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\begin{document}
\title{The Low Activity Phase of Some Dirichlet Series}
\date{ESI Preprint N. 313, March 1996}
\author{Pierluigi Contucci\thanks{Department of Mathematics, Kerchof
Hall, University of Virginia, Charlottesville Va 22903, U.S.A..
e-mail: contucci@virginia.edu} \and
Andreas Knauf\thanks{Technische Universit\"at,
Fachbereich 3 - Mathematik, MA 7--2,
Strasse des 17. Juni 135, D-10623 Berlin, Germany.
e-mail: knauf@math.tu-berlin.de
}}
\maketitle
%
%
\begin{abstract}
We show that a rigorous statistical mechanics description of some
Dirichlet series is possible.
Using the abstract polymer model language of statistical mechanics and
the polymer expansion theory we
characterize the {\it low activity} phase by the suitable
exponential decay of the truncated correlation functions.
\end{abstract}
%
\section{Introduction}
%
The idea to relate number theory and equilibrium statistical mechanics
or, more precisely, zeta functions and partition functions, is now already
quite old.
One motivation for pursuing this idea lies in the probabilistic
aspects of the prime number distribution. Statistical mechanics
as an intrinsically probabilistic theory is hoped to be an appropriate language
for these phenomena. The book \cite{KAC} by Kac nicely presents
this kind of probabilistic reasoning.
More concretely, the formulation of the famous Lee-Yang
theorem was influenced by a paper \cite{Po} by P\'{o}lya on the
Riemann zeta function. In that paper P\'{o}lya took the asymptotics of the
Fourier transformed zeta function and proved for its inverse
Fourier transform the 'Riemann hypothesis', saying that the non-real
zeroes have real part $\eh$.
As described by Kac in \cite{Po},
the method of P\'{o}lyas proof inspired the first version of the Lee-Yang
theorem (which says that the partition function of
ferromagnetic Ising models has only
zeroes on the unit circle of the activity plane).
This lead to the natural question whether inversely the Riemann hypothesis
or simpler number-theoretical questions could be proven
by some statistical mechanics method.
In recent years two approaches have been followed.
In one of them the Riemann zeta function $\zeta(s)$ itself was
interpreted as partition function of a system of {\it interacting
primes} at inverse temperature $s$,
see Julia \cite{JUL1,JUL2} and Bost and Connes \cite{CB,CB2}.
In the last-mentioned paper the system was shown to exhibit
a phase transition at $s=1$ with type I states (resp. type III)
at low (resp. high) temperature.
In the second approach mentioned the quotient $\zeta(s-1)/\zeta(s)$
is interpreted as a partition function at the inverse temperature $s$,
see Cvitanovi\'{c} \cite{Cv},
Knauf \cite{KN1,KN2,KN3,KN4}, Contucci and Knauf \cite{CK},
and Guerra and Knauf \cite{GK}.
It was shown that the partition function described a spin chain with
asymptotically translation-invariant long-range ferromagnetic
interaction (the {\em num\-ber-theoretical spin chain}).
The point $s=2$ corresponds to a phase transition where magnetization jumps
form 0 to 1.
Although there exist versions of the Lee-Yang theorem
predicting zero-free half planes in the
inverse temperature plane, unfortunately these
theorems cannot be applied to the above spin chain,
since its interaction includes multi-body terms.
In this paper, using the general polymer model approach of
statistical mechanics, we propose a criterium to
interpret a large class of Dirichlet series as grand canonical
partition functions of hard-core interacting systems.
The criterium involves a finite-volume approximation
and a precise notion of activity.
We present two possible polymerizations: the first is based
on the notion of Euler product and works for
multiplicative arithmetical functions, the second covers
a wider class of cases.
We show that the natural thermodynamical quantities of the
polymer model, like correlation functions, carry a deep number
theoretical meaning being the probability of suitable
divisibility properties.
In order to control the behaviour of the correlation functions
we apply the polymer expansion technique by means of the
Kirkwood-Salsburg iterative equations: the low activity
expansion theory enables us to prove the
exponential decay of all the truncated correlation functions and
provides, in general, a full analytical control of the low
temperature phase.
This shows that the language of polymer models
is not only formally but also analytically adequate to describe
the considered class of Dirichlet series.
Our approach clarifies the statistical mechanics
meaning of the absolute convergence theory for the Dirichlet series
and introduces new perspectives on it; moreover it has the merit
to point out the natural limits of each polymerization. The
polymerizations treated in this work, like similar
techniques in number theory, provide
an approximation of the Dirichlet function which works well
for large real part of the complex plane but it results to be too
non-uniform elsewhere, especially on the critical strip.
We believe that, in order to obtain new analytical results from
the number theoretical point of view, one has to search
for different polymerizations, for instance the high temperature ones,
or better to explore more subtle strategies like the
{\it rearrangement} procedure for polymer models (see \cite{BR1,BR2})
which, in some cases, provide a good control of the asymptotic
behaviour of the correlation functions in the interesting regions
of the phase space.
All these ideas can be improved and tested with the study of the number
theoretical spin chain: the interacting objects are there not directly
related to primes and could suggest different types of polymerization
based on groups of spins (see Guerra and Knauf \cite{GK}).
Moreover the approximant family $\varphi_k$ \cite{KN1,Cv} of the Euler
totient function could really be seen as a systematic way to rearrange
the Euler totient function $\varphi$ thought as a {\it bare}
interaction. We will return on these question elsewhere.
{\bf Notation.}
Sums resp.\ products over empty sets equal zero resp.\ one.
We write $\bN:=\{1,2,3,...\}$ for the integers,
$\bN_0:=\bN\cup\{0\}$
and
$\bP\equiv \{2,3,5,...\}$ for the primes.
If $n$ divides $m$ we write $n|m$ and
the symbol $\sum_{n|m}$ denotes a sum over all the divisors of $m$;
$(n,m)$ is the greatest common divisor of $m$ and $n$.
%
\section{The Polymer Expansion}
%
Statistical Mechanics seeks to describe the collective behaviour of
a large number of similar particles. One assumes that
these particles are enclosed in a finite region $\Lambda\subset S$ of
space $S$ (typically $S=\bR^d$ or $S=\bZ^d$)
and then considers the thermodynamic limit $\Lambda\nearrow S$.
The mutual interaction between the particles in a
configuration $\sigma$ is encoded
by their total energy $H_\Lambda(\sigma)$.
At inverse temperature $s$ the probability
of that configuration is given by
$\exp(-s H_\Lambda(\sigma))/Z_\Lambda(s)$,
\begin{equation}
Z_\Lambda(s) := \sum_\sigma \exp(-s H_\Lambda(\sigma))
\label{def:partition:general}
\end{equation}
being the
partition function for volume $\Lambda$.
So the basic objects of statistical mechanics are the
Boltzmann factors $\exp(-s H_\Lambda(\sigma))$ of the configurations.
Whereas the above Gibbs probability measures for the finite volume
$\Lambda$ are real-analytic in the parameter $s$, in the
thermodynamic limit $\Lambda\nearrow S$ nonanalyticities arise
which are called phase transitions.
Different asymptotic Gibbs measures may then be
compatible with a given interaction
and inverse temperature.
This phenomenon is typical for random fields, i.e.\ random functions
in several variables, and is of central interest in today's theory
of probability.
Thus one basic problem of statistical mechanics consists in
determining regions in parameter space (e.g., in the $s$ plane)
where intensive quantities like the free energy
$|\Lambda|^{-1} \ln (Z_\Lambda(s))$ stay analytic in the
thermodynamic limit.
Many of the techniques employed in that context
recently turned out to be related, the common ground being the abstract
{\em polymer model} formulation
(see Glimm and Jaffe \cite{GJ}, Simon \cite{Si} and
Koteck\'{y} and Preiss \cite{KP}).
In the abstract setting one starts with a denumerable set
$P\equiv \{\gamma_1, \gamma_2, ...\}$ whose elements are called
{\em polymers} and with an assigned reflexive symmetric relation
of {\it incompatibility} between each two of them.
In the concrete
application of a two-dimensional Ising model, the polymers may be the contours
enclosing a region of constant spin direction, or the subgraphs
of the nearest neighbour graph, depending on whether one is
interested in small or large temperatures;
the incompatibility between two of them is simply the mutual
overlapping.
Thus one may associate to a $k$-{\em polymer}
$X:=\{\gamma_1,\ldots,\gamma_k\}\in P^k$ an undirected graph
$G(X)=(V(X),E(X))$
with vertex set $V(X):=\{1,\ldots,k\}$, vertices $i\neq
j$ being connected by the edge $\{\gamma_i,\gamma_j\}\in E(X)$ if
$\gamma_i$ and $\gamma_j$ are incompatible.
Accordingly the $k$-polymer $X$ is called connected if $G(X)$ is
path-connected and (completely) disconnected if it has no edges
($E(X)=\emptyset$).
The corresponding subsets of $P^k$ are called $C^k$ resp.\ $D^k$,
with $D^0:=P^0:=\{\emptyset\}$ consisting of a single element. Moreover
$P^{\infty}:= \bigcup_{k=0}^{\infty} P^k$ with the subsets
$D^{\infty}:= \bigcup_{k=0}^{\infty} D^k$ and
$C^{\infty}:= \bigcup_{k=1}^{\infty} C^k$.
We write $|X|:=k$ if $X\in P^k$; indicating with $X(\gamma)$ the
multiplicity of $\gamma$ in $X$ it results $k=\sum_i X(\gamma_i)$.
It is useful to define the function $c(X):=\prod_i X(\gamma_i)!$.
We will indicate with a hat the
abelianized set: for instance $\hat{P}^\infty$ is the set of abelian
words (which we also call polymer configurations) which arises if one
identifies $k$-polymers $X=\{\gi_1,\ldots,\gi_k\}$,
$Y=\{\delta_1,\ldots,\delta_k\}\in P^\infty$ if
$\delta_{\pi(i)} = \gamma_i$ for some permutation $\pi$.
Statistical weights or activities $z:P\ar\bC$ of the
polymers are multiplied to give the activities
$z^X:=\prod_{i=1}^k z(\gamma_i)$ of $k$-polymers.
The thermodynamical properties of the model are defined through the
partition function
\begin{equation}
Z = \sum_{X\in {\hat{D}^{\infty}}} z^X.
\label{eq:parti}
\end{equation}
We observe that no multiple occurence of a polymer is allowed
since the incompatibility relation is reflexive; for this reason the
sum is finite when $P$ has finite cardinality which corresponds
to a finite volume in the concrete cases.
It has to be stressed that the polymer models are not statistical
mechanics models in the usual form (\ref{def:partition:general}).
They are useful devices to study the true models
in the {\it low activity} regime of the phase diagram: for this reason
a given model is often mapped into different polymer models according to
each different phase regime.
One is mainly interested in the $|P|\to\infty$ limit (thermodynamic
limit) for the mean values of the configuration functions $h$:
\begin{equation}
_{z}=
{
\sum_{X\in \hat{D}^{\infty}}
h(X)z^{X}
\over
\sum_{X\in \hat{D}^{\infty}}z^{X}
},
\label{def:m}
\end{equation}
and especially for the correlation functions
\begin{equation}
\rho_{z}(Y)=<\chi_{Y}>_{z}
\label{def:cor}
\end{equation}
where $\chi_Y$ is the characteristic function of $Y$.
Often the dependence of the correlation
functions on the activity is studied in terms of parameters like,
for instance in statistical mechanics, the inverse temperature
or a magnetic field.
An important quantity to be studied
in the thermodynamic limit is the free energy density, or pressure,
which turns out to be (see for instance \cite{GMM,GJ}), up to a
suitable normalization factor
\begin{equation}
\ln(Z) =
\sum_{X\in \hat{P}^{\infty}}
{n^T(X)\over c(X)}
z^X,
\label{def:fr}
\end{equation}
with $n^T(X):=n_+(X)-n_-(X)$, $n_\pm (X)$ being the number of
subgraphs of $G(X)$ connecting all the vertices of $G(X)$
with an even resp.\ odd number of edges. The structure of the factors
$n^T$ implies that the previous sum is actually supported only on
$\hat{C}^{\infty}$.
We notice that, although in the partition sum only compatible
configurations of polymer may appear, the free energy contains
contribution from all the configuration
and also coincident polymers (multiplicities) are allowed.
Formula (\ref{def:fr}) is important from the conceptional
as well as from the analytical
point of view. It is complemented by the so-called tree estimate
$|n^T(X)| \leq |\tau(G(X))|$, where $\tau(G)$ denotes the
set of maximal subtrees of the connected graph $G$.
This inequality is useful, since there exist techniques to estimate
the number of subtrees. As an example, a theorem by Cayley
says that the complete (all edges present) graph $K(k)$ with $k$
vertices contains $|\tau(K(k))| = k^{k-2}$ maximal trees.
It's easy to check that the simplest example
$P=\{p\}$, i.e., $Z = 1+z$, of a polymer model
reduces the formula (\ref{def:fr}) to the Taylor expansion
for the logarithm
$\ln(Z) = \sum_{k=1}^\infty \frac{(-1)^{k-1}}{k} z^k$,
since $n^T(K(k)) = (-1)^{k-1} (k-1)!$ (the last formula showing,
by the way, that the tree estimate is non-optimal).
So even for a finite cardinality of $P$ one needs
bounds on the activities to ensure convergence of the free energy.
In the statistical mechanics applications such bounds are given
in terms of energy (or activity) and entropy bounds.
%
\section{Dirichlet Series}
%
A basic object of analytic number theory is the Dirichlet series
consisting of terms of the form $e^{-s\lambda_n}$ whose
exponents $\{\lambda_n\}_{n\in\bN}$ being a real-valued sequence
strictly increasing to $\lim_{n\ar\infty} \lambda_n=\infty$.
A formal series of the form
\begin{equation}
\sum_{n=1}^{\infty} a(n) e^{-s\lambda_n}
\label{def:general:diri}
\end{equation}
with complex coefficients $a(n)$ and argument $s$ is called a
general Dirichlet series. In this context functions
$A:\bN\ar\bC$ are called {\em arithmetical functions}.
Dirichlet series have abscissae $\sigma_a$, $\sigma_c$
of {\em absolute} resp.\ {\em conditional} convergence.
For $\lambda_n:=n$ eq.\ (\ref{def:general:diri}) is a power series in
$x:=e^{-s}$ so that $\sigma_a$ and $\sigma_c$
coincide.
For $\lambda_n:=\ln(n)$ eq.\ (\ref{def:general:diri})
is called an ordinary Dirichlet series, and we write it in the form
\begin{equation}
Z_a(s) := \sum_{n=1}^{\infty} a(n) n^{-s}.
\label{def:ordinary:diri}
\end{equation}
In that case $0\leq \sigma_a - \sigma_c \leq 1$.
The simplest choice $a(n):=1$ of coefficients leads to the
Riemann zeta function
$\zeta(s)=\sum_{n=1}^{\infty} n^{-s}$ with inverse
$1/\zeta(s) = \sum_{n=1}^{\infty} \mu(n) n^{-s}$,
with the M\"{o}bius function $\mu$ (see Appendix).
In that case $\sigma_a = 1$ and $\eh\leq \sigma_c\leq 1$, the
Riemann hypothesis being $\sigma_c=\eh$.
Many Dirichlet series arising in number theory can be written as
an Euler product
\begin{equation}
\sum_{n=1}^{\infty} a(n) n^{-s} = \prod_{p\in\bP} f_p(p^{-s}).
\label{def:euler:product}
\end{equation}
By the fundamental theorem of arithmetic this is the case exactly if
the arithmetical function $n\mapsto a(n)$ is {\em multiplicative}, that is,
(it is not identically zero) and
\[ a(mn) = a(m) a(n) \quad \mbox{if } \gcd(m,n)=1. \]
Then $f_p(x) = \sum_{k=0}^{\infty} a(p^k) x^{-k}$. For example,
$\zeta(s) = \prod_{p\in\bP} (1-p^{-s})^{-1}$.
The product
\begin{equation}
Z_{f}(s)Z_{g}(s)=Z_{f*g}(s).
\label{def:dirip}
\end{equation}
of Dirichlet series has coefficients
\begin{equation}
f*g(n)=\sum_{d|n}f(d)g({n\over d}).
\label{def:dp1}
\end{equation}
which are given by the Dirichlet convolution product $f*g$ of the
arithmetical functions of the factors.
With pointwise addition and Dirichlet multiplication the set of
arithmetical functions becomes an associative algebra with unit $I$,
$I(n) = \delta_{1,n}$ (a so-called monoid).
It is easy to prove that when $f(1)\neq 0$ a Dirichlet inverse $f^{(-1)}$
exist.
Dirichlet series are used in number theory in order to make use of
analytic tools in the theory of prime numbers.
As an example, the prime number theorem states that the number
$\pi(x)=|\{p\in\bP\mid p\leq x\}|$ of primes smaller than $x$
is asymptotic to $x/\ln x$. This can be shown by analyzing
$\zeta'(s)/\zeta(s)$ for ${\rm Re}(s) = 1$, that is on the line containing
the pole.
%
In order to give a statistical mechanics interpretation of
(some) Dirichlet series as polymer partition functions we have to
identify the sums (\ref{def:ordinary:diri}) and (\ref{eq:parti}).
This can be done, of course, in many ways: the main point is that
in the partition sum each polymer can only have simple multiplicity.
We propose two type of polymerization: the first works for
multiplicative arithmetical function and is based on the notion of
Euler product, the second is more general. We introduce both of them
because the first admits a special treatment in the convergence theorems
leading to better convergence estimates (see Appendix B).
\begin{enumerate}
\item Multiplicative Polymerization.
\begin{itemize}
\item
If we now interpret $\zeta(s) = \sum_{n=1}^{\infty} e^{-s\ln(n)}$
as a partition function for an infinite system with state space $n\in\bN$
and energies $\ln(n)$, then $\zeta'(s)/\zeta(s)$ is minus the expectation
of the internal energy. Moreover, in the notation of the previous section
\begin{eqnarray*}
\zeta(s) &=& \prod_{p\in\bP} (1-p^{-s})^{-1} =
\sum_{X\in \hat{D}^{\infty}} z_s^X,
\end{eqnarray*}
taking the primes as the polymers ($P:=\bP$), assuming different primes to
be compatible and setting the activities $z_s(p):=1/(p^s-1)$; moreover
\[ \zeta'(s)/\zeta(s) = \frac{d}{ds} \ln(\zeta(s))
= - \sum_{p\in\bP} \ln(p)\cdot z_s(p) .\]
%
\item
Alternatively one may consider the set $P := \{p^n \mid p\in\bP,n\in\bN\}$
of prime powers as
polymers with the activities $z_s(x) := x^{-s}$ for $x\in P$ and call
$p_1^{n_1},p_2^{n_2} \in P$ incompatible iff $p_1=p_2$.
Then, again, $\zeta(s)$ can be written as a polymer model (\ref{eq:parti})
and thus its logarithm may be written using formula (\ref{def:fr}).
\end{itemize}
%
Clearly this kind of game can be played with any Dirichlet
series having an Euler product (\ref{def:euler:product}).
Then for the first choice $P=\bP$ of polymers the activities
are $z_s(p) := f_p(p^{-s})-1$, whereas $z_s(p^k) := a(p^k)p^{-sk}$
in the second case.
It is clear that when the multiplicative arithmetical function
$a:\bN\ar\bC$ is a square-free function (that is, it vanishes on integers
containing squares), then both polymer model interpretations
lead to the same activity
$z_s(p) = a(p)e^{-s\ln p}$, $z_s(p^k) = 0$ for $k>1$.
The function $a$ plays the role of an interaction.
\item General Polymerization.
\begin{itemize}
\item Square-Free Case.
A large class of
{\it square-free} Dirichlet series admit the interpretation
of polymer models where each prime number
is considered a polymer. For instance we can consider the family of
arithmetical functions $\phi=\li f$ where $f$ is multiplicative (and
possibly positive to have a genuine probabilistic framework) and the
function $\li$ is defined as
\begin{equation}
\li(n)=\cases{1,&if $n=1,p$;\cr
\prod_{pp'|n}g(p,p'),&otherwise,\cr}
\label{def:765}
\end{equation}
where $g(p,p')$ takes values $0,1$ and is a symmetric
function vanishing on the diagonal. We stress that the previous conditions
define a {\it class} of matrices (of entries $g(p,p')$) and
correspondingly a class of square-free arithmetical functions $\phi$ not
necessarily multiplicative.
Two primes with $g(p,p')=0$ are called incompatible; two integers are
incompatible if there are two incompatible primes in the respective
decomposition.
Some examples of incompatibility are the following:
$p,p'$ are compatible polymers when
\begin{itemize}
\item $p\neq p'$;
\item $|p-p'|> const$;
\item $|p-p'|>\log\sqrt{pp'}$.
\end{itemize}
The first case corresponds to the square-free function $|\mu|$ (see
appendix) in which the only interaction is the Fermi statistic; the
relative zeta function is $Z_{|\mu|}(s)={\zeta(s) \over \zeta(2s)}$.
The interest of the third case will be clear in the section on convergence.
The fundamental theorem of arithmetic on the unique decomposition
of an integer into primes permits the formal identification of the
function
\begin{equation}
Z_{\phi}(s)=\sum_{n\in\bN}\phi(n)n^{-s},
\label{def:sqfrpf}
\end{equation}
with a partition function of a polymer system in which each prime
has activity $z_{s}(p)=f(p)e^{-s\log p}$ and the function $\li$
play the role of the hard-core interaction.
\item Non Square-Free case.
An important observation is that to treat the case of non square-free
Dirichlet series we have simply to change the polymer
identification: the polymers are now the prime powers
$P := \{p^n \mid p\in\bP,n\in\bN\}\equiv \{2,3,4,5,7,8,9,...\}$
with the activities $z_s(x) := f(x)x^{-s}$ for $x\in P$. The class
of arithmetical function treated in this way is defined by $\phi=f \li$
where $f$ is multiplicative and $\li$
\begin{equation}
\li(n)=\cases{1,&if $n=1,p^{k}$;\cr
\prod_{x,x'\in P, xx'|n}g(x,x'),&otherwise,\cr}
\label{def:eug}
\end{equation}
where $g(x,x')$ takes values $0,1$ and is a symmetric
function vanishing on all the couples $(x,x')=(p^{k},p^{k'})$. The
previous conditions define a {\it class} of matrices and
correspondingly a class of arithmetical functions $\phi$ in general
not square-free nor multiplicative.
Also in this case there is plenty of examples; for instance the second
and third example of the previous polymerization can be rephrased
exactly in this one. The simplest example is just the Riemann zeta
function: it corresponds to the element of the previous class in which
$f(n)=1$ and $g(x,x')=1$ if $(x,x')\neq (p^{k},p^{k'})$
which says that two polymers are incompatible when they are power
of the same prime and they are compatible otherwise.
This means that the Riemann zeta function admits the
interpretation of the partition function of an hard-core interacting
polymer system.
\end{itemize}
\end{enumerate}
%
\section{The Hard-Core Models}
%
Let consider, for simplicity, the square-free case with
$\phi=\li$. We introduce now a family of
approximating functions $\li_k$ depending on a integer $k$; the
meaning of this approximation is just the finite volume approximation
in statistical mechanics which manifest itself with a finite number
of polymers.
The corresponding partition function becomes a finite series
for each $k$ and the problem to control the thermodynamical limit
for the correlation functions concerns the possibility to obtain
bounds which are uniform in $k$.
We first
define the $k$-th set of square-free integers $\bN_k$ as the integers
of the form
\begin{equation}
n=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k},\mbox{ where }
\alpha_i=0\mbox{ or }1,
\label{def:nk}
\end{equation}
and $p_1,...,p_k$ are the first $k$ prime numbers.
Then, for instance, $\bN_{0}=\{1\}$, $\bN_{1}=\{1,2\}$,
$\bN_{2}=\{1,2,3,6\}$,
$\bN_{3}=\{1,2,3,5,6,10,15,30\}$, etc, and
$|\bN_k|=2^{k}$.
Now we define:
\begin{equation}
\li_k(n)=\cases{\li(n),&if $n\in\bN_{k}$;\cr
0,&otherwise.\cr}
\label{def:kint}
\end{equation}
It is easy to prove that:
\begin{equation}
\li_k(n)=\li (n),\mbox{ for }n\le p_k,
\label{th:iaprox}
\end{equation}
and
\begin{equation}
\li_k(n)=0,\mbox{ for }n> p_1\cdots p_k.
\label{th:saprox}
\end{equation}
The origin of this approximation is quite simple: we consider the
natural numbers
progressively generated by prime numbers; the nature of the function
$\li$ implies that
for each generation only a finite quantity of integers gives a
contribution.
{\it Remark:} The above mechanism induces in general a one-to-one
correspondence between the functions of the variable $(\alpha_1,...,\alpha_k)$
and the $k$-th approximation of a square-free arithmetical functions.
It clearly
turns out that the approximating zeta function
admits the
interpretation of a grand canonical partition function for a system of
$k$ particles interacting via a hard-core two-body potential:
\begin{equation}
Z_{\li_k}(s)=\sum_{n\in\bN}\li_{k}(n)n^{-s}
=\sum_{\alpha}
\prod_{i}p_i^{-s\alpha_i}
\prod_{i_{k}(s)={\sum_{\alpha} f(\alpha)\prod_{i}p_i^{-s\alpha_i}
\prod_{i_{k}(s),
\label{def:corre}
\end{equation}
with $i_11$; for each of them one could actually
improve the general convergence strategy we are going to present.
Our first goal is to express the correlation function at the
temperature $s$ as a zeta function:
\begin{equation}
<\alpha_{i_1}\cdots\alpha_{i_r}>_{k}(s)=
\sum_{l}(l)l^{-s},
\label{def:cor1}
\end{equation}
where $n=p_{i_1}\cdots p_{i_r}$.
The algebraic properties of the Dirichlet convolution imply that
eq.\ (\ref{def:cor1}) can be solved in the arithmetical function
$$ and the solution is:
\begin{equation}
=n^{-s}(\li_k^{(-1)}*D_n\li_k),
\label{eq:afc}
\end{equation}
where we have introduced the operation $D_n$ as
\begin{equation}
D_nf(k)=f(nk).
\label{def:deri}
\end{equation}
One immediately realizes that the arithmetic function corresponding
to the correlations of a square-free model {\it is not}
square-free. This is because the Dirichlet inverse
operation does not conserve the square-free property and it is the main
motivation to introduce a formalism able to handle generic polymer
configurations with the suitable convolution. Moreover it also implies
that, even for finite $k$, the correlations zeta function is no more a
finite series; we want to show how it is possible to control its
properties in the limit $k\to \infty$ using the statistical mechanics
method of the iterative equations. This will provide a statistical
mechanics meaning to the limiting correlations and a new point of view
in the study of some number theoretical quantities.
The idea, which is a central one in statistical mechanics, is to study
the ``interaction'' between one particle and the remaining ones or,
in number theoretical terms, to have some control on the non
multiplicativity of the $\li$.
Defining the function
\begin{equation}
\Gamma_n:=\li_k^{(-1)}*D_n\li_k,
\label{eq:logder}
\end{equation}
we consider an
integer of the form $pn$, where p is a prime compatible with n
(otherwise $\Gamma_{pn}=0$). By definition we have
\begin{equation}
\Gamma_{pn}(l)=\sum_{d|l}\li_k^{(-1)}(d)\li_k(pn{l\over d});
\label{eq:delta}
\end{equation}
indicating $\sum_{k\subset n}$ a sum over all the divisors of $n$ counted
with multiplicity (see appendix), we first observe that
\begin{equation}
\li_k(pn{l\over d})=\li_k(n{l\over d})
{\sum_{r\subset{l\over d}}}^{p}\lambda(r),
\label{eq:qm1}
\end{equation}
where the ${\sum}^{p}$ means a sum over all square-free integers build on
$p$-incompatible primes and the function $\lambda$ is the Liouville
function defined by $\lambda(n)=(-1)^{\Omega(n)}$ where $\Omega$ is the
number of prime factors counted with multiplicity.
Since $\li(p)=1$ the previous formula gives an evaluation of how much
the interaction $\li$ deviates from a completely multiplicative function;
it can be proved, for instance, observing that the factor $G(p,h)$
defined by
\begin{equation}
\li(ph)=\li(h)G(p,h)
\label{eq:qm2}
\end{equation}
is
\begin{equation}
G(p,h)=\prod_{p'\subset h}g(p,p')=\prod_{p'\subset h}((g(p,p')-1)+1)=
{\sum_{r\subset h}}^{p}(-1)^{\Omega(r)},
\label{def:gi}
\end{equation}
which is the (\ref{eq:qm1}) since the integer $pn$ is supposed to be
compatible. Substituting the (\ref{eq:qm1}) inside
the (\ref{eq:delta}) we have
\begin{equation}
\Gamma_{pn}(l)=\sum_{d|l}\li_k^{(-1)}(d)\li_k(p)\li_k(n{l\over d})
{\sum_{r\subset{l\over d}}}^{p}\lambda(r),
\label{eq:ad}
\end{equation}
and interchanging the summation order
\begin{equation}
\Gamma_{pn}(l)=
{\sum_{r\subset l}}^{p}\lambda(r)
\sum_{d|{l\over r}}\li^{(-1)}_k(d)\li_k({nl\over d}),
\end{equation}
which is, up to renaming the sets,
\begin{equation}
\Gamma_{pn}(l)={\sum_{r\subset l}}^{p}\lambda(r)
\Gamma_{nr}({l\over r}).
\label{eq:ks}
\end{equation}
This is the iterative equation we want to consider. In order to
control its solutions we observe that, by inspection, it lives
naturally as equation for the {\it two-variable} arithmetical function
$\Gamma$;
moreover defining the ``index'' of the quantity $\Gamma_{n}(l)$ as
$\Omega(nl)$, the equations (\ref{eq:ks}) can be solved iteratively
observing that they allow to
compute the family of index $\Omega(nl)+1$ in terms of that whose
index is $\Omega(nl)$. This fact not only makes it possible to study
the iterative solutions with the initial condition $\Gamma_1(1)=1$
but it also gives hints on the suitable Banach space structure to be
introduced in order to make use of the contraction principle.
%
\section{The Contraction Regime for the Iteration}
%
In our number theoretical context we can introduce the seminorms
for the family of the $(l)$ with $\Omega(nl)=m$, depending on
a parameter $\delta$ to be optimized at the end,
\begin{equation}
N_m(\delta)=\sup_{n,1\le\Omega(n)\le m}\sum_{l,\Omega(nl)=m}
|(l)|l^{-s}n^{(s-\delta)};
\label{def:norm}
\end{equation}
we claim that, for suitable values of $\delta$, this norm is contractive for
the iterative equations.
The proof is along the following lines. Using the
(\ref{eq:ks}) we observe that it holds the bound
\begin{equation}
\sum_{l,\Omega(nl)=m}|(l)|l^{-s}(pn)^{(s-\delta)}\le
\sum_{l,\Omega(nl)=m}p^{-s}{\sum_{r\subset l}}^{p}
r^{s}|({l\over r})|l^{-s}(pn)^{(s-\delta)},
\label{st:1}
\end{equation}
since the Liouville function is bounded in modulus by one. It follows that
\begin{equation}
\sum_{l,\Omega(nl)=m}|(l)|l^{-s}(pn)^{(s-\delta)}
\le N_m(\delta)p^{-\delta}
{\sum_{r}}^{p}r^{-(s-\delta)}
\label{st:2}
\end{equation}
where the last sum runs over the square-free $r$ build on $p$-incompatible
primes. We also observe that, since $r$ runs over square-free integers,
\begin{equation}
{\sum_{r}}^{p}e^{-(s-\delta)\log r}\le
\sum_{n=0}^{\infty}{1\over n!}
\left({\sum_{p'}}^{p}e^{-(s-\delta)\log p'}\right)^{n}.
\label{st:3}
\end{equation}
Making use of the entropy bound it is possible to control the sum on
$p$-incompatible
primes observing that it can be written as
\begin{equation}
\sum_{v=1}^{\infty}\sum_{p',v\le\log p'^{T}(s):=
<\alpha_{i_1}\alpha_{i_2}>(s)-<\alpha_{i_1}>(s)<\alpha_{i_2}>(s).
\end{equation}
This function represents
the deviation from the independence of the two events ``$p_{i_1}$
divides an integer'' and ``$p_{i_2}$ divides an integer''.
As for the simple correlations an easy computation shows that it is
possible to express them as Dirichlet series of a suitable
arithmetical function:
\begin{equation}
<\alpha_{i_1},\alpha_{i_2}>^{T}(s)=\sum_l
(p_{i_1}p_{i_2})^{-s}
(\Gamma_{p_{i_1}p_{i_2}}-\Gamma_{p_{i_1}}*\Gamma_{p_{i_2}})l^{-s}.
\label{def:tpct}
\end{equation}
We recognize that the two point truncated expectation is the Dirichlet
series of the arithmetical function given by the second order of the
formal logarithm of the functions $\Gamma$ with respect to the
{\it lower variable}:
\begin{equation}
\Gamma^{T}:={\rm Log}\Gamma.
\label{def:ct}
\end{equation}
Let us clarify the geometrical meaning of the operations
which naturally appear considering the simple and the truncated
correlation functions. First we notice that for the operation $D_n$
it holds (see \cite{RUELLE}) the Leibnitz rule with respect to the
circle product (see appendix):
\begin{equation}
D_n(f\circ g)=D_n f\circ g+f\circ D_n g.
\label{th:leib}
\end{equation}
From it one can easily prove that the operation $\partial_n$ defined by
\begin{equation}
\partial_n f (k)=f(nk){c(nk)\over c(k)},
\label{der:multip}
\end{equation}
plays the role of a {\it multiple} derivative with respect to the Dirichlet
product since it fulfills the composition
rule $\partial_{n_1}\partial_{n_2}=\partial_{n_1n_2}$ and, when $n$ is
a prime number, the Leibnitz rule with respect to the Dirichlet
multiplication. This can be seen observing that defining
the operation $\DI$ from the set
of the one-variable to that of the two-variable arithmetical functions by
\begin{equation}
(\DI f)(n,k)=f(nk){c(nk)\over c(n)c(k)};
\label{def:cohom}
\end{equation}
it holds for it the important property:
\begin{equation}
\DI (f*g)= \DI f * \DI g,
\label{th:om}
\end{equation}
where the convolution at the right hand side is the two variable
Dirichlet convolution.
In particular it holds the
\begin{equation}
\partial_p {\rm Exp}f=\partial_p f*{\rm Exp}f,
\label{def:idesp}
\end{equation}
and
\begin{equation}
\partial_p {\rm Log}g=g^{(-1)}*\partial_p g.
\label{def:idlog}
\end{equation}
Since $\li$ is a square-free function $D_n \li=\partial_n \li$ we
have, with $\li^{T}:={\rm Log}\li$,
\begin{equation}
\partial_n \li^{T}=\Gamma^{T}_n.
\label{def:trcofe}
\end{equation}
Choosing $n=p$ it holds $\partial_p \li^T=\Gamma_p$ which is
\begin{equation}
\li^T(pn)={c(n)\over c(pn)}\Gamma_p(n).
\label{def:rlz}
\end{equation}
This relation enable us
to obtain a bound, inside our contraction regime, on a quantity which
represents the free energy density centered around the prime $p$:
\begin{equation}
\sum_{l=1, p|l}^{\infty}\li^T(l)l^{-s}.
\label{def:fedp}
\end{equation}
In fact applying the (\ref{def:rlz}) and the contraction scheme for
the norm one has:
\begin{equation}
|\sum_{l=1, p|l}^{\infty}\li^T(l)l^{-s}|\le
\sum_{k=1}^{\infty}\sum_{l,\Omega(pl)=k}|\Gamma_{p}(l)|p^{-s}l^{-s}=
\label{def:fedpb}
\end{equation}
\begin{equation}
\le p^{-(s-\bar{\delta})}\sum_{k=1}^{\infty}e^{-\rho k}=
e^{-(s-\bar{\delta}) \log p}{e^{-\rho}\over {1-e^{-\rho}}};
\label{th:czmp}
\end{equation}
which is the claimed exponential decay in terms of the polymer volume
(notice that $s>\bar{\delta}$ in the contraction regime).
In the same way it is possible to obtain the decay for the multiple
truncated correlations functions; let us show it for the two-point
case.
From (\ref{def:tpct}) we have
\begin{equation}
|<\alpha_{i_1},\alpha_{i_2}>^{T}(s)|\le (p_{i_1}p_{i_2})^{-s}
\sum_{l=1}^{\infty}(|\Gamma_{p_{i_1}p_{i_2}}(l)|+
|\Gamma_{p_{i_1}}*\Gamma_{p_{i+2}}(l)|)l^{-s}.
\end{equation}
The term with the convolution product of the right hand side is
bounded using the (\ref{th:czmp}) and the multiplicative property
of the relative Dirichlet series. For the first terms one has:
\begin{equation}
\sum_{l=1}^{\infty}|\Gamma_{p_{i_1}p_{i_2}}(l)|l^{-s}=
(p_{i_1}p_{i_2})^{\bar{\delta}}
\sum_{k=2}^{\infty}\sum_{l,\Omega(p_{i_1}p_{i_2})=k}(l)l^{-s}
(p_{i_1}p_{i_2})^{(s-\bar{\delta})}=
\end{equation}
\begin{equation}
\le (p_{i_1}p_{i_2})^{\bar{\delta}-s}\sum_{k=2}^{\infty}e^{-k\rho}=
e^{-(s-\delta)\log(p_{i_1}p_{i_2})}{e^{-2\rho}\over {1-e^{-\rho}}}.
\label{th:czm}
\end{equation}
Summing the two contributions we obtain
\begin{equation}
|<\alpha_{i_1},\alpha_{i_2}>^{T}(s)|\le e^{-(s-\bar{\delta})
\log(p_{i_1}p_{i_2})}\left( {e^{-2\rho}\over {1-e^{-\rho}}}+
{e^{-2\rho}\over {(1-e^{-\rho})^{2}}}\right)
\label{th:tpczt}
\end{equation}
which is the desired result. In the same way one can obtain the same
exponential decay for all the other truncated correlation functions.
%
%
\appendix
%
\section{Some Arithmetical Functions}
%
Some arithmetical functions considered on this work are:
the identity for the pointwise multiplication:
\begin{equation}
u(n)=1\quad \forall n,
\end{equation}
the identity for the Dirichlet product
\begin{equation}
I(n)=\cases{1,&if $n=1$;\cr
0,&otherwise,\cr}
\label{def:id}
\end{equation}
the identity map from $\bN$ to $\bN$
\begin{equation}
N(n)=n,
\label{def:mapid}
\end{equation}
and the ``square" function:
\begin{equation}
Q(n)=\cases{1,&if $n$ is a square;\cr
0,&otherwise,\cr}.
\label{def:squre}
\end{equation}
In terms of them it is easy to express other important
functions: the M\"obius function
\begin{equation}
\mu=u^{-1},
\label{def:mu}
\end{equation}
its absolute value
\begin{equation}
|\mu|=u*Q^{(-1)},
\label{def:amf}
\end{equation}
the Liouville function
\begin{equation}
\lambda=\mu*Q=|\mu|^{(-1)}.
\label{def:lioi}
\end{equation}
%
\begin{figure}
\begin{tabular}{||l|cccccccccc||} \hline
Function &1&2&3&4&5&6&7&8&9&10 \\ \hline
$Q$ &1&0&0&1&0&0&0&0&1&0 \\
$Q^{(-1)}$ &1&0&0&-1&0&0&0&0&-1&0 \\
$\lambda$ &1&-1&-1&1&-1&1&-1&-1&1&1 \\
$\mu$ &1&-1&-1&0&-1&1&-1&0&0&1 \\
$\omega_3$ &1&1&1&0&1&1&0&0&0&1 \\
$\omega_3^{(-1)}$ &1&-1&-1&1&-1&1&0&-1&1&1 \\
$D_2\omega_3$ &1&0&1&0&1&0&0&0&0&0 \\
$\omega_3^{(-1)}*D_2\omega_3$ &1&-1&0&1&0&0&0&-1&0&0 \\ \hline
\end{tabular}
\end{figure}
It can be useful to introduce another convolution product:
considering an integer as an unordered sequence of primes
$n\equiv \{p_{i_1},...,p_{i_1},p_{i_2},...,p_{i_2},...,p_{i_n}\}$ the
natural definition of convolution is the sum over all the subsequences
\begin{equation}
f\circ g(n)=\sum_{d\subset n}f(d)g({n\over d}),
\label{def:rup}
\end{equation}
where, for instance, the set of subsequences of $4$ is $\{1,2,2,4\}$.
It is easy to see that it is related to the Dirichlet one by:
\begin{equation}
f\circ g(n)=\sum_{d|n}f(d)g({n\over d})c(d,{n\over d}),
\label{def:rupr}
\end{equation}
where $c(l,m)={c(lm)\over c(l)c(m)}$
with $c(p_{i_1}^{\alpha_{i_1}}\cdots p_{i_k}^{\alpha_{i_k}})=
\prod_{j=1}^{k}\alpha_{i_k}!$.
This property is equivalent to the fact that the $\circ$-product
plays the role of the convolution for the deformed zeta
functions with the non-character activity $\tilde{z}(n)={n^{-s}\over c(n)}$
\begin{equation}
\tilde{Z}_f(s)=\sum_{n}f(n)\tilde{z}(n),
\label{def:zetaru}
\end{equation}
i.e. it holds
\begin{equation}
\tilde{Z}_{f}(s)\tilde{Z}_{g}(s)=\tilde{Z}_{f\circ g}(s).
\label{def:ruep}
\end{equation}
For both the convolution products it is possible to define the powers of a
function and, in some cases, power series like exponential and logarith:
Defining the sets of arithmetical functions
$\Ar_0$ and $\Ar_1$ respectively by the conditions
$f(1)=0$
and $f(1)=1$ it is possible to construct
well defined power series in the convolution products; in particular
the arithmetical
function corresponding to the exponential
for $f\in \Ar_0$ is
\begin{equation}
{\rm Exp} f= \sum_{k=0}^{\infty}{f^{(k)}\over k!}
\label{eq:esp}
\end{equation}
and the logarithm,
for $h\in \Ar_1$
which is, defining $h=I+{\tilde h}$,
\begin{equation}
{\rm Log} h=\sum_{k=0}^{\infty}(-1)^{k}{{\tilde h}^{(k)}\over k}.
\label{eq:tr}
\end{equation}
It is easy to see that the operation ${\rm Exp}$: $\Ar_0\to \Ar_1$ and
${\rm Log}$: $\Ar_1\to \Ar_0$ are mutually inverse.
From a combinatorial point of view the main advantage to consider the
circle product is that it permits
to define the exponential of a function as the sum over the {\it partitions}.
Example: the $\circ$-exponential of $f$ in $12=2^2\cdot 3$ is in fact
\begin{equation}
{\rm Exp}(f)(12)=f(12)+2f(2)f(6)+f(3)f(4)+f(2)f(2)f(3);
\label{def:compa}
\end{equation}
On the other hand the Dirichlet exponential implies the important
{\it formal} property:
\begin{equation}
Z_{f}(s)=exp(Z_{{\rm Log} f}(s)),
\label{th:clex}
\end{equation}
which permits to obtain the {\it free energy} series expansion
starting from the partition function series expansion on a Dirichlet series.
%
\section{Convergence in the Multiplicative Case}
%
First keeping within the context of general polymer models,
we set $\psi(X):=\phi(X) z^X$
so that the partition function equals
$Z = \sum_{X\in\hat{P}^{\infty}} \psi(X)$.
Then the probability that the $k$-polymer $X$ is present is defined by
\begin{equation}
\rho(X) := \frac{\sum_{Y\in\hat{P}^{\infty}} \psi(Y\cdot X)}
{\sum_{Y\in\hat{P}^{\infty}} \psi(Y)} =
\sum_{Y\in\hat{P}^{\infty}} \Delta_X(Y)
\label{prob:kpolymer}
\end{equation}
with $\Delta_X(Y) = (\psi^{-1}*D_X \psi)(Y)$.
The terms $\Delta_X(Y)$ meet the following
recursive equation w.r.t.\ addition
of a polymer $\gi\in P$ to $X$:
\begin{equation}
\Delta_{\gi\cdot X}(Y) = z(\gi){\protect \sum_{S\subset Y}}^{\gi}
(-1)^{|S|} \Delta_{X\cdot S}(Y/S).
\label{eq:recursive}
\end{equation}
Here the superscript $\gi$ means that summation is restricted to
multi-polymers $S$ of $Y$ which are incompatible with $\gi$.
We now express the correlation function
\[ \rho_k(X) = <\alpha_{i_1}\cdots\alpha_{i_r}>_{k}(s)=
\sum_{Y\in\bN} \Delta^k_X(Y) \]
at the inverse temperature $s$ as a series in the activities.
By definition $\Delta^k_X(Y) = (\li_k^{-1}\ * D_X\li_k)(Y)\cdot z^X z^Y$,
with $D_n$ defined in (\ref{def:deri}).
Now by (\ref{eq:recursive}) for a prime $p\in \bP_k$
\begin{eqnarray}
\Delta^k_{p\cdot X}(Y)
&=& z_s(p){\protect \sum_{S|Y}}^p (-1)^{|S|} \Delta^k_{S\cdot X}(Y/S) \nonumber\\
&=& z_s(p){\protect \sum_{S: p|S|Y}} \lambda(S) \Delta^k_{S\cdot X}(Y/S),
\label{eq:p:ite}
\end{eqnarray}
since $S\in\bN$ is incompatible with $p\in\bN$ iff $p|S$.
Furthermore for an integer $S$ of the form
$S=\prod_i p_i^{\alpha_i}$ by definition
$|S| = \sum_i \alpha_i = \Omega(S)$.
Moreover, the Liouville function $\lambda$ is defined by
$\lambda(S) = (-1)^{\Omega(S)}$, showing (\ref{eq:p:ite}).
Eqs.\ (\ref{eq:p:ite}) are the iterative equations we want to consider.
Defining the ``index'' of the quantity $\Delta^k_X(Y)$ as
$\Omega(XY)$, the equations (\ref{eq:p:ite}) can be solved iteratively observing that
give the the family of index $\Omega(XY)+1$ in terms of that whose
index is $\Omega(XY)$. This fact not only makes it possible to study
the iterative solutions with the initial condition $\Delta^k_1(1) = 1$
but it also gives hints on the suitable Banach space structure to be
introduced in order to make use of the contraction principle.
%
%
In our number-theoretical context the seminorms $N_m^\delta$
have the form
\begin{eqnarray}
N_m^\delta(\rho_k) &=& \sup_{X\in\bN} \sum_{Y\in\bN,\Omega(XY)=m}
|\Delta^k_X(Y)| e^{-(\ln(a)+\delta)v(X)}\nonumber \\
&=& \sup_{X\in\bN} \sum_{Y\in\bN,\Omega(XY)=m}
|\Delta^k_X(Y)| X^{(A'-\delta')}
\label{nt:seminorm}
\end{eqnarray}
with $A' = -\ln(a)/\ln 2$, $\delta' = \delta/\ln 2$ and $v(X)= \ln(X)/\ln(2)$.
In the multiplicative case $\li=|\mu|$ we can improve the convergence
estimate
to the optimal value.
%
\begin{eqnarray}
N_m^\delta(\rho_k) &=& \sup_{X\in\bN} \sum_{Y\in\bN,\Omega(XY)=m}
|\li_k^{-1} * D_X\li_k(Y)| |z^X z^Y| e^{-(\ln(a)+\delta)v(X)}\nonumber \\
&=& \sup_{X\in\hat{P}_k^\infty} \sum_{Y\in\hat{P}_k^\infty,\Omega(XY)=m}
|\li^{-1} * D_X \li(Y)| |z^X z^Y| e^{-(\ln(a)+\delta)v(X)}.
\label{k:away}
\end{eqnarray}
Remember that $\bP_k$ consists of of the first $k$ primes. So
\[\hat{P}_k^\infty = \{ n\in\bN \mid p\in\bP \mbox{ and } p|n \Rightarrow p\in \bP_k \}.\]
Since we have assumed $\li=|\mu|$, $\li^{-1} = \lambda$.
So
\begin{eqnarray*}
\li^{-1} * D_X \li(Y)
&=& \sum_{d|Y} \lambda(Y/d) |\mu|(Xd) \\
&=& \sum_{d|Y, (d,X)=1} \lambda(Y/d) = \left\{
\begin{array}{ll}
\lambda(Y) & , p|Y\Rightarrow p|X (p\in\bP) \\
0 & , \mbox{otherwise.}
\end{array} \right.
\end{eqnarray*}
Now if there is a $m$-independent bound $b$ on the minimal number of
prime factors of an $X$ which attains the supremum in (\ref{nt:seminorm}),
then $m\mapsto N_m^\delta(\rho_k)$ converges exponentially fast to zero,
since then there are only ${\cal O}(b^m)$ terms in the sum (\ref{nt:seminorm}).
Then we are done.
So we can assume w.l.o.g.\ that the maximal number of prime factors
of the $X$ grows with $m$. Now since $m+1\geq 2$, the
$\tilde{X}\in\bN$ which attain the supremum in $N_{m+1}^\delta(\rho_k)$
are unequal 1 so that we can write
them in the form $\tilde{X_0}=pX$ and assume that
$p\in\bP$ is the largest prime factor.
Then we use the recursion relation (\ref{eq:recursive}):
\begin{eqnarray}
N_{m+1}^\delta(\rho_k) &=& \sup_{\tilde{X}\in\bN}
\sum_{Y\in\bN,\Omega(\tilde{X}Y)=m+1}
|\Delta^k_{\tilde{X}}(Y)| \tilde{X}^{(A'-\delta')}\nonumber \\
&=& \sum_{Y\in\bN, \Omega(XY)=m}
|\Delta^k_{pX}(Y)| (pX)^{(A'-\delta')}\nonumber \\
&=& \sum_{Y\in\bN, \Omega(XY)=m} |z_s(p)\sum_{S: p|S|Y} \lambda(S)
\Delta^k_{XS}(Y/S)| (pX)^{(A'-\delta')}\nonumber \\
&\leq& p^{-\delta'} \sum_{Y\in\bN, \Omega(XY)=m}
\sum_{S: p|S|Y} |\Delta^k_{XS}(Y/S)| (SX)^{(A'-\delta')} S^{(\delta'-A')}
\nonumber \\
&=& p^{-\delta'}\sum_{S: p|S} S^{(\delta'-A')} \sum_{Y:S|Y,\Omega(XY)=m}
|\Delta^k_{XS}(Y/S)| (SX)^{(A'-\delta')}\nonumber \\
&\leq& p^{-\delta'}\sum_{S: p|S} S^{(\delta'-A')} N_{m}^\delta(\rho_k)
\nonumber \\
&=& p^{-A'}\zeta(A'-\delta') N_{m}^\delta(\rho_k)
\label{long}
\end{eqnarray}
Now we assume that $A'>1$ and $\delta'=\eh (A'-1)$. Then as $m$ and thus $p$
become large, the constant $c:=p^{-A'}\zeta(A'-\delta')$ in
\[ N^\delta_{m+1}(\rho) \leq c N^\delta_{m}(\rho) \]
coming from (\ref{long}) is getting strictly smaller than one,
implying convergence.
In other words, we have absolute convergence if $|z(p)|\leq p^{-1-\vep}$
for some $\vep>0$. This is clearly optimal.
\\[1cm]
%
%\newpage
%
{\bf Acknowledgement.} We thank the Erwin Schr\"odinger Institute,
Wien, for the hospitality and the financial support. One of us (P.C.)
thanks Prof. D. Brydges for many enlightening discussions on
polymer models; during this work P.C. has been supported by a Nato-Cnr
advanced fellowship and, partially, by University of Virginia.
%
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