/x)
\frac{v^2}{v^2-x^2}\frac{dv}{v},
\endaligned
\end{equation}
for sufficiently small $x>0$.
Because $|\phi(u)| \leq C u^2$,
the right hand side of $\eqref{624}$ is bounded as $x \to 0$.
The fourth term and the fifth term on the right hand side are zero
for sufficiently small $x>0$ because the support of $h$ is compact.
Hence we obtain
\begin{equation}\label{625}
\aligned
\sum_{n=1}^\infty & \Lambda(n) h(xn) \phi(n) \\
=& \int_0^\infty h(xu) \phi(u)du
 -\sum_{\zeta^\ast(\rho)=0} \int h(xu) \phi(u) u^\rho \frac{du}{u}
 +O(1).
\endaligned
\end{equation}
Lemma 6 follows from $\eqref{621}$ and $\eqref{625}$.
\end{pf}

\begin{lem}
Let $\phi \in C_L(N)$ with $N \geq 2$ and let $\widetilde{S}_2(x)$ be as above.
Then
\begin{equation}\label{626}
\tilde{S}_2(x)=O(x^{-\frac{1}{2}-\varepsilon})
\end{equation}
for sufficiently small $x>0$.
Further we obtain
\begin{equation}\label{627}
\widetilde{S}_2(x)
= \mu A_\phi \int_0^\infty h(xu^2) u^{\mu} \frac{du}{u}
 +O(x^{ -\frac {\nu}{2} -\varepsilon})
\end{equation}
for any $\varepsilon >0$, when
\begin{equation}\label{628}
\sum_{p \leq T,\, p \not\in S_\phi} c_2(p) \log p = A_\phi T^{\mu} +O(T^{\nu +\varepsilon})
\end{equation}
holds for some constant $A_\phi$ and ${\rm Re}(\mu) > \nu >0$.
\end{lem}

\begin{pf}
Suppose that the support of $h(u) \in C_0^\infty$ is contained in $[a,b]$.
Then, from $\psi(p^m) \ll p^{m \varepsilon}$, we have
\begin{equation}\label{629}
\sum_{p \not\in S} h(xp^2) c_{2}(p) \log p
\ll \sum_{ \sqrt{\frac{a}{x}} \leq p \leq \sqrt{\frac{b}{x}} }p^{2\varepsilon}
\ll x^{-\frac{1}{2}-\varepsilon}.
\end{equation}
This is the first assertion.
By using partial summation we have
\begin{equation}\label{630}
\widetilde{S}_2(x)
= -\int_0^\infty 2xu h^\prime (xu^2)
 \left( \sum_{p \leq u,\, p \not\in S_\phi} c_2(p) \log p  \right) du.
\end{equation}
Applying the assumption of the Lemma we find that
the right hand side of $\eqref{630}$ is equal to
\begin{align*}
-\int_0^\infty & 2xu h^\prime (xu^2)
 ( A_\phi u^{\mu} +O(u^{\nu+\varepsilon})) du \\
&= \mu A_\phi \int_0^\infty h(xu^2) u^{\mu} \frac{du}{u}
+O(\int_0^\infty 2xu^2 |h^\prime(xu^2)|u^{\nu+\varepsilon}\frac{du}{u}) \\
&= \mu A_\phi \int_0^\infty h(xu^2) u^{\mu} \frac{du}{u}
+O( x^{-\frac{\nu + \varepsilon}{2}} ).
\end{align*}
\end{pf}

\begin{lem}
Let $\phi \in C_L(N)$ with $n \geq 2$ and let $\widetilde{S}_3(x)$ be as above.
Then
\begin{equation}\label{631}
\widetilde{S}_3(x) = O(1).
\end{equation}
\end{lem}

\begin{pf}
By using partial summation we have
\begin{equation}\label{632}
\widetilde{S}_3(x)
= -\int_0^\infty 2xu^2 h^\prime (xu^2)
 \left( \sum_{p \leq u,\, p \in S_\phi} c_2(p) \log p  \right) \frac{du}{u}.
\end{equation}
From the proof of Lemma 4, we have $c_2(p) \ll  p^{2\varepsilon}$.
Hence
\begin{equation}\label{633}
\sum_{p \leq u,\, p \in S_\phi} c_2(p) \log p   \ll
\sum_{p \in S_\phi} p^{2 \varepsilon} \log p  =O(1).
\end{equation}
From $\eqref{632}$ and $\eqref{633}$ we have $\widetilde{S}_3(x) =O(1)$.
\end{pf}

\subsection{Proof of Theorem 1.}
Let $\phi \in C_L(1)$.
From the definition of $C_L(1)$, $L_\phi(s)$ belongs to ${\cal S}$ with  $\theta=\varepsilon$.
Hence we obtain
\begin{equation}\label{634}
S(x)
=
m_\phi \cdot \int_0^\infty h(xu) du
-\sum_{L_\phi^\ast(\rho)=0} \int_0^\infty h(xu) u^\rho \frac{du}{u}+O(1)
\end{equation}
by Lemma 2.
Here we note that
$\Lambda_{\phi}(n)=\Lambda(n)\phi(n)$ for $\phi \in C_L(1)$.
Therefore
\begin{equation}\label{635}
S(x) =\sum_{n=1}^\infty \Lambda(n)h(xn)\phi(n).
\end{equation}
By applying Proposition 1 to $\zeta(s)$ and $u \mapsto h(xu)\phi(u)$, we obtain
\begin{equation}\label{636}
S(x) = \int_0^\infty h(xu) \phi(u) du
- \sum_{\zeta^\ast(\rho)=0} \int_0^\infty h(xu) \phi(u) u^\rho \frac{du}{u} + O(1),
\end{equation}
in a way similar to the proof of Lemma 2.
Theorem 1 follows from $\eqref{634}$ and $\eqref{636}$. $\Box$

\subsection{Proof of Theorem 2.}
Let $\phi \in C_L(N)$ with $N \geq 2$.
From the definition of $C_L(N)$ and Lemma 4, $L_\phi(s)$ belongs to ${\cal S}$ with  $\theta=\varepsilon$.
Hence we obtain
\begin{equation}\label{637}
S(x)
=m_\phi \cdot \int_0^\infty h(xu) du
-\sum_{L_\phi^\ast(\rho)=0} \int_0^\infty h(xu) u^\rho \frac{du}{u}
+O(1)
\end{equation}
by Lemma 2. On the other hand we have
\begin{equation}\label{638}
S(x) = \widetilde{S}(x)+O(x^{-\frac{1}{3}-\varepsilon})
\end{equation}
by Lemma 3.
Together with Lemmas $5,6,7$ and $8$ we obtain
\begin{equation}\label{639}
S(x) =
\int_0^\infty h(xu) \phi(u) du
- \sum_{\zeta^\ast(\rho)=0} \int_0^\infty h(xu) \phi(u) u^\rho \frac{du}{u}
+ J_{\phi}(x,h) + O(x^{-\frac 13 -\varepsilon})
\end{equation}
where $J_{\phi}(x,h)=\tilde{S}_2(x)$.
Theorem 2 follows from $\eqref{637}$, $\eqref{639}$ and Lemma 7. $\Box$

\section{ Proof of Theorem 3 and Theorem 4.}

Theorem 3 and Theorem 4 are proved by an argument quite similar to the proof of Theorem 1 and Theorem 2,
so we only describe the outline of the proof. \\

Define the numbers $\Lambda_{\phi \otimes \psi}(n)$ by
\begin{equation}\label{701}
-\frac{L_{\phi \otimes \psi}^\prime(s)}{L_{\phi \otimes \psi}(s)}=\sum_{n=1}^\infty
\frac{ \Lambda_{\phi \otimes \psi}(n) }{n^s}.
\end{equation}
Then we have
\begin{align}\label{702}
\aligned
\,&\Lambda_{\phi \otimes \psi}(n)= \\
&
\begin{cases}
\left( \sum_{i=1}^M \alpha_{\phi}(p,i)^m \right)
\left( \sum_{j=1}^N \alpha_{\psi}(p,j)^m \right) \log  p
 &\text{if $n=p^m$ with $p \not\in S$,  $m \geq 1$,} \\
0&\text{if $n$ is not a power of a prime,}
\end{cases}
\endaligned
\end{align}
and the following Lemma 9 is proved similarly to Lemma 1.

\begin{lem}
Let $(\phi, \psi) \in C_L(M) \times C_L(N)$ be a Rankin-Selberg pair.
Then
\begin{align}\label{703}
\sum_{p \not\in S} \sum_{{l \leq m}\atop{p^m\leq T}}
\Lambda_{\phi \otimes \psi}(p^m)
\ll_{\varepsilon,l} T^{\frac{1}{l}+\varepsilon }
\end{align}
for $l \geq 1$, and
\begin{align}\label{704}
\sum_{p \in S} \sum_{m=1}^\infty
\Lambda_{\phi \otimes \psi}(p^m)=O(1).
\end{align}
\end{lem}
\begin{pf}
First we note the relation
$$
b_\phi(p^m)= \frac{\Lambda_{\phi}(p^m)}{\log p^m}=\frac{1}{m} \sum_{i=1}^M \alpha_{\phi}(p,i)^m
$$
for $p \not\in S_{\phi}$ and
the estimate $b_\phi(n) \ll_{\varepsilon} n^{\varepsilon}$ for $\phi \in C_L$ which follows by Lemma 4.
From these we have
\begin{equation}\label{705}
\Lambda_{\phi \otimes \psi}(p^m) =
\left( \sum_{i=1}^M \alpha_{\phi}(p,i)^m \right)
\left( \sum_{j=1}^N \alpha_{\psi}(p,j)^m \right)
\log  p  \ll_{\varepsilon} p^{m\varepsilon}
\end{equation}
for $p \not\in S$, $(\phi, \psi) \in C_L(M) \times C_L(N)$.
Hence $\eqref{703}$ is obtained by the same arguments as in the proof of Lemma 4.
Because $L_{\phi \otimes \psi}(s) \in {\cal S}$,
\begin{equation}\label{706}
\Lambda_{\phi \otimes \psi}(p^m) = b_{\phi \otimes \psi}(p^m) \log p^m
\ll p^{m\theta}\log p^m \ll p^{m(\theta+\varepsilon)},
\end{equation}
where $b_{\phi \otimes \psi}(n)$ are given by
$\log L_{\phi \otimes \psi}(s)=\sum_{n=1}^\infty b_{\phi \otimes \psi}(n)n^{-s}$.
Therefore
\begin{equation}\label{707}
\sum_{p \in S} \sum_{m=1}^\infty \Lambda_{\phi \otimes \psi}(p^m)
 \ll \sum_{p \in S} \sum_{m=1}^\infty p^{m(\theta+\varepsilon)}
 = \sum_{p \in S} \frac{ p^{\theta+\varepsilon} }{1-p^{\theta+\varepsilon}}
 =O(1).
\end{equation}
\end{pf}

For any fixed $h\in C_0^\infty$ and $x>0$ we consider the sum
\begin{equation}\label{708}
S(x):=S_{\phi,\psi}(x;h):=\sum_{n=1}^\infty \Lambda_{\phi \otimes \psi}(n) h(xn).
\end{equation}
Theorem 3 or Theorem 4 is proved by computing the sum $S(x)$ in two ways.
Applying Proposition 1 to $L_{\phi \otimes \psi}(s)$ and $u \mapsto h(xu)$, we have
\begin{equation}\label{709}
S(x)
=
m_{\phi \otimes \psi} \int_{0}^{\infty} h(xu)du
-\sum_{L_{\phi \otimes \psi}^\ast (\rho)=0}
\int_{0}^{\infty} h(xu) u^{\rho} \frac{du}{u} +O(1)
\end{equation}
for sufficiently small $x>0$.
By using Lemma 9 and partial summation, we obtain
\begin{equation}\label{710}
S(x)= \sum_{p \not\in S} \sum_{m=1}^2 \Lambda_{\phi \otimes \psi}(p^m) h(xp^m) + O(x^{-\frac{1}{3}-\varepsilon}).
\end{equation}
Take
\begin{equation}\label{711}
\widetilde{S}(x) := \sum_{p \not\in S} \sum_{m=1}^2 \Lambda_{\phi \otimes \psi}(p^m) h(xp^m),
\end{equation}
and divide the sum $\widetilde{S}(x)$ into two parts as
\begin{equation}\label{712}
\aligned
\widetilde{S}(x)
&=\sum_{p \not\in S} \sum_{m=1}^2 \Lambda_\phi(p^m) h(xp^m) \psi(p^m)\log p
+ \sum_{p \not\in S} \Lambda_\phi (p^2) h(xp^2) c_{\psi,2}(p) \\
&=: \widetilde{S}_{1}(x) + \widetilde{S}_{2}(x),
\endaligned
\end{equation}
say. For $\widetilde{S}_{1}(x)$ we obtain
$$
\widetilde{S}_{1}(x)=
\sum_{n = 1}^{\infty} \Lambda_\phi(n) h(xn) \psi(n) + O(x^{ -\frac{1}{3} -\varepsilon})
$$
by Lemma 9.
Applying Proposition 1 to $L_\phi(s)$ and $u \mapsto h(xu)\psi(u)$,
we obtain
\begin{equation}\label{713}
\widetilde{S}_{1}(x)
=
m_\phi \int_{0}^{\infty} h(xu)\psi(u)du
-\sum_{L_\phi^\ast(\rho)=0} \int_0^\infty h(xu) \psi(u) u^\rho \frac{du}{u}
+O(x^{-\frac{1}{3}-\varepsilon}).
\end{equation}
For $\widetilde{S}_{2}(x)$ we find that
\begin{equation}\label{714}
\tilde{S}_2(x)=O(x^{-\frac{1}{2}-\varepsilon})
\end{equation}
for sufficiently small $x>0$.
In fact, when the support of $h(u) \in C_0^\infty$ is contained in the interval $[a,b]$,
\begin{equation}\label{715}
\sum_{p \not\in S} \Lambda_\phi (p^2) h(xp^2) c_{\psi,2}(p)
\ll \sum_{ \sqrt{\frac{a}{x}} \leq p \leq \sqrt{\frac{b}{x}} }p^{2\varepsilon}
\ll x^{-\frac{1}{2}-\varepsilon}
\end{equation}
by Lemma 4 and $\psi(p^m) \ll p^{m \varepsilon}$.
This implies $\eqref{714}$.

Here we note that
\begin{equation} \label{716}
\aligned
\Lambda_\phi(p^2)
& = \left( \sum_{i=1}^M \alpha_{\phi}(p,i)^2 \right) \log p \\
& =
\left[
\left( \sum_{i=1}^M \alpha_{\phi}(p,i) \right)^2
-2 \sum_{1 \leq i<j \leq M} \alpha_{\phi}(p,i) \alpha_{\phi}(p,j)
\right]
\log p \\
& =\Lambda_{\phi \otimes \phi}(p) + 2 c_{\phi,2}(p)\log p.
\endaligned \end{equation}
By using $\eqref{716}$ we divide $\widetilde{S}_{2}(x)$ into two parts as
\begin{align*}
\widetilde{S}_{2}(x)
& =\sum_{p \not\in S} \Lambda_{\phi \otimes \phi}(p) h(xp^2)c_{\psi,2}(p)
  + 2\sum_{p \not\in S} h(xp^2) c_{\phi,2}(p) c_{\psi,2}(p) \log p \\
&=:\widetilde{S}_{3}(x) + 2\widetilde{S}_{4}(x),
\end{align*}
say.
By partial summation $\widetilde{S}_{3}(x)$ is expressed as
\begin{equation}\label{717}
\widetilde{S}_{3}(x)
=-\int_0^\infty
\left( \sum_{p \leq u} \Lambda_{\phi \otimes \phi}(p) c_{\psi,2}(p) \right)
(h(xu^2))^\prime du.
\end{equation}
If $\eqref{thm4:1}$ holds, we have
\begin{equation}\label{718}
\widetilde{S}_{3}(x)= \mu_1 A_1
\int_0^\infty h(xu^2) u^{\mu_1} \frac{du}{u}
+O(x^{-\frac{\nu_1}{2}-\varepsilon})
\end{equation}
by substituting $\eqref{thm4:1}$ into $\eqref{717}$.
Also, by using the integral expression
\begin{equation}\label{719}
\widetilde{S}_{4}(x) =
-\int_0^\infty \left(\sum_{p \leq u} c_{\phi,2}(p) c_{\psi,2}(p) \log p \right) ( h(xu^2) )^\prime du
\end{equation}
we obtain
\begin{equation}\label{720}
\widetilde{S}_{4}(x)= \mu_3 A_3 \int_0^\infty h(xu^2) u^{\mu_3} \frac{du}{u}
+O(x^{-\frac{\nu_3}{2} -\varepsilon}),
\end{equation}
if estimate $\eqref{thm4:3}$ holds.
Combining $\eqref{709}$, $\eqref{713}$, $\eqref{718}$ and $\eqref{720}$ we obtain the first half of Theorem 4.
Also we obtain the first half of Theorem 3 by the same equations, if we replace $c_{\psi,2}(p)$ by $0$.

To prove the latter half of Theorem 4, we divide $S_4(x)$ into four parts as
\begin{equation}\label{721}
\aligned
\widetilde{S}(x)
=&\sum_p \sum_{m=1}^2 h(xp^m)\phi(p^m)\psi(p^m)\log p + \sum_p h(xp^2)\phi(p^2)c_{\psi,2}(p)\log p \\
 &+\sum_p h(xp^2)\psi(p^2)c_{\phi,2}(p)\log p +\sum_p h(xp^2)c_{\phi,2}(p)c_{\psi,2}(p)\log p \\
=&\sum_p \sum_{m=1}^2 h(xp^m)\phi(p^m) \psi(p^m) \log p + \sum_p h(xp^2) \phi(p)^2 c_{\psi,2}(p)\log p \\
 &+\sum_p h(xp^2) \psi(p)^2 c_{\phi,2}(p) \log p +3\sum_p h(xp^2)c_{\phi,2}(p)c_{\psi,2}(p)\log p \\
=&\sum_p \sum_{m=1}^2 h(xp^m)\phi(p^m) \psi(p^m) \log p + \sum_p h(xp^2) \Lambda_{\phi \otimes \phi}(p) c_{\psi,2}(p) \\
 &+\sum_p h(xp^2) \Lambda_{\psi \otimes \psi}(p) c_{\phi,2}(p) +3\sum_p h(xp^2)c_{\phi,2}(p)c_{\psi,2}(p)\log p \\
=&:\widetilde{S}_{5}(x)+\widetilde{S}_{6}(x)+\widetilde{S}_{7}(x)+\widetilde{S}_{8}(x),
\endaligned
\end{equation}
say.
By a way similar to the proof of $\eqref{714}$ we obtain
\begin{equation}\label{722}
\widetilde{S}_{l}(x)=O(x^{-\frac{1}{2}-\varepsilon}) \quad (l =6,7,8)
\end{equation}
for sufficiently small $x>0$. Further we find that
\begin{equation}\label{723}
\aligned
\widetilde{S}_{6}(x) &= \mu_1 A_1 \int_0^\infty h(xu^2) u^{\mu_1} \frac{du}{u}
+O(x^{-\frac{\nu_1}{2} -\varepsilon}), \\
\widetilde{S}_{7}(x) &= \mu_2 A_2 \int_0^\infty h(xu^2) u^{\mu_2} \frac{du}{u}
+O(x^{-\frac{\nu_2}{2} -\varepsilon}) \\
\widetilde{S}_{8}(x) &= 3 \mu_3 A_3 \int_0^\infty h(xu^2) u^{\mu_3} \frac{du}{u}
+O(x^{-\frac{\nu_3}{2} -\varepsilon})
\endaligned
\end{equation}
hold, if the estimates $\eqref{thm4:1}$, $\eqref{thm4:2}$ and $\eqref{thm4:3}$
hold respectively.
We omit the process of calculations  for $ \widetilde{S}_{6}(x)$, $\widetilde{S}_{7}(x)$ and $\widetilde{S}_{6}(x)$,
because thay are calculated in almost the same way as that for $\widetilde{S}_{2}(x)$ or $\widetilde{S}_{4}(x)$.
For $\widetilde{S}_{5}(x)$ we have
\begin{equation}\label{724}
\widetilde{S}_{5}(x)=\sum_{n=1}^\infty \Lambda(n)\phi(n)\psi(n)h(xn)+O(x^{-\frac{1}{3}-\varepsilon}).
\end{equation}
By applying Proposition 1 to $\zeta(s)$ and $u \mapsto h(xn)\phi(n)\psi(n)$ we have
\begin{equation}\label{725}
\aligned
\sum_{n = 1}^\infty & \Lambda(n)\phi(n)\psi(n)h(xn)  \\
&=\int_{0}^{\infty} h(xu)\phi(u)\psi(u)du
-\sum_{\zeta^\ast(\rho)=0}
\int_0^\infty h(xu) \phi(u) \psi(u) u^{\rho} \frac{du}{u}
+O(1)
\endaligned
\end{equation}
for sufficiently small $x>0$.
Combining $\eqref{709}$, $\eqref{723}$ and $\eqref{725}$ we obtain the latter half of Theorem 4.
Also we obtain the latter half of Theorem 3 by the same equations, if we replace $c_{\psi,2}(p)$ by $0$.
\hfill $\Box$

\section{Additional Topics}

\subsection{Explicit equations}
Our theorems in $\S 3$ are asymptotic results.
We can also obtain a result which is an explicit version of our theorems in $\S3$,
if we use an interpolation function of
\begin{equation}\label{801}
\omega_\phi(n):=
\begin{cases}
\sum_{i=1}^{n_p} \alpha_{\phi}(p,i)^m & \text{if $n=p^m$ with $m \geq 1$,} \\
0 & \text{otherwise}.
\end{cases}
\end{equation}
However there is a possibility that such interpolation functions are not so useful for applications.
At least it seems that
a well-chosen interpolation function of Dirichlet coefficients is more useful than
an interpolation function of $\eqref{801}$ for some specific purposes.
This is one reason why we adopt asymptotic formulas as main results.

Anyway we will establish our explicit identities.
The key of the following results are the equations
\begin{align}\label{802}
\Lambda_\phi(n) &= \Lambda(n)\omega_{\phi}(n), \\
\Lambda_{\phi \otimes \psi}(n) &=\Lambda_{\phi}(n)\omega_\psi(n) = \Lambda(n) \omega_\phi(n) \omega_\psi(n).
\end{align}
\begin{thm}
Let $\phi \in C_L(M)$ and
let $\Omega_{\phi}(u)$ be an interpolation function of $\omega_{\phi}(n)$,
that is, $\Omega_{\phi}(n)=\omega_{\phi}(n)$ for any non-negative integer $n$.
Then we have the following explicit identitiy
\begin{align}
\aligned
 m_\phi  \, \widehat{h}(0) & -\sum_{L_\phi^\ast(\rho)=0}\widehat{h}(\rho) + m_\phi \, \widehat{h}(1) \\
 & -\sum_{n=1}^\infty \overline{\Lambda_\phi(n)} n^{-1} h(n^{-1})
  -( 2 \log Q_\phi+d_\phi \,C_E)\,h(1)-\sum_{j=1}^{r_\phi} W_{\lambda_j(\phi),\mu_j(\phi)}(h), \\
=\, \widehat{ h_\Omega }(0) & -\sum_{\zeta^\ast(\rho)=0}\widehat{ h_\Omega }(\rho) +   \widehat{ h_\Omega }(1) \\
 & -\sum_{n=1}^\infty \Lambda(n) n^{-1} h_\Omega (n^{-1})
   -(\log \pi+\,C_E)\,h_\Omega (1) - W_{\frac{1}{2},0}( h_\Omega ),
\endaligned
\end{align}
where $h_\Omega(u):=h(u)\Omega_{\phi}(u)$ and $W_{\lambda,\mu}(\cdot)$ is the functional defined in $\eqref{505}$.
\end{thm}
\begin{thm}
Let $(\phi,\psi) \in C_L(M) \times C_L(N)$ be a Rankin-Selberg pair.
Then we have the following explicit identities
\begin{align}
\aligned
 m_{\phi \otimes \psi} \, \widehat{h}(0)
& -\sum_{L_{\phi \otimes \psi}^\ast(\rho)=0}
\widehat{h}(\rho) + m_{\phi \otimes \psi} \, \widehat{h}(1) \\
 & -\sum_{n=1}^\infty \overline{\Lambda_{\phi \otimes \psi}(n)} n^{-1} h(n^{-1})
  -( 2 \log Q_{\phi \otimes \psi}+d_{\phi \otimes \psi} \,C_E)\,h(1)
  -\sum_{j=1}^{r_{\phi \otimes \psi}} W_{\lambda_j(\phi \otimes \psi),\mu_j( \phi \otimes \psi)}(h), \\
=\, m_\phi  \, \widehat{h_{\Omega_\psi}}(0) & -\sum_{L_\phi^\ast(\rho)=0}\widehat{h_{\Omega_\psi}}(\rho)
+ m_\phi \, \widehat{h_{\Omega_\psi}}(1) \\
 & -\sum_{n=1}^\infty \overline{\Lambda_\phi(n)} n^{-1} h_{\Omega_\psi} (n^{-1})
  -( 2 \log Q_\phi+d_\phi \,C_E)\,h_{\Omega_\psi}(1)-\sum_{j=1}^{r_\phi} W_{\lambda_j(\phi),\mu_j(\phi)}(h_{\Omega_\psi} ), \\
=\, \widehat{ h_{\Omega_\phi \Omega_\psi} }(0) & -\sum_{\zeta^\ast(\rho)=0}\widehat{ h_{\Omega_\phi \Omega_\psi} }(\rho)
 +   \widehat{ h_{\Omega_\phi \Omega_\psi} }(1) \\
 & -\sum_{n=1}^\infty \Lambda(n) n^{-1} h_{\Omega_\phi \Omega_\psi} (n^{-1})
   -(\log \pi+\,C_E)\,h_{\Omega_\phi \Omega_\psi}(1) - W_{\frac{1}{2},0}( h_{\Omega_\phi \Omega_\psi} ),
\endaligned
\end{align}
where $h_{\Omega_\phi}(u):=h(u)\Omega_\phi(u)$ and $h_{\Omega_\phi \Omega_\psi} (u):=h(u)\Omega_\phi(u)\Omega_\psi(u)$.
\end{thm}
Theorem 7 is obtained by calculating the sum $\sum_{n=1}^\infty  \Lambda_\phi(n) h(n)$ in two ways.
Theorem 8 is obtained by calculating the sum $\sum_{n=1}^\infty  \Lambda_{\phi \otimes \psi}(n) h(n)$ in three ways.
These processes are very similar to the proofs of Theorem 1 $\sim$ Theorem 4,
therefore we omit the details of their proofs.
We deal with one way to construct $\Omega(u)$ in the next section.

\subsection{ One way of the construction of an interpolation function.}
In this part we give a way to construct an interpolation function by using Fourier series.
Let $a:{\Bbb N} \to {\Bbb C}$ be a function on natural numbers.
When $a(\cdot)$ has polynomial order,
we define the function $f_a$ by
\begin{equation}
f_{a}(z):=\sum_{n=1}^\infty a(n) e^{2\pi i n z }.
\end{equation}
Since $a(\cdot)$ has polynomial order,
$f_a(z)$ converges absolutely on the upper half-plane ${\rm Im}(z)>0$.
By using $f_a$, we define $A(u)$ by
\begin{equation}
A(u):=A(u;y,\eta):= e^{2\pi uy} \int_\eta^{\eta+1} f_a(x+iy) e^{-2\pi i u x}dx,
\end{equation}
for some fixed $y>0$ and $\eta \in {\Bbb R}$, or
\begin{equation}
A(u):=A(u;\eta):= e^{2\pi} \int_\eta^{\eta+1} f_a(x+iu^{-1}) e^{-2\pi i u x}dx,
\end{equation}
for some fixed $\eta \in {\Bbb R}$.
From the definition, $A(u)$ satisfies $A(n)=a(n)$.

For $\phi \in C_L(n)$, we can construct the interpolation function $\Phi(u)$ of $\phi(n)$,
since $\phi(n) \ll_{\varepsilon} n^{\varepsilon}$.
Of course $\Phi \not\equiv \phi$ as a function on $(0,\infty)$ in general.
Similarly we can construct the interpolation function $\Omega_{\phi}(u)$ of $\omega_\phi(n)$,
since $\omega(n) \ll_\varepsilon n^{\theta+\varepsilon} $ for any fixed $\varepsilon >0$ from Lemma 4.

\subsection{Symmetries of zero-sums.}
In $\S 8.2$, we gave one way to construct an interpolation function.
However, there is no reason that the interpolation in $\S 8.2$ is a canonical one.
Actually, there are infinity many possibilities of interpolation functions,
when we restrict them to the class of smooth functions.
However the non-existence of canonical interpolation is not unfortunate.
The existence of several different interpolation functions gives a symmetry of zero-sums.
Let $\phi_\chi(\cdot)$, $\psi_\chi(\cdot)$ be two different interpolation functions
of a primitive Dirichlet character $\chi$ mod $q$,
let $\phi_f(\cdot)$, $\psi_f(\cdot)$ be two different interpolation functions
of Fourier coefficients of $ f \in S_k(1)$
and let $\phi_g(\cdot)$, $\psi_g(\cdot)$ be two different interpolation functions
of Fourier coefficients of $ g \in S_k(1)$.
They give several ``symmetries'' of the sums $\sum_{\zeta^\ast(\rho)=0}$, $\sum_{L^\ast(\rho,\cdot)=0}$ etc.
For example, for a suitable test function $h \in C_0^\infty$, we have
\begin{equation*}
\aligned
\sum_{\zeta^\ast(\rho)=0}
\int_0^\infty h(xu)\phi_\chi(u) u^\rho \frac{du}{u}
\sim
\sum_{\zeta^\ast(\rho)=0}
\int_0^\infty h(xu)\psi_\chi(u) u^\rho \frac{du}{u},
\endaligned
\end{equation*}
\begin{equation*}
\aligned
\sum_{\zeta^\ast(\rho)=0}
\int_0^\infty h(xu) \phi_f(u)u^\rho \frac{du}{u}
\sim
\sum_{\zeta^\ast(\rho)=0}
\int_0^\infty h(xu) \psi_f(u)u^\rho \frac{du}{u},
\endaligned
\end{equation*}
and
\begin{equation*}
\aligned
\sum_{\zeta^\ast(\rho)=0} &
 \int_0^\infty h(xu)\phi_f(u)\phi_g(u)u^\rho \frac{du}{u}
 \sim \sum_{\zeta^\ast(\rho)=0} \int_0^\infty h(xu) \phi_f(u) \psi_g(u) u^\rho \frac{du}{u} \\
& \sim \sum_{\zeta^\ast(\rho)=0} \int_0^\infty h(xu) \psi_f(u) \phi_g(u) u^\rho \frac{du}{u}
 \sim \sum_{\zeta^\ast(\rho)=0} \int_0^\infty h(xu) \psi_f(u) \psi_g(u) u^\rho \frac{du}{u},
\endaligned
\end{equation*}
for sufficiently small  $x > 0$.
Furthermore we find that
\begin{equation*}
\aligned
\sum_{\zeta^\ast(2\rho)=0} \int_0^\infty h(xu) u^\rho \frac{du}{u}
\sim
 \sum_{\zeta^\ast(\rho)=0} \int_0^\infty h(xu) \phi_\theta(u) u^\rho \frac{du}{u}
\endaligned
\end{equation*}
where $\phi_\theta$ is an interpolation function of the coefficients of
$\theta(z)=\sum_{n=1}^\infty e^{i n^2 z}$, and
\begin{equation*}
\aligned
\sum_{L^\ast(\rho,f)=0}
& \int_0^\infty h(xu) \phi_g(u) u^\rho \frac{du}{u}  \sim \sum_{L^\ast(\rho,f)=0}
 \int_0^\infty h(xu) \psi_g(u) u^\rho \frac{du}{u} \\
& \sim \sum_{L^\ast(\rho,g)=0}
 \int_0^\infty h(xu) \phi_f(u) u^\rho \frac{du}{u} \sim \sum_{L^\ast(\rho,g)=0}
 \int_0^\infty h(xu) \psi_f(u) u^\rho \frac{du}{u}.
\endaligned
\end{equation*}
These are consequences of Theorem 1 $\sim$ Theorem 4.

\subsection{An interpretation in terms of distributions.}

In this part we give an interpretation of our results in a local-global view point.
Throughout this part, we denote by ${\cal Z}(F)$ the set of all zeros of $F$ .

Let $\phi \in C_L(N)$ and let $L_{\phi,p}(s)^{-1}$ be the reciprocal $p$-th Euler factor of $L_{\phi}(s)$.
Denote by $\zeta_p(s)^{-1}$ the reciprocal $p$-th Euler factor of $\zeta(s)$.
As explained in $\S 1$, in the local point of view,
the relation between ${\cal Z}({L_{\phi,p}^{-1}})$ and ${\cal Z}(\zeta_p^{-1})$
is very simple.
For $p \not\in S_{\phi}$, the reciprocal $p$-th Euler factor $L_{\phi,p}(s)^{-1}$ is expressed as
\begin{equation}
L_{\phi,p}(s)^{-1} =\prod_{i=1}^N (1-\alpha_{\phi}(p,i)p^{-s})
\end{equation}
Hence, if we assume the general Ramanujan conjecture, $\alpha_{\phi}(p,i)$ can be written as
\begin{equation}
\alpha_{\phi}(p,i)=e^{i \theta_{\phi}(p,i)}
\end{equation}
for some $ \theta_{\phi}(p,i) \in {\Bbb R}$.
This shows that ${\cal Z}({L_{\phi,p}^{-1}})$ is
the union of $n$-piece translation of ${\cal Z}(\zeta_p^{-1})$.
Therefore we find that
\begin{equation}\label{811}
\sum_{ L_{\phi,p}(\rho)^{-1}=0}  \widehat{h}(\rho)
=\sum_{i=1}^N \sum_{\zeta_p(\rho)^{-1}=0}
\widehat{h}\left( \rho + \sqrt{-1} \,\frac{ \theta_{\phi}(p,i)}{\log p} \right)
\end{equation}
for any $h \in C_0^\infty$.
Similarly we obtain
\begin{equation}\label{812}
\sum_{ L_{\phi,\infty}(\rho)^{-1}=0}  \widehat{h}(\rho)
=\sum_{j=1}^{r_{\phi}} \sum_{\zeta_\infty(\rho)=0}
\widehat{h}\left( \frac{\rho + \mu_j(\phi)}{\lambda_j(\phi)} \right)
\end{equation}
where $L_{\phi,\infty}(s)^{-1}$ is the reciprocal $\Gamma$-factor of $L_\phi(s)$.

Next we show a relation between the global zeros and the local zeros.
From Proposition 1, we obtain
\begin{equation}\label{813}
m_\phi  \, \widehat{h}(0) -\sum_{L_{\phi}^\ast(\rho)=0}\widehat{h}(\rho) + m_\phi \, \widehat{h}(1)
=\sum_{p} W_\phi(h,p) + W_{\phi}(h,\infty),
\end{equation}
where
\begin{equation}\label{814}
W_\phi(h,p)=\sum_{m=1}^\infty (\Lambda_\phi(p^m) h(p^m) +\overline{\Lambda_\phi(p^m)} p^{-m}h(p^{-m}))
\end{equation}
and $W_{\phi}(h,\infty)$ is given by $W_{\lambda, \mu }(\cdot)$.
Moreover, Poisson's summation formula yields
\begin{equation}\label{815}
W_\phi(h,p)=\sum_{L_{\phi,p}(\rho)^{-1}=0} ( \widehat{h}(\rho) + \widehat{h}(1-\overline{\rho})).
\end{equation}

Together with $\eqref{811}$, $\eqref{812}$, $\eqref{813}$ and $\eqref{815}$ we obtain
\begin{align}\aligned
\widehat{h}(1) & -\sum_{L_{\phi}(\rho)=0} \widehat{h}(\rho) +\widehat{h}(0) \\
=& \sum_{p} \sum_{ L_{\phi,p}(\rho)^{-1}=0 } \left[ \widehat{h}( \rho )
 + \widehat{h} ( 1 -\rho )\right]
 + \sum_{L_{\phi,\infty}(\rho)^{-1}=0} \widehat{h}( \rho ) \\
=& \sum_{p \not\in S_{\phi} }\sum_{i=1}^N \sum_{\zeta_p(\rho)^{-1}=0}
\left[ \widehat{h}\left( \rho + \sqrt{-1} \,\frac{ \theta_{\phi}(p,i) }{\log p} \right)
 + \widehat{h}\left( 1-\rho - \sqrt{-1} \,\frac{ \theta_{\phi}(p,i) }{\log p} \right) \right] \\
&+ \sum_{p \in S_{\phi}} \sum_{ L_{\phi,p}(\rho)^{-1}=0 } \left[ \widehat{h}( \rho ) + \widehat{h} ( 1 -\rho )\right]
+ \sum_{j=1}^{r_{\phi}} \sum_{\zeta_\infty(\rho)^{-1}=0}
\widehat{h}\left( \frac{\rho + \mu_j(\phi)}{\lambda_j(\phi)} \right).
\endaligned \end{align}
This implies that there exists some relation between ${\cal Z}(L_{\phi})$ and ${\cal Z}(\zeta)$.

Under the Riemann hypothesis for $L_\phi(s)$ and $\zeta(s)$,
we can show that
\begin{equation}\label{dis:1}
\sum_{L_{\phi}^\ast (\rho)=0} \frac{X^\rho}{\rho} \sim
\sum_{\zeta^\ast(\rho)=0} \int_0^X \phi(u)u^{\rho}\frac{du}{u}
\quad (X \to +\infty),
\end{equation}
if $\int_0^X \phi(u) du =o(\sqrt{X})$, by a way similar to the proof of Theorem 2.
By acting the differential operator $X\frac{d}{dX}$ on both side of $\eqref{dis:1}$,
we obtain
\begin{equation}\label{dis:2}
\sum_{ L_{\phi}^{\ast}(\rho)=0} X^\rho \quad
\sim
\sum_{\zeta^\ast (\rho)=0 } \phi(X) X^\rho
\quad (X \to +\infty).
\end{equation}
Unfortunately, two series in $\eqref{dis:2}$ do not converge for any $X$,
hence $\eqref{dis:2}$ have no strict meaning as a function in $X$.
However, if we define the distributions as
$$
\int_0^\infty h(u)\sum_\rho u^{\rho}\frac{du}{u}=\sum_\rho \widehat{h}(\rho)
$$
for a test function $h \in C_0^\infty$,
the relation $\eqref{dis:2}$ can be interpreted as a relation of two distributions.
That is, at least in the level of distribution,
the relation $\eqref{dis:2}$ shows that
${\mathcal Z}(L_{\phi})$ is the ``translation'' of
${\mathcal Z}(\zeta)$ by $\phi(\cdot)$.
The local relations $\eqref{811}$, $\eqref{812}$ are extended to
the global relation $\eqref{dis:2}$ via the interpolation function $\phi(\cdot)$.
We may say that
the original RH for $\zeta(s)$ implies the RH of $L_\phi(s)$
in the level of distribution.
This suggests that if the original RH is false, the RH for automorphic $L$-functions is also false.
To show this rigorously, we need to establish the Sprindzuk type theorem as in~\cite{Sp75}.
Althogh it is one of the most important applications of our results,
we postpone such a study to a forthcoming work.


\section*{Finite-Size Scaling  and Long-Range  Interactions}

\vskip 5mm N.S. Tonchev

 \vskip 5mm

 {{\it Inst. of Solid State Physics, Bulg. Acad. of Sci.,
Tzarigradsko shauss$\acute{e}$e 72, 1784 Sofia, Bulgaria.}
\\

}
\end{center}
\vskip 5mm

\begin{abstract}
The present review is devoted to the problems of finite-size
scaling due to the presence of long-range interaction decaying at
large distance as $1/r^{d+\sigma}$, where $d$ is the spatial
dimension and the long-range parameter $\sigma>0$. Classical and
quantum systems are considered.
\end{abstract}
\vskip 8mm

\subsection*{1. Finite-size systems and critical phenomena}

  A common wisdom is that the singularities
in the thermodynamic functions at a critical point may occur only
in the  thermodynamic limit. If the system is fully finite (as any
real system is), no such singularities exist and, strictly
speaking, no phase transitions occur. These facts naturally give
rise to the following questions:

a) Why the bulk theory turns out to be adequate to the
experimental evidence for finite  objects?

b) How the singularities appear  when no phase transitions occur
in any fully finite system?

The answer to the first question lies in the fact that a
macroscopic body is close, under some circumstances, to the
idealized thermodynamic limit.

The answer to the second question becomes evident if one recalls
that the limit function of a sequence of analytic functions needs
not to be analytic. Therefore, what happens is that in the
thermodynamic limit some thermodynamic functions of the system
develop singularities which are attributes of phase transitions.
So, in a finite system one expects an appreciable rounding of the
critical point singularities.

 The bulk correlation length $\xi_{\infty}(T)$
measures the distance over which particles in a system are
significantly correlated. Then, as the temperature $T$ approaches
the critical temperature $T_c$, $\xi_{\infty}(T)$ diverges. When
$\xi_{\infty}(T)$ attains a magnitude of the order of the
characteristic size $L$ of the finite system, then the boundary
particles at the opposite sides of the system become well
correlated, and ordering cannot continue to build up further in
the restricted dimensions. It is certainly reasonable to expect
that the rounding of the phase transition is controlled by the
criterion: $\xi_{\infty} (T)\sim L.$

The description of this rounding and crossover was first
formulated by M. Fisher (1972) and is called Finite-Size Scaling
(FSS)( see e.g.\cite{brankov00}).

An important application of FSS is to analyze numerical data
obtained from simulations on relatively small finite systems, and,
thereby to obtain knowledge of the bulk systems.

Finite-size effects are much  stronger in systems of particles
with long-range (LR) interactions because every particle directly
feels the influence of the boundaries.

 To study finite systems, one must at the start address two basic
issues, namely the overall geometry of the system and the specific
nature of the boundary conditions.

\textsc{Geometry}: We will use the general notation
 $L^{d-d'}\times\infty^{d'}$ and distinguish
three important cases ($d$,$d'$ are spatial dimensions): $ d'=0$
-- a fully finite system,
 $d'=1$ --
  a system with the geometry of a cylinder,
and $d'=(d-1)$ --
 a system with slab (or film) geometry.

\textsc{Boundary conditions}: Here and bellow we will consider
periodic boundary conditions (p.b.c.). Close to the bulk critical
point
 for a $d$-dimensional system with a finite
size $L$ one has for the free energy density
\begin{equation}
\label{perbc}
 f_{L}^{(p)}(T) = f(T)+ O\left({\mathrm e}^{-L/\xi (T)}\right).
\end{equation}
where $f(T)\equiv f_{\infty}(T)$ is the bulk free energy density.
In this case there are no surfaces, edges and corners.

The experimental observability of FSS effects is hampered by the
fact that nature does not provide us systems with p.b.c. The rapid
exponential approach to the thermodynamic limit is one of the
reasons why p.b.c. are preferred in Monte Carlo simulations. One
observes that the l.h.s. in the above expansions  is a regular
function of $T$  whereas the r.h.s. is not: the bulk density
$f(T)$, for example, is singular at the critical point $T_c$.
Therefore, Eq.(\ref{perbc}) can hold only {\it away} from $T_c$.
Obviously, for $T\simeq T_c$ one needs an alternative,
 (FSS) formulation.

\subsection*{2. Finite-size scaling hypothesis} Scaling hypotheses are
made in the form of  statements about properties of
 thermodynamic quantities in terms of homogeneous functions.

 To simplify the further
discussion we consider that: the studied system is below its upper
critical dimension $d_{u}$, the infinite  system has a second
order phase transition
 at a critical temperature $T_{c}$, and the  system has length $L$
in all finite directions. The violation of these restrictions will
cause complications and
 modifications of the standard FSS hypothesis \cite{brankov00}.

 For a
 physical quantity $\textit{P}$ with singular behavior in the
thermodynamic limit the content of the standard FSS hypothesis is
to assume the existence of a scaling function $\textit{P}_{L}(T)$
such that:
\begin{equation}\label{FSS1}
\textit{P}_{L}(T)\simeq\textit{P}_{\infty}(T)\textit{F}_{L}(L/\xi(T)).
\end{equation}
As far as $L$ is finite, $\textit{P}_{L}(T)$ must be a regular
function of $T$. The (universal) function $\textit{F}_{L}(x)$ must
compensate the singularity of $\textit{P}_{\infty}(x)$.
Eq.(\ref{FSS1}) may be written in equivalent form:

\begin{equation}\label{FSS2}
 P_{L}(t)\simeq L^{p_x/\nu}{\cal F}_{L}(tL^{1/\nu}),
\end{equation}
where  $p_x$ - the critical exponent of the observable $ P_{L}$,
$\nu$  - the critical exponent of the correlation length
 and $t=(T-T_{c})/T_{c}$.
The hypothesis is expected to be valid when $L$ is large and $T$
approaches $T_{c}$.

 What is the common status of the above hypothesis? One can usually meet
 the following statements,
e.g. J. Cardy, in \cite{cardy96}:
 "FSS  is by now well-established theoretically, at least for
systems with short-range interactions ...", or  quite recently
 Chen
and Dohm in \cite{CD02} :
 "The finite-size scaling hypothesis asserts ... in \textit{the absence} of
LR  interaction ..."

We would like to point out the restrictions  related with the
nature of the interaction - one insists that  it must be
short-range (SR). In this situation the problem is to find out:
 does the FSS take place if the interaction is of the
LR type? As we will explain bellow there is a positive answer to
this question.

We will consider interactions which decay algebraically at large
distances with the standard notation

\begin{equation}\label{01}
u(r)\sim-1/r^{d+\sigma},\qquad\sigma>0.
\end{equation}

For $\sigma\leq0$ the interactions are \textit{nonintegrable}, and
so, under standard definitions, the thermodynamic limit does not
exist. The case of \textit{nonintegrable} interaction is beyond
the scope of the present study.

The LR interaction (\ref{01}) enters the expressions of the theory
only through its Fourier transform \cite{brankov00}. The
corresponding small - ${\mathbf q}$ expansion of the Fourier
transform has the general form
\begin{equation}\label{1}
v({\mathbf q}) = v_{0} + v_{2}{\mathbf q}^{2} +v_{\sigma}
{|\mathbf q|}^{\sigma}+w({\mathbf q}) \qquad  0< \sigma \neq 2,
\end{equation}
with $w({\mathbf q})/{|\mathbf q|}^\sigma\to 0$, for ${\mathbf
q}\to 0$,  i.e. in the long-wavelength approximation. Further on
we will formally relate $\sigma=2$, to the SR  interaction since
then~(\ref{1}) is the Fourier transform of an interaction decaying
exponentially with distance.

Depending on whether in~(\ref{1}) $\sigma $ is less or bigger than
2 we will speak about {\it leading} or {\it subleading} LR term
respectively.

\textsc{Nota bene}: The \textit{subleading} LR term does not
affect the bulk critical behavior, but influences the finite-size
one (for details see \cite{dantchev01,chamati2001,D01}).

 The defined above LR interaction has
to be a mimicry of the real one. Nevertheless, from the pure
theoretical point of view the considered LR interaction seems to
be of specific interest. The reason is that renormalization group
 (RG) predictions obtained on the basis of the
$\epsilon$-expansion, can be verified on ideal testing ground. The
upper and lower critical dimensions are $2\sigma$ and $\sigma$
respectively. Since $\sigma$ is a continuous parameter
 the value of $\epsilon=2\sigma-d$ or $\epsilon=d-\sigma$
would be small enough for integer values of the dimensionality.
This  places us in the rare situation when the outcome of the
computer simulations, obtained by means of the Monte Carlo method,
can be directly compared to the
 predictions of the $\epsilon$-expansion with $\epsilon$ arbitrarily small.

 But along this line of consideration there are some problems.

 Firstly, although the general scheme  of the $\epsilon$-expansion has been widely
 accepted, the part of it related to LR systems has not been well understood
until now (see \cite{CT03} and references therein).

Secondly, the study of LR systems leads to prohibitively large
computational requirements. Still recently a novel Monte Carlo
algorithm which has an efficiency that is independent of the
number of interactions per  spin has been announced
\cite{LB95,LB97}.

Thirdly, Monte Carlo analyzes of the RG predictions typically
apply FSS concepts. However, specific problems arise in the proper
generalization of the FSS concepts in the case under consideration
\cite{BD91,BT92}.

\subsection*{3. Problems with the correct definitions of the
correlation length}

By definition the correlation length is
\begin{equation}
\xi_{1}(T)=-\lim_{R\rightarrow\infty}[R/\ln
G_{\infty}(\mathbf{R};T)].
\end{equation}
Alternatively, one may consider the second moment of the bulk pair
correlation function and define the effective correlation radius:
\begin{equation}
\xi_{2}(T)=\left[\sum_{\mathbf{R}}R^{2}G_{\infty}(\mathbf{R};T)/G_{\infty}(\mathbf{R};T)
\right]^{1/2}.
\end{equation}

These two most commonly used definitions of the bulk correlation
length are unambiguous and equivalent (up to a constant) in the
case of exponential with the distance $R=|\mathbf{R}|$ decay of
the bulk pair correlation function $G_{\infty}(\mathbf{R};T)$. A
distinctive feature of the LR interaction is that the function
$G_{\infty}(\mathbf{R};T)$ decays as $R^{-d+\sigma}$ when
$R\rightarrow\infty$ for $T>T_{c}$,  and both definitions yield
$\xi_{1}(T)=\xi_{2}(T)=\infty $ if $\sigma<2$.

Anyhow complications with correlations that decay as power laws in
a whole domain of thermodynamic parameters, rather than only at
special points, arise even in the bulk case, provoking the idea of
"generic scale invariance"
 in the theory of phase transitions (see, e.g.
\cite{BKV04}). The FSS under consideration encounters the same
phenomena mixed with specific boundary effects. One can overcome
the difficulty with the absence of good definition of the
correlation length (when $\sigma<2$) following the ideas proposed
in \cite{BD91,BT92}, where instead of $\xi(T)$ a bulk
\textit{characteristic length} $\lambda(T)$ is used. This
characteristic length determines the length-scale of variation of
the correlation function and diverges at the critical point.

A finite system has three characteristic length scales: a linear
size $L$, a length $\lambda_{L}(T)$ which determines the scale of
variation of the correlations, and a microscopic length $a$ (e.g.,
$a$ is the lattice spacing). The approach proposed in \cite {BT92}
 involves the following steps:

 \textsc{First step}: It consists of three assumptions.

The first one defines the finite-size characteristic length
$\lambda_{L}(T)$ from the large-distance asymptotic behavior of
the finite-size pair correlation function $G_{L} ({\bf R},T)$.

The second assumption is the homogeneity of any thermodynamic
quantity of the finite-size system, such that its bulk limit is singular at
$T = T_{c}$, as a function of the two dimensionless ratios $\lambda_{L}(T)/a$
and $L/a$.

The third assumption is the existence of finite thermodynamic limits
for the characteristic length and the pair correlation
function.

\textsc{Second step}: It concerns the relationship between the
finite-size length\index{characteristic length!finite-size|(}
$\lambda_{L}(T)$ and its bulk limit\index{characteristic
length!bulk} $\lambda_{\infty}(T)$; the corresponding homogeneity
assumption  involves the dimensionless ratios
$\lambda_{\infty}(T)/a$ and $L/a$.

\subsection*{4. Mathematical problems}\label{app}

\textsc{Classical case}:  In the case of SR interaction the
following replacement is used as an indispensable part of the FSS
calculations  (see, e.g. \cite{brankov00}).

\begin{equation}\label{Scr}
\sum_{\mathbf{q}}\frac{1}{m^{2}+|\mathbf{q}|^{2}}=\int_{0}^{\infty}dt\exp(-m^{2}t)
\left[\sum_{q}\exp(-q^{2}t)\right]^{d},
\end{equation}
where $\mathbf{q}$ and $q$ are d-dimensional and one-dimensional
discrete vectors, respectively. This is the so called Schwinger
parametric representation. The aim of this replacement is
two-fold: to reduce the $d$-dimensional sum to  the corresponding
effective one-dimensional one, and  to give  the dimensionality
$d$ the status of a continuous variable.

In the case of leading  $q^{\sigma}$ term  in (\ref{1}), one
cannot  use the Schwinger representation in its familiar form.
Then the following generalization of (\ref{Scr}) has been
suggested \cite{BT88}:
\begin{equation}\label{BTr}
\sum_{\mathbf{q}}\frac{1}{m^{2}+|\mathbf{q}|^{\sigma}}=
m^{\frac{4-2\sigma}{\sigma}} \int_{0}^{\infty}dt
Q_{\sigma}(m^{4/\sigma}t)\left[\sum_{q}\exp(-q^{2}t)\right]^{d},
\end{equation}
where the function $Q_{\sigma}(t)$ for $0<\sigma<2$ is given by
\begin{equation}\label{F}
Q_{\sigma}(t)= \int_{0}^{\infty}dy\exp(-ty){\tilde
Q}_{\sigma}(y),\qquad {\tilde
Q}_{\sigma}(y)=\frac{1}{\pi}\frac{\sin(\sigma\pi/2)y^{\sigma/2}}
{1+2y^{\sigma/2}\cos(\sigma\pi/2)+y^{\sigma}}.
\end{equation}

>From (\ref{BTr}) and (\ref{F}) one
obtains
\begin{equation}\label{int1}
\sum_{\mathbf{q}}\frac{1}{m^{2}+|\mathbf{q}|^{\sigma}}= \int_{0}^{\infty}dt{\tilde
Q}_{\sigma}\left(\frac{t}{m^{\frac{4-2\sigma}{\sigma}}}\right)
\sum_{\mathbf{q}}\frac{1}{tm^{2}+|\mathbf{q}|^{2}}.
\end{equation}
The identity (\ref{int1}) demonstrates the possibility to  reduce
the LR case  to the SR case with an integration over an additional
parameter. So, all the mathematical  machinery developed for the
SR case, in principle, may be used. Let us note however that such
nonlocal expressions are unsuitable for explicit calculations. It
is more convenient to use the relation \cite{B89}:

\begin{equation}\label{MLFa}
Q_{\sigma}(x)=x^{\sigma/2 -
1}E_{\sigma/2,\sigma/2}(-x^{\sigma/2}),
\end{equation}
where $E_{\alpha,\beta}(z)$
 is an entire
function of the Mittag-Leffler type defined by the power series
\begin{equation}\label{defMLa}
E_{\alpha,\beta}(z)=\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma(\alpha
k+\beta)} \qquad \alpha>0.
\end{equation}
 and to study the finite-size properties of the system resulting from the analytical properties
 of $E_{\alpha,\beta}$.

\textsc{Pure quantum case}: At $T=0$, the quantum counterpart of
(\ref{BTr}) is

\begin{equation}\label{qScr1}
\sum_{\mathbf{q}}\frac{1}{\left[m^{2}+|\mathbf{q}|^{\sigma}\right]^{1/2}}=
\frac{2}{\pi}\int_{0}^{\infty}dp\sum_{\mathbf{q}}\frac{1}{m^{2}+p^{2}+|\mathbf{q}|^{\sigma}},
\end{equation}

 Equation (\ref{qScr1}) displays the well known
property of the pure quantum case. The auxiliary variable $p^{2}$
acts effectively as an extra dimension. Indeed the pure quantum
system corresponds to a $d+1$ dimensional classical system with
the geometry of a cylinder $L^{d}\times\infty^{1}$. Introducing
$m^{2}(p):=m^{2}+p^{2}$ with the help of Eq.(\ref{int1}) from
Eq.(\ref{qScr1}) one gets

\begin{equation}\label{qScr11}
\sum_{\mathbf{q}}\frac{1}{\left[m^{2}+|\mathbf{q}|^{\sigma}\right]^{1/2}}=
\frac{2}{\pi}\int_{0}^{\infty}dt\int_{0}^{\infty}dp{\tilde
Q}_{\sigma}\left(\frac{t}{m(p)^{\frac{4-2\sigma}{\sigma}}}\right)
\sum_{\mathbf{q}}\frac{1}{tm(p)^{2}+|\mathbf{q}|^{2}}.
\end{equation}
The integral representation (\ref{qScr11}) illustrates  the
mathematical difficulties  appearing in the pure quantum case. It
is shown that by two additional integrations the problem can be
effectively reduced to the classical SR case.

On the other hand, the following modification of (\ref{BTr}) has
been prosed \cite{C94}
\begin{equation}\label{BTrH}
\sum_{\mathbf{q}}\frac{1}{\left[m^{2}+|\mathbf{q}|^{\sigma}\right]^{1/2}}=
m^{\frac{4-\sigma}{\sigma}} \int_{0}^{\infty}dt
C_{\sigma}(m^{4/\sigma}t)\left[\sum_{q}\exp(-q^{2}t)\right]^{d},
\end{equation}
where
\begin{equation}\label{MLF}
C_{\sigma}(x)=x^{\sigma/4 -1}G_{\sigma/2,\sigma/4}(-x^{\sigma/2}),
\end{equation}
i.e. the function $C_{\sigma}(x)$ is related  to the entire
function defined by the power series
\begin{equation}\label{defML}
G_{\alpha,\beta}(z)=\frac{1}{\pi^{1/2}}\sum_{k=0}^{\infty}\frac{\Gamma(
k+1/2)z^{k}}{\Gamma(\alpha
k+\beta)k!} \qquad \alpha>0,\beta>0.
\end{equation}

Some analytical properties of $G_{\sigma/2,\sigma/4}(z)$ (see also
\cite{CT2000})  may be established with the help of the identity
\begin{equation}
G_{\sigma/2,\sigma/4}(z)=\frac{2}{\pi}\int_{0}^{\infty}E_{\sigma/2,\sigma/2}
\left(-(z+p^{2})\right)dp,
\end{equation}
obtained  from Eq.(\ref{qScr1}) and the relation of its l.h.s and
r.h.s. with the functions $G_{\sigma/2,\sigma/4}(z)$ and
$E_{\sigma/2,\sigma/2}(z)$, respectively.
 \subsection*{5. Theoretical models}

Here we will mention the models and the related works in which
leading LR interactions are considered in the context of FSS.

The mean spherical model  was considered in
\cite{BD91,BT92,BT88,B89}, \cite{FP86}-\cite{CD04}. The
Hamiltonian of the model is
\begin{equation}\label{model1}
{\cal H}=-\frac12\sum_{ij}{\cal J}_{ij}\ {\cal S}_i{\cal
S}_j-H\sum_i{\cal S}_i,
\end{equation}
where ${\cal S}_i$ is the spin variable at site $i$, ${\cal
J}_{ij}$ is the interaction matrix between spins at sites $i$ and
$j$, and $H$ is an ordering external magnetic field. The spins
 obey the
spherical constraint $\sum_i\langle{\cal S}_i^2\rangle=N,$ where
$\langle\cdots\rangle$ denotes the standard thermodynamic average
taken with the Hamiltonian ${\cal H}$ and $N$ is the the total
number of spins.

 The quantum version of model (\ref{model1})  was considered in \cite{CDT00}.
 The Hamiltonian of the model is
\begin{equation}\label{modelq}
{\cal H}=\frac{g}{2}\sum_{i}{\cal P}_{i}^{2}-\frac12\sum_{ij}{\cal J}_{ij}\ {\cal S}_i{\cal
S}_j +\frac{\mu}{2}\sum_i{\cal S}_i^{2}-H\sum_i{\cal S}_i,
\end{equation}
where ${\cal S}_i$ are spin operators at site i, the operators
${\cal P}_{i}$ are "conjugated" momenta and $[{\cal S}_i,{\cal
S}_i']=i\delta_{ii'},[{\cal P}_{i},{\cal P}_{i'}]=i\delta_{ii'}$
(with $\hbar=1$). Model (\ref{modelq}) in the large-n limit is
equivalent to the quantum ${\cal O}(n)$ nonlinear sigma model.

The $d$-dimensional ${\cal O}(n)$-symmetric model  was considered
in \cite{KT91}-\cite{LB02}. The model is defined by
\begin{equation}\label{modela}
{\cal H}\left\{\varphi\right\}=\frac12\int_V d^dx\left[
\left(\nabla^{\sigma/2}\varphi\right)^2+r_0\varphi^2
+\frac12u_0\varphi^4\right],
\end{equation}
where $\varphi$ is a shorthand notation for the space dependent
$n$-component field $\varphi(x)$, $r_0=r_{0c}+t_0$ ($t_0\propto
T-T_c$) and $u_0$ are model constants. The first term  denotes
$k^\sigma|\varphi(k)|^2$ in the momentum representation. A
strongly anisotropic version of (\ref{modela}) with exponents
depending on the direction was considered in \cite{CGGP}.

The quantum version of model (\ref{modela}) is
\begin{equation}\label{modelq}
{\cal H}\left\{\varphi\right\}=\frac12\int_{0}^{1/T}d\tau\int_V
d^dx\left[ \left(\nabla^{\sigma/2}\varphi\right)^2+r_0\varphi^2
+\frac12u_0\varphi^4\right],
\end{equation}
where $\varphi$ is a shorthand notation for the space-time
dependent $n$-component field $\varphi(x,\tau)$ \cite{CT2000}.

\subsection*{6. Verification of FSS }
\subsubsection*{Classical phase transitions}

Analytically the case  $\sigma<d<2\sigma$ has been investigated
first exactly in the spherical limit ($n=\infty$)
\cite{BD91,BT92,BT88,B89}, \cite{FP86}-\cite{BT} and by the RG
methods for {$n=1$} \cite{luijten99}, $n\geq 1$
\cite{chamati01,chamati02}. Finite-size effects are predicted in
both  regions of first order \cite{FP86,SP89,BD91,BT92} and second
order phase transitions
\cite{BT88,BT9,B89,BT92,luijten99,chamati01,chamati02}. As a
result in the latter case it has been shown that the
  FSS   relations can be written in  form
 equivalent to Eq.(\ref{FSS2}).
Some thermodynamic quantities, for instance, the shift of the
critical temperature due to the finite-size effects \cite{CT96},
the susceptibility \cite{BT,BT9}, the pair correlation function
\cite{BD91,SP89} and the Binder's cumulant at the bulk critical
temperature $T_c$ \cite{luijten99,chamati01,chamati02} and above
it \cite{chamati01,chamati02} as a function of $ \sqrt{\epsilon}$
up to ${\cal O}(\epsilon^{3/2})$ have been obtained.

Away from the critical region, the critical behavior of the
systems is dominated by its bulk critical behavior

 \textsc{Nota bene}: A distinctive feature of the LR case is that for
$tL^{1/\nu}\gg1$ the finite-size corrections are not exponentially
small as in the SR case; they vary instead as algebraic power of
the variable $tL^{1/\nu}$ \cite{SP89,BD91,chamati01}.

Some problems exist with the comparison of RG analytical results
and numerical data. It has been shown \cite{luijten99} (see also
\cite{luijten2001}) using Monte Carlo simulation (for $n=1$), that
the Binder's cumulant ratio is linear in  $\epsilon$. The
analytical evaluation of the Binder's cumulant for the ${\cal
O}(n)$ symmetric $\varphi^4$ model, however, showed that it is
linear in $\sqrt\epsilon$. A possible way to resolve this
controversy between the Monte Carlo method and the analytical
results is to carry on finite-size calculations to higher loop
order \cite{luijten99}. This could ameliorate the analytical
results, which would be comparable to those obtained by numerical
simulations. Here we would like to mention that higher loop
corrections that are dealt with through minimal subtraction scheme
(see \cite{BZ85a,luijten99,chamati01}) and $\epsilon$-expansion
have not been performed, even for the more simple case of SR
interaction and therefore much knowledge is still absent.

\subsubsection*{Quantum phase transitions}

Quantum phase transitions \cite{SS} occur at zero temperature as a
function of some \textit{nonthermal} control parameter such as
composition or pressure and are driven by quantum fluctuations. In
addition to being of an interest for various low-temperature
phenomena, quantum phase transitions are important because they
are believed to play a crucial role in quantum information
science, e.g. in the phenomenon known as entanglement - the
resource that enables quantum computation and communications (see
\cite{HNO} and refs. therein).

The main feature of quantum systems is the coupling between
statics and dynamics. As a result the dynamical critical exponent
$z$ plays an important role in scaling of the "imaginary time
dimension". For nonzero temperature this extra "dimension" extends
only over a finite interval $L_{\tau}$. Here $L_{\tau}\sim1/T$ is
"finite-size" in the temporal (imaginary time) direction and so
the temperature's influence in the quantum critical point my be
determined by FSS effects \cite{brankov00}.

Unlike classical models, where scaling can be done uniformly for
all ``spatial'' dimensions, quantum models are anisotropic in
general, and therefore the ``space'' and ``imaginary-time''
directions will not scale in the same fashion. According to the
general hypothesis of the FSS theory extended here to quantum
(anisotropic) systems, a physical quantity ${\cal A}
\left(r,L,T\right)$, which is  singular  in the thermodynamic
(bulk) limit at the quantum critical point ($r=0$), will scale
like
\begin{equation}\label{FSSA}
{\cal A}(r,T,L)=b^p {\cal
A}_s\left(rb^{1/\nu},Tb^z,bL^{-1}\right).
\end{equation}
Here, $r$ measures the distance from the quantum critical point.
In the scaling form~(\ref{FSSA}) $p$ corresponds to the
engineering dimension $d+z$ of the system in the case when the
scaling function refers to the singular part of the free energy.
For the other physical quantities of interest it is   the critical
exponent measuring the divergence of the bulk thermodynamic
function ${\cal A}$ at the critical point divided by $\nu$ (for
the correlation length $p=1$, for the susceptibility
$p=\gamma/\nu$, etc). Depending on the choice of the RG rescaling
factor $b$ we obtain different scaling functions ${\cal A}_s$,
which are related among each other by some appropriate change of
the scaling variables \cite{CT2000}.

 The fact that the
inverse temperature can be used as an additional size in the
imaginary-time direction creates  strong anisotropy in the system
and in general, one needs a shape--dependent formulation  of the
FSS \cite{CT2000,CGGP}. This  will lead to the establishment of
some changes in the scaling properties of the finite quantum
system. In this  case we can consider the {\it quantum to
classical} and the {\it finite-size to the bulk} system
\cite{CT2000}, and as well as other \cite{CGGP} crossover
phenomena.

\subsubsection*{Results on the Casimir effect}

The correlation function serves as a measure of how fluctuations
at one point are correlated with fluctuations at another point.
The confinement conditions, imposed on a system with correlation
function decaying as \textit{power} law in space induce a LR force
between the surfaces limiting the system. One can generally call
this phenomenon Casimir effect. There can be different mechanisms
 leading to this phenomenon that are related with the
 scale invariance or mentioned above "generic scale invariance".
 Prominent examples
are systems at critical points or systems with spontaneously
broken global continuous symmetry that leads to massless modes:
"spine waves" or Goldstone bosons (statistical-mechanical Casimir
effect \cite{FG78}), and  the discussed for the first time by
H.G.B. Casimir (1948) constrained zero-point vacuum fluctuations
of the electromagnetic field (quantum - mechanical Casimir effect
).

 In a system with geometry $L^{1}\times
\infty^{d-1}$ at or bellow the critical point a LR
  ``statistical-mechanical Casimir
force'' \cite{brankov00,FG78,krech94} appears. According to the
standard FSS hypothesis, near the critical point of the bulk
system one expects for this force  (the magnetic field H=0)
\begin{equation}\label{CFH}
F_{{\rm Casimir}} (T,L)\simeq L^{-d}X(\kappa tL^{1/\nu}),
\end{equation}
where $X(x)$ is an universal function and $\kappa$ is a
nonuniversal metric factor. In the case of SR interaction the
universal scaling structure of Eq.(\ref{CFH}) has been confirmed
by RG and exact calculations \cite{brankov00,krech94}.

The investigation of systems with LR interaction possesses some
peculiarities: due to the
 character of the interaction there exists a natural attraction between the surfaces bounding
 the system. In this case only results  within the spherical
 model with p.b.c. are obtained \cite{CD04}.
It is
 possible to formulate  some general statements:
the Casimir force is always negative, (i.e. it is a force of
attraction
 between the surfaces bounding the system) for any T and $\sigma\geq1$,
as well as for any  $T\geq T_{c}$ if $\sigma<1$; the behavior of
the Casimir force depends strongly on the range of interaction
$\sigma$, being (at $T=T_{c}$) a monotonically increasing function
of $\sigma$; in the neighborhood of  $T=T_{c}$, the Casimir force
is an increasing function of T if $\sigma>1$, and possesses a
complex behavior if $\sigma\leq1$.

For the Casimir force
 there is an exact expression obtained in the
framework of the quantum spherical model (see \cite{brankov00})
\begin{equation}
F_{{\rm Casimir}}(T,\lambda;L)\simeq L^{-(d+z)}X_{{\rm
Casimir}}(x,\rho ), \label{eq12}
\end{equation}
with scaling variable $x=L^{1/\nu }\left(\lambda^{-1} -\lambda
_c^{-1}\right), \ \mbox{and} \  \rho =L^z/L_\tau,  L_\tau =\lambda
/T.$ Here
  $\lambda $ is the quantum parameter that governs the
  transition near the quantum critical point $\lambda _c$.

The obtained exact expression for the {\it universal} scaling
function $X_{{\rm Casimir}} (x_,\rho )$ gives the possibility of
analysis including issues as the sign of the Casimir force and its
monotonicity  as a function of the temperature and parameter
 $\sigma$ \cite{CDT00}.

The author thanks J. Brankov, H. Chamati and D. Danchev for
valuable discussions. Support by Bulgarian NSF, Grant F-1402, is
acknowledged.