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\documentstyle[12pt]{article}
\title{Algebraic proof of Riemann hypothesis}
\author{Andrzej M\c{a}drecki\thanks{Institute of Mathematics,
Wroc{\l}aw University of Techology, 50-370 Wroc{\l}aw , Poland}}
\pagestyle{normal}
\newtheorem{pr}{Proposition}
\newtheorem{de}{Definition}
\newtheorem{th}{Theorem}
\newtheorem{lem}{Lemma}
\newtheorem{re}{Remark}
\def\thede{\arabic{section}.\arabic{de}}
\def\theequation{\arabic{section}.\arabic{equation}}
\newfont{\lll}{msbm10 scaled 1095}
\def\LC{\mbox{\lll \char67}}
\def\LN{\mbox{\lll \char78}}
\def\LQ{\mbox{\lll \char81}} 
\def\LR{\mbox{\lll \char82}}
\def\LZ{\mbox{\lll \char90}}
\begin{document}
\maketitle

{\bf Abstract}. A concise and elementary proof of the Riemann hypothesis
is given.\\
\section{Introduction}
In 1859 {\bf Bernhard Riemann}[Rie] formulated his famous conjecture (now
called the {\bf Riemann hypothesis} (RH) for short) on the {\bf
nontrivial} solutions of the classical {\bf zeta equation}:
\begin{equation}
\;\;\;\;\zeta(s)\;=\;0\;;\;s\in \LC
\end{equation}
, where $\zeta$ is the {\bf classical Riemann zeta function}
(cf.e.g.[Ch], [KV]).
Henceforce we denote the complex number field by $\LC$.

We start by recalling the well-known functional equation (f.e.
in short) for $\zeta$. It seems that the role of that intermediate equation
has been rather ignored and - maybe - here lies the reason that (RH)
has been open for so long. It is almost sure that the (below) f.e. was
always considered as an indirect step to the obtaining of the following
{\bf Riemann functional equation} (R.f.e. in short)
(cf.[Ch,Sect.II.1,(9)]).

Let $G = G(x) = e^{-\pi x^{2}}$  be the standard  {\bf gaussian fixed point} of
the {\bf Fourier transform} ${\cal F}$ on $\LR$, which moreover belongs
to the {\bf Schwartz space} ${\cal S}(\LR)$. Then the following {\bf
triplet functional equation} (t.f.e. for short) holds 
\begin{equation}
\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s)=\frac{1}{s(s-1)}+\int_{1}
^{+\infty}(x^{s/2-1}+x^{-s/2-1/2})\theta(x)dx
\end{equation}
, where $s\in \LC-\{0,1\}$. Thus, the equation (1.2) combines $\zeta$ (which
is "locally" defined by the {\bf Dirichlet series})
\begin{displaymath}
\;\;\;\;\zeta(s)\;:=\;\sum_{n=1}^{+\infty}\frac{1}{n^{s}},\;\;Re(s)>1,
\end{displaymath}
with three important meromorphic functions:\\
(i) the {\bf Gamma function} $\Gamma$ (defined "locally" by the
integral)
\begin{equation}
\;\;\Gamma(s):=\int_{0}^{+\infty}t^{s-1}G(\sqrt{\frac{t}{\pi}})dt;\;\;\;Re(s)
>0
\end{equation}
 (Henceforth we denote the real part of a complex number $s$ by $Re(s)$),\\
(ii) the {\bf theta-function} $\theta$ defined locally as the series 
\begin{displaymath}
\;\;\;\;\theta(s)\;:=\;\sum_{n=1}^{+\infty}G(n\sqrt{s});\;\;\;Re(s)>0.
\end{displaymath}

Finally we see that with $\zeta$ is associated the {\bf rational
function}
(defined globally by the sum):\\
\begin{equation}
(iii)\;\;\;W(s) :=\frac{1}{s(s-1)};\;\;\; s \in \LC- \{0,1\}
\end{equation}
now observe that the nature of the functions $\Gamma$ and $\theta$ is
deeply analytic , whereas $W$ has a {\bf simple algebraic character}.
(It was communicated to me by H.Iwaniec that $s(s-1)$ is a quite special
and very important expressions : they are the {\bf eigenvalues of the
Laplace operator} on a hiperbolic manifold).
Since we shall consider a few additional functional equations for $\zeta$, 
we call the f.e. (1.2) the{\bf triplet functional equation} (t.f.e. in short)
for $\zeta$.

The triplet $(\Gamma,\theta,W)$ will be called the {\bf
$\zeta$-triplet} of functions, and below we derive the most important -
the following {\bf algebraic type conjecture} on $\zeta$, that $I(s) :=
Im(W(s))$ is mainly responsible for the form of the Riemann hypothesis
(RH).
The t.f.e. imposes in a natural way some additional notation:
\begin{equation}
\;\Gamma^{*}(s)\;:=\pi^{-s/2}\Gamma(\frac{s}{2}) ;
\end{equation}
and
\begin{equation}
\zeta^{*}(s)\;:=\;\pi^{-s/2}\Gamma(\frac{s}{2})\zeta(s)\;=\Gamma^{*}(s)
\zeta(s).
\end{equation}
Under these notations the t.f.e immediately implies the following {\bf
Riemann functional equation} (R.f.e. in short)
\begin{equation}
\;\;\;\zeta^{*}(1-s)\;=\;\zeta^{*}(s); \;\;\;s\in \LC,
\end{equation}
which shows the importance of the -so called - {\bf "modular"} (or
$\zeta^{*}$-invariant) map $m(s)
:= 1-s$ on $\LC$. The modular map $m$ gives an "open"
symmetry of $\zeta^{*} : \zeta^{*}\circ m = \zeta^{*}$. Opposite to
that, we have got some very "hidden" (and hence deep) {\bf Galois
symmetry} of $\zeta^{*}$ - which is crucial in our proof of (RH).

Let $G = Gal(\LC/\LR) = \{id_{\LC},c\}$ be the {\bf Galois group} of
$\LC$, i.e. $id_{\LC}$ denotes the identity map of $\LC$ and $c$ is the
{\bf complex conjugation}
\begin{displaymath}
c(z)\;=\;c(u+iv)\;:=\;u-iv.
\end{displaymath}
Moreover , if we introduce the {\bf affinic-antyconjugation} $a:\LC
\longrightarrow \LC$ by the formula:($a$ is included into $m$))
\begin{displaymath}
a(u+iv)\;:=\;(1 - u) + iv
\end{displaymath}
, then $m = a\circ c = c\circ a$ (i.e. $a$ {\bf commutes} with $c$).
It is almost obvious - that such an important {\bf invariant} of the field
$\LC$ - like $G$ (although very simple) , must have some connections
with (RH).

Observe that the Riemann hypothesis can be written as the
following implication:
\begin{displaymath}
(RH)\;\;\;if\;\zeta(s)=0\;and\;Im(s) \ne 0\;, then\;Re(s)=1/2.
\end{displaymath}

The complex rational function $W$ of one variable (being the inversion
of the quadratic polynomial $P_{W}(s) = s^{2}-s$) without constant term 
naturally determines {\bf two real, rational functions of two real variables}:
the more important (and fundamental in fact) function :
\begin{equation}
\;\;Im(W)(s)\;=\;\frac{Im(s)(1-2Re(s))}{\mid s(s-1)\mid^{2}},\;s\in
 \LC-\{0,1\}
\end{equation} 
(as usuall $Im(s)$ and $\mid s \mid$ denote the imaginary part and
module of a complex number $s\in \LC$), and the less important function
\begin{equation}
\;\;Re(W)(s)\;:=\;\frac{Re(s)(Re(s)-1)-Im^{2}(s)}{\mid
s(s-1)\mid^{2}},\; s\in \LC -\{0,1\}.
\end{equation}
In particular (1.8) and (1.9) determine two polynomials of two
variables $I,R \in \LZ[U,V]$ : the {\bf fundamental}
\begin{equation}
I(U,V):=V(2U-1)=-Im(W)(s)\mid s(s-1)\mid^{2}=I(s),\;s=U+iV,
\end{equation}
and the less important
\begin{equation}
R(U,V):=U(U-1)-V^{2}=Re(W)(s)\mid s(s-1)\mid^{2}=R(s),\;s=U+iV.
\end{equation}
Observe that $I$ is a linearly-affine form of $(U,V)$ whereas $R$
is quadratic and $\partial^{2}_{xy}I\ne 0$, whereas $\partial^{2}_{xy}R
= 0$ , so there is some essential algebraic and analytic difference
between $I$ and $R$.

Observe that the set of all {\bf poles} of $\zeta$ :
$\zeta^{-1}(\LC) = \{s\in \LC:\zeta(s) = \infty\}$ is contained in the
intersection $R(\LC)\cap I(\LC)$, i.e.
\begin{equation}
\;\;\;\;\;\zeta^{-1}(\LC)\subset R(\LC)\cap I(\LC)
\end{equation}
where $R(\LC) = \{s\in \LC : R(s) = 0\}$ and $I(\LC) = \{s\in \LC :
I(s)=0)\}$.

According to the form (1.10) of $I$ we can obtain the following
remarkable implication (in this paper this implication will be called
the {\bf Minor Riemann Hypothesis} ((MRH) in short), according to its
formal resemblance (likeness) to (RH):
\begin{equation}
(MRH)\;\;\;\;If\;I(s)=0\;and\;Im(s)\ne 0\;then,\;Re(s)=1/2.
\end{equation}

In this paper we will see that there exists a "dual" hypothetical
statement to (1.12): the set of {\bf all zeros of $\zeta$} : $\zeta(\LC)
=\{s\in \LC: \zeta(s)=0\}$ is contained in the set $I(\LC)$, i.e. 
\begin{equation}
\;\;\;\;\;\;\zeta(\LC)\subset I(\LC).
\end{equation}

The inclusions (1.12) and (1.14) are very surprising. They suggest that
 {\bf topological information} about isolated points of the
meromorphic function $\zeta$ is written de facto in the {\bf algebraic
$\LR$-varieties} $I(\LC)$ and $I(\LC)\cap R(\LC)$, where $I$ and $R$ are
polynomials from $\LZ[U,V]$ of degree 2.\\
(Obviously only zeros and poles are - in fact - interesting, if we
consider a meromorphic function from the "topological" point of view).

The aim of this paper is thus to explain the deeper sense of the {\bf
Riemann hypothesis} as some relationship between the cycles : the
holomorphic manifold $\zeta(\LC)$ and the algebraic cycle $I(\LC)$ of
the affine variety $\LR^{2}$. We are going to show that a formal
similarity between (RH) and (MRH) is not only  "optical," but there exists
some {\bf unexpected} (and hence {\bf deep}) relation between the {\bf
arithmetic} of the polynomial $I$ from $\LZ[U,V]$ over $\LR$ and the
{\bf arithmetic} of $\zeta$ over $\LC$). To be more exact, when
thinking over these ideas it becomes more and more apparent to us that the
Riemann hypothesis on $\zeta$ could be reduced to the formal
consequence of (MRH).

In this paper we will show - what was also quite unexpected for us -
that the reduction mentioned above can be obtained by using only simple
calculations of integrals and is quite elementary. (At this moment it
would be useful to recall the very short Faltings proof [Fa] of the
{\bf theorem on the finitness of the number of $l$-adic characters of
representations of the Galois group of the algebraic closure of an
algebraic field}, which was the "psychological catalyst" for our proof.
Although commonly it was considered as very hard to prove ( e.g.
Serre [Se1] has showed that in the case of elliptic curves it follows
from the {\bf generalized Riemann hypothesis}) Faltings has given
a proof "on one page" using the old {\bf Hermite theorem} from the Galois
theory of finite extensions and the technical {\bf Nakayama lemma} (it
is also necessary to apply the well-known {\bf Chebotarev density
theorem}(cf.e.g. [Se1]))).

The imaginary part of t.f.e. obviously has the form
\begin{equation}
(Im(t.f.e.))\;Im(\zeta^{*}(s))
=\int_{1}^{+\infty}(x^{\frac{(u-2)}{2}}-x^{-\frac{(u+1)}{2}})sin
(\frac{vlnx}{2})\theta(x)dx-\frac{I(s)}{\mid s(s-1) \mid^{2}};
\end{equation}
$s = u+iv$.

If we delete the "perturbation" $\theta = \theta(x)$ from the
integral in $Im(t.f.e.)$ and change the variables according to the
substitution $x=e^{2y}$, then we obtain
\begin{displaymath}
\int_{0}^{+\infty}(e^{uy}-e^{(1-u)y}) sin(vy)dy\;\;=\;\;\frac{I(s)}{\mid
s(s-1)\mid^{2}}\;;\;s=u+iv.
\end{displaymath}
The above formula (or rather {\bf rationality} of the integral as a function
of $s$) was the direct catalyst of our proof.

Finally, all this leads to the following main result of
this paper - the {\bf Riemann hypothesis functional equation} (R.h.f.e in short)
with the {\bf rational term} $I$ and the {\bf action of the {Galois
group}} $Gal(\LC/\LR)$.
We have just mentioned that such an important {\bf invariant} of $\LC$ as
$Gal(\LC/\LR)$ must have some relation with (RH) and it is really so:
RH is an immediately consequence of (R.h.f.e.) (similarly as in the
case of Matti Pitkanen's analytic proof of (RH) [Pi] , RH is an
immediate consequence of the {\bf spectral Hilbert-Polya conjecture}
proved by him). 
Observe that in fact the "secret action" of $Gal(\LC/\LR)$ is written
in the following $(Rhfe_{G})$.

The fundamental equation mentioned above has the form: there exists
such multiplicators $F_{1}(s), F_{2}(s)$ and $f_{1}(s),f_{2}(s)$ with
$f_{1}-f_{2} \ne 0$ that
\begin{equation}
(R.h.f.e_{G})\;\;Im(\sum_{g\in Mod(\LC)}(F_{1}\zeta)(g(s))\;+\;
(F_{2}\zeta)(c(g(s)))\;=\;\frac
{(f_{1}(s)-f_{2}(s))}{\vert s(s-1)\vert^{2}}I(s).
\end{equation}

(Assume that for some $s$, with $Im(s)\ne 0$ and $Re(s)\ne 1/2$, is
$\zeta(s) = 0$. Then also $\zeta(c(s)) = 0$ and the left-hand side of
$(R.h.f.e._{G})$ is zero whereas the right-hand side {\bf is not}. This proves
(RH)).

$(R.h.f.e._{G})$ is the crowning of the {\bf algebraic type}
conjecture, which we derived during our first visit in U.S.A. ten years ago 
and introductory discused with Chris Burdzy. 
That equation thus proves the {\bf main algebraic conjecture} posed by us ten
years ago : 
\begin{equation}
(MAC)\;\;\;(SRH)\Longrightarrow (RH).
\end{equation}
In [M\c{a}2] and [M\c{a}3] we proved (MAC) in a quite different manner. In
[M\c{a}2] we have proved that (MAC) is an immediate consequence of the
{\bf existence} of the so called {\bf Hodge measure $H_{2}$} (i.e.
$\sigma$-additive positive measure on the Banach algebra
$C_{0}[0,1]^{2}$, where $C_{0}[0,1]$ is the Banach algebra of all
continuous functions $f$ on $[0,1]$ with $f(0) =1$), which gives the
following {\bf Abstract Hodge Decomposition} $(AHD_{2}$) of the 2-dimensional 
{\bf Green function}:
\begin{equation}
(AHD_{2})\;\;\;\mid s \mid^{-2}\;=\;\int_{C_{0}[0,1]\times C_{0}[0,1]}
e^{[s,c]}dH_{2}(c),\;\;;s\in \LC^{*}.
\end{equation}
Here $[.,.]$ is some (rather complicated) bilinear form on
$C[0,1]\times C[0,1]$ (the so called Mikolasch-Koch error). We can
succintly say that existence of $AHD_{2}$ means that the 2-dimensional Green
function is a {\bf functional Laplace transform} of $H_{2}$, i.e. $\mid
s \mid^{-2} = \hat{H_{2}}(s)$ (cf.[M\c{a}2,Th.1]).
 
$(AHD_{2})$ implies in an elementary way the $(R.h.f.e._{2})$ of the
form (cf.[M\c{a}2, Th.2])
\begin{equation}
(Rhfe_{2})\;\;\;\; -\frac{Im(\zeta^{*}(s))}{Tr_{2}(M_{G}M_{s})}\;=\;I(s),
\end{equation}
where the "analytic Riemann trace" $Tr_{2}(M_{G}M_{s})$ is given by the formula
\begin{equation}
\;\;Tr_{2}(M_{G}M_{s})\;:\;\sum_{n=0}^{+\infty}\sum_{j=0}^{+\infty}
\frac{(-\pi n^{2})^{j}}{j!}\cdot\frac{(4j+1)}{\mid
(s-2j)(s+2j-1)\mid^{2}},
\end{equation}
and  $0< Tr_{an}(M_{G}M_{s}) <+\infty$.
(It is very convenient to choose a {\bf non-canonical} scenario:
$0^{0}=1$).
However an anonymous referee from Crelle's Journal in his report [CJR]  on
[M\c{a}2] dated February 26, (1999) discovered numerical counterexamples to
$(R.h.f.e._{2})$. He denoted $h(s)=\zeta^{*}(s)/\mid \zeta^{*}(s)\mid$ and
obtained:
\begin{displaymath}
h(0.6+15.4i)\;=\;0.9999989257...+i0.0014657814...,
\end{displaymath}
\begin{displaymath}
h(0.9+20i)\;=\;0.7882729297...+i0.6153257577....
\end{displaymath}
The reason and explanation of this contradiction is that the proof of
$(Rhfe_{2})$ is strongly based on the existence of the Hodge measure $H_{2}$.
However, as shortly after was observed by M.Bo\.zejko, {\bf on the ground
of the classical logic}, the measure $H_{2}$ {\bf cannot exist}, since
$\hat{H_{2}}(s)$ is {\bf positive definite} in the Laplace-Haenkel sense on
$\LC^{*}$ whereas the Green function  $\mid .\mid^{-2}$ is not.
Herceforth this is refered to as the {\bf Bo\.zejko paradox}.

However, if we reject the {\bf Tertium non Datur} (TnD in short), (then
also ad absurdum method of proof), i.e. we agree to work on the
ground of {\bf Brouwer intuicionistic logic}, then $H_{2}$ exists.
In the construction of $H_{2}$, in a secret way appears famous
diagonal type construction from the {\bf Godel theorem}. Constructions
have been known for a hundred years which exist only on the level of Brouwer 
logic (or the set theory is contradictory !): 
for example {\bf diagonal Brouwer set} of integers $D\LN$ (for which
the statement $(n\in D\LN)\lor (notin D\LN)$) or the {\bf Specker sequence} 
$\{s_{n}\}$ (monotonic, bounded and divergent) (cf.[Br],[Sp],[Ric],[ML]).

Since we could not check directly the correctness of the computer calculations
and did not entirely believe them, we sent the paper to the Acta
Arithmetica Journal. An anonymous referee from AA (cf.[AAR]) wrote :"we see
therefore that the claimed "Riemann hypothesis equation" $(Rhfe_{2})$ 
together with $0<Tr_{2}(M_{G}M_{s})<\infty$ implies both the Riemann hypothesis
and its negation. This shows obviously that at least one of the formulae cannot
be true". We add: "obviously" on the ground of {\bf classical logic}. 
Moreover, we see that on the ground of Brouwer logic , (RH) is exactly the 
statement which {\bf violates} (TnD) itself.

The series of {\bf de Branges' papers} devoted to the proof of (RH): [dB1-dB5]
has been known since 1994. His "proof" was in many points very similar to our
proof in [M\c{a}2]: he also uses some integral transforms (in the adic case)
and shows that (RH) is a consequence of the positivity of his-
so-called {\bf de Branges trace} :$Re<F(z),F(z+i)>_{{\cal H}(E)} \ge 0$. More
exactly : let $E(z)$ be an entire function satysfying $\mid E(c(z))\mid
<\mid E(z) \mid$ for $z$ in the upper half-plane. A Hilbert space of entire
functions ${\cal H}(E)$ is the set of all entire functions $F(z)$ such that
$F(z)/E(z)$ is square integrable on the real axis and such that
\begin{equation}
\;\;\;\mid F(z)\mid^{2} \le \mid\mid F \mid\mid^{2}_{{\cal H}(E)} K(z,z)
\end{equation}
for all complex $z$, where the inner product of the space is given by
\begin{displaymath}
\;\;\;<F(z),G(z)>_{{\cal
H}(E)}\;:=\;\int_{-\infty}^{+\infty}\frac{F(x)c(G(x))}{\mid E(x)
\mid^{2}}dx
\end{displaymath}
for all elements $F,G \in {\cal H}(E)$ and where
\begin{displaymath}
\;\;K(w,z) \;=\;\frac{E(z)c(E(w))-c(E(c(z)))E(w)}{2\pi i(c(w)-z)}
\end{displaymath}
is the reproducing kernel function of the space ${\cal H}(E)$, that is,
the identity
\begin{displaymath}
\;\;\;F(w)\;=\;<F(z),K(w,z)>_{{\cal H}(E)}
\end{displaymath}
holds for every complex $w$ and for every element $F\in {\cal H}(E)$.
The above identity is obtained by using Cauchy's integration formula in
the upper half-plane (cf.[dB1]), and the condition (1.2) is made so that
Cauchy's formula applies to all functions in the space ${\cal H}(E)$.
In other words ${\cal H}(E)$ is a RKHS (Reproducing Kernel Hilbert
Space).
According to de Branges results for $E(z)$ and some additional "technical
conditions"(cf.[dB2-dB5]), if
\begin{equation}
\;\;\;Re<F(z),F(z+i)>_{{\cal H}(E)}\;\;\;\ge\;0
\end{equation}
for every element $F(z)\in {\cal H}(E)$ with $F(z+i)\in {\cal H}(E)$,
then the zeros of $E(z)$ lie on the line $Im(z) =-1/2$ (i.e. satisfies
(RH)) and moreover  $Re<c(E)^{\prime}(w)E(w+i)/2\pi i> \;0\ge$  when $w$ 
is a zero of $E(z)$ (i.e. belongs to the algebraic variete $E(\LC)$).
Finally he applied the theory in the case : $E(z) =
(iz-1)iz\zeta^{*}(1-iz)$ to obtain (RH).

However Conrey and Li in [CL] showed the falsity of de Branges's approach to
(RH). They used MATHEMATICA and numerically proved that de Branges'
{\bf positivity conditions}, which imply the generalized Riemann
hypothesis, are not satisfied by defining functionsof the reproducing
kernel Hilbert spaces associated with the Riemann zeta function
$\zeta(s)$ and the Dirichlet $L$-function $L(s,\chi _{4})$
(cf.[CL,(3.2),(3.4)and Section 3.2]).

My explanation of this phenomen is the same as above : the adic
construction of de Branges {\bf violates} (TnD) and exists only on the level
of Brouwer logic!

In [M\c{a}3] we proved the existence of a measure quite different from
the $H_{2}$ {\bf Hodge measure} $H_{0}$. The measure $H_{0}$ also {\bf exists
only} on the level of Brouwer logic (according to the {\bf Hardy-Littlewood
theorem}, which asserts that on the critical line $Re(s)=1/2$ there exists
infinitely many zeros of zeta : $\mid \{s\in \LC: \zeta(s) = 0\}\mid =+\infty$ ).

$H_{0}$ gives the Abstract Hodge Decomposition $(AHD_{0})$ of {\bf
sub-Dirac} functions $D_{0}(t)$ , i.e. such {\bf positive} functions on
$\LR$, which satisfy the following three conditions : $(i) D_{0} =1,
(ii) support(D_{0})(0)\subset[-1,1]$ and $(iii) \int_{\LR}D_{0}(t)dt\le 1$. Then on the
a functional space $\Omega_{0}$ of {\bf fixed points $\omega$ of} ${\cal F}$
with $\omega(0) =1$ (endowed with a Banach space structure) there
exists a Borel probability measure $H_{0}$, which gives $(AHD_{0})$ mentioned
above (with the {\bf bilinear} kernel):
\begin{equation}
(AHD_{0})\;\;\;\;D_{0}(t)\;\;=\;\;\int_{\Omega}\omega(t)dH_{0}(\omega)\;\;\;
;t\in\LR.
\end{equation}
In other words Dirac functions (not distributions) $D_{0}$ are
integral transforms of Hodge measures $H_{0}$ w.r.t. bilinear kernels on the
space of fixed points of Fourier transforms, i.e.
$D_{0}(t) = \hat{H_{0}}(t), t\in \LR$.

We thus see the crucial role of the set $\Omega_{0}$ for the Riemann hypothesis
problem. This role is also explored in this paper.
(At this moment we remark that the Riemann hypothesis in the case of
{\bf Hasse-Artin zetas} associated with the algebraic varieties over finite
fields (Weil conjectures - Grothendieck-Deligne theorems) - is a
consequence of the {\bf Lefschetz fixed points formula} in Weil cohomologies.
Hence the surprising role of the general fixed point theory (cf.e.g. [DG])
in the Riemann problem).
In [M\c{a}3] we also proved that $(AHD_{0})$ implies the existence of a functional
equation of the  following form: there exists such a sequence of symmetric
sub-Dirac functions $D^{n}_{0}(t)$  that
\begin{equation}
(R.h.f.e._{0})\;\lim_{n\rightarrow \infty}\int_{\Omega_{0}}
Im((\Gamma(\omega+(D_{0}^{n}+\hat{D}^{n}_{0}))\zeta)(s))dH_{0}(\omega)\;=\;
\frac{H_{0}(\Omega_{0})I(s)}{\mid s(s-1)\mid^{2}}
\end{equation}
, which obviously immediately implies (RH).

Brouwer logic results need - it seems - some explanations. Firstly, we
propose the following explanation: according to the famous {\bf Godel theorem}
on the non-solvalibity of axiomatic theories [Go] we cannot prove that even
integer arithmetic is not contadictory. Hence, we can only believe that it
is so! But belief is only belief - and for example - I believe that it is
true in the case of arithmetic. But I also believe that in the axiomatic Zermelo-
Frankel-Choice-axiom (ZFC in short) - from set theory  - (a contradiction is
possible).  The paradoxes of Cantor and Russel in early "naive" and non-
axiomatic Cantor set theory shows that it is possible.
The consructions from [M\c{a}2] and [M\c{a}3] use only standard maths methods
of set and measure theory. So there are no natural reasons to reject
these constuction and maybe therefore is now time to acknowledge that Brouwer logic
is as good as classical logic (as was done -for example -in informatics
(cf.e.g. [De]). Obviously we have to pay some price for such a weakness of classical 
logic : (RH) is both true and false !, i.e. $v(p_{RH}\land \sim
p_{RH})=1$, where $v(p)$ denotes the classical logic value of the logical
statement $p$.

During the recent edition of this paper M. Bo\.zejko informed me that
{\bf Matti Pitkanen} [Pi] proved the {\bf Hilbert-Polya conjecture}
on $\zeta$ using methods from conformal field theory (originated
in quantum physics). In his proof one of the main roles is played by the
function $\mid z \mid^{2}$ although in this case as eigenvalues of the
(semi-) Hermitian operator $H$, which gives the Hilbert-Polya conjecture.
An idea for proving the Riemann hypothesis was to give a {\bf spectral
interpretetation} of the zeros. That is, if the zeros can be
interpreted as the eigenvalues of $1/2+iT$, where $T$ is a {\bf
Hermitian} operator on some Hilbert space, then since the zeros of a
Hermitian operator are real, the Riemann hypothesis follows. This idea
was originally put forth by Polya and Hilbert (cf.e.g. [Ed]), and serious 
support for this idea was found in the resemblence between the "explicit formulae"
of prime number theory, which go back to Riemann and Von Mangoldt, but
whch were formalized as a duality principle by Weil, on the one hand,
and the Selberg trace formula on the other.

At the beginning of this year, when I heard a short measure theoretic proof of the
following combinatorial conjecture proved by M.Morayne in [Mo]:if the 
{\bf chromatic number} $\chi(\LR^{2})$ of the complex plane is equal
$7$,then the underlying graph must have 6800 apexes, then I realized that if there exists
a short proof of (RH), then it is possible on the ground of the
classical logic. In this case we do not need the complicated machinery of
Hodge measures and decompositions, based on set theory, which can
leads to a contradiction (and in addition we must to work on the ground of
Brouwer logic). In February, the possibility of such a proof was
showed by Matti Pitkanen in [Pi]. So, the paper [Mo] was the direct motivation 
of this article.
  
According to {\bf Poincare} [Po] the unique thing, which we must demand
from an object which exists in mathematics is {\bf non-contradictivity}
(although in the light of resent results it is also problematic).
Moreover that requirement could not be kept, as in the light of Godel's
results, it was not possible to keep the requirement of the completness of
the majority of axiomatic systems.

In the approach of Poincare mentioned above, the Cantor set theory was "bad",
because it lead to the well-known contradictions : the Cantor and Russell
paradoxes, " but contradictions which can expose themselves - we cannot
forecast"[Po, BookII. Section III(Maths and Logic. Introduction)].
Obviously, if the early Cantor set theory is not bad, then we must reject TnD
- what Brouwer has done (for many Brouwer has poured out "the baby with
the water"!).

Obviously ZFC-axioms remove contradictions mentioned above ( of Cantor
and Russel type) but we do not know if some new paradoxes "do not expose
themselves" [Po].

Now ( at the beginning of 3rd millennium) we know , that it really is so
- e.g. the fundamental examples mentioned here from constructive maths.

I think that one of the reasons that (RH) was an open problem for so long -
is the problems considered above, together with the famous "crisis of
fundaments of maths" since the beginning of the XX-centaury. This crisis
has still not expired.

At the end of this long introduction we will try also to explain why in
the title of this paper the word "algebraic" figures. That is in
contadiction to the completely analytic proof of (RH) of Matti Pitkanen
[Pi], our proof uses purely {\bf algebraic objects} such as : $I(s),$
$s(s-1), \mid. \mid^{2}, Gal(\LC/\LR)$ and {\bf Cramer systems}. Moreover it also
refers to - in some sense - the purely algebraic (cohomological) proof of
the {\bf Weil conjectures} ( the Dwork-Grothendieck and Deligne's theorems).

\section{Three important transforms : Mellin, Fourier and $\theta$
associated with the Riemann hypothesis}

For a large class of {\bf $\Gamma$-admissible} functions
$f:\LR_{+}\longrightarrow \LC$ the {\bf Mellin
transform} M(f) (or {\bf Gamma $\Gamma(f)$ associated with $f$ }) is
well-defined as
\begin{equation}
\Gamma(f)(s)\;:=\int_{0}^{+\infty}x^{s-1}f(x)dx\;=:\;M(f)(s)\;\;;Re(s)>0.
\end{equation}
(The {\bf Schwartz space} ${\cal S}(\LR)$ of rapidly decreasing functions
constitutes a small class of $\Gamma$-admissible functions).

Recall that $f : \LR\longrightarrow \LC$ belongs to ${\cal S}(\LR)$,
iff for each positive integer $k, l$
\begin{displaymath}
\;\;\;p_{k,l}(f)\;:=\;sup_{x \in \LR} \mid x^{k}
f^{(l)}(x)\mid\;<\;+\infty .
\end{displaymath}
The {\bf Frechet space topology} on ${\cal S}(\LR)$ is given by a family
of seminorms $\{p_{k,l}: k,l \in \LN\}$ (it is a locally convex
topology).
All we need on $\Gamma(f)$ is given in the following
\begin{lem}
If $f\in {\cal S}(\LR)$, then $\Gamma(f)(s)$ is a well-defined complex number
for all $s$ with $Re(s)>0$.
\end{lem}
{Proof}. Obviously 
\begin{displaymath}
\Gamma(f)(s)\;=\;\int_{0}^{1}f(x)x^{s-1}dx\;+\;\int_{1}^{+\infty}f(x)x
^{(s-1)}dx .
\end{displaymath}
Moreover
\begin{displaymath}
\int_{0}^{1}\mid f(x) x^{(s-1)}\mid dx \le p_{u-\epsilon,0}(=sup_{x\in
\LR}\mid x^{u-\epsilon}f(x)\mid)\int_{0}^{1}\frac{dx}{x^{1-\epsilon}}<+\infty,
\end{displaymath}
and
\begin{displaymath}
\int_{1}^{+\infty}\mid f(x)x^{s-1}\mid dx <\;p_{u+\epsilon,0}(f)\int_{1}
^{+\infty}\frac{dx}{x^{1+\epsilon}}<\infty,
\end{displaymath}                          
if $Re(s)>0$.

Also for another large class of {\bf $\theta$-admissible} functions
$f : [1,+\infty)\longrightarrow \LC$ {\bf theta function}
$\theta(f)$  {\bf associated with} $f$ is defined as the series
\begin{equation}
\theta(f)(x)\;:=\;\sum_{n=1}^{+\infty}f(nx)\;=\;\int_{\LN^{*}}
f(nx)dc(x)\;\;;x>0,
\end{equation}
where $dc$ denotes the {\bf calculating measure} on $\LN^{*}$.

By ${\cal F}f$ we denote the {\bf Fourier transform} of $f$ (for ${\cal
F}$-admissible functions):
\begin{equation}
{\cal F}(f)(x)\;:=\;\int_{\LR}e^{2\pi ixy}f(y)dy\;=:\;\hat{f}(x)\;\;;x\in
\LR.
\end{equation}

The Schwartz space ${\cal S}(\LR)$ is {\bf admissible} for the all
integral transforms considered above: $M, \theta, {\cal F}$.\\

In this paper more convenient will be work with the {\bf plus-Fourier
transform ${\cal F}_{+}$} defined as
\begin{equation}
\;\;{\cal F}_{+}(f)(x)\;:= \;\int_{0}^{+\infty}e^{2\pi
ixy}f(y)dy\;=:\hat{f}_{+}(x)\;;\;x\in \LR_{+}.
\end{equation}

\section{Quasi-fixed points of ${\cal F}$ and (RH)}

This section underline the role and importance of the nation of a
quasi-fixed point of ${\cal F}$ and the Riemann hypothesis problem.

A complex function $\omega$ on $\LR$ is called a {\bf fixed point of} ${\cal
F}$, if it is an {\bf eigenvector} of ${\cal F}$ with an {\bf
eigenvalue} equal to $1$, i.e. 
\begin{displaymath}
\;\;\;{\cal F}(\omega)\;=\;\hat{\omega}\;=\;\omega.
\end{displaymath}
Probably, the best known fixed point of ${\cal F}$ is the {\bf standard
gaussian fixed point} $G$ of ${\cal F}$ , given by the formula:
\begin{equation}
\;\;\;G(x)\;:=\;e^{-\pi x^{2}}\;\;\;x\in \LR.
\end{equation}
In fact $G$ is {\bf only} an one function from the family of {\bf
gaussian fixed points} $G_{\sigma}$ of ${\cal F}$, where
\begin{equation}
G_{\sigma}(x)\;:=\;\frac{(\sigma^{-1/2}e^{-\pi x^{2}/\sigma}+e^{-\pi
\sigma x^{2}})}{1+\sigma^{-1/2}}\;;\;x\in \LR.
\end{equation}
We see that $G_{\sigma}(0) =1, G_{\sigma} \in {\cal S}(\LR)$ and ${\cal
F}$.
But, subsequently, $G_{\sigma}$ is a special case of the more general
construction, which is based on the WELL-KNOWN fact that Fourier
transform is an {\bf idempotent} (a part of {\bf Plancherel theorem})
on the class of ${\cal F}$ -admissible symmetric functions.
More exactly,if $f$ is a ${\cal F}$-admissible and {\bf symmetric}
(i.e. $f(-x) = f(x)$) function on $\LR$, then according to the {\bf
Plancherel theory}9cf.e.g. [Ha, Th.3.13]
\begin{equation}
\;\;\;\omega\;:=\;f\;+\;\hat{f},
\end{equation}
is a {\bf fixed point} of ${\cal F}$.
The definition (?) is a special case of the fundamenthal nation which
we are going to explore in this paper. Jest to najistotniejsze
uogolnienie, "gwozdz programu", esencja (ekstrakcja)z fixed punktu
We propose the following definition 
\begin{de}(Of a quasi-fixed point of ${\cal F}$).\\
Let $f:\LR \longrightarrow \LC$ be {\bf symmetric} and ${\cal
F}$-admissible. Let $\lambda \in \LC$
be arbitrary parameter. We say that a function $F_{\lambda} =
Q_{\lambda}(f) :\LR \longrightarrow \LC$ is a {\bf quasi-fixed point of
{\cal F}} (associated with $f$ and the {\bf spectral} parameter
$\lambda$) iff
\begin{equation}
(QF)\;\;F_{\lambda}(x)\;=\;Q_{\lambda}(f)(x)\;:=\;f(x)+\lambda \hat{f}(x),
\end{equation}
$x \in \LR$.
\end{de}
\begin{re}
Let $\omega$ be a {\bf fixed point} of ${\cal F}$. Then obviously
$\omega$ is a {\bf quasi-fixed point} of ${\cal F}$ :
\begin{displaymath}
\omega\;=\; Q_{1}(\ omega/2) \;=\;\frac{\omega}{2}+
\hat{\frac{\omega}{2}}.
\end{displaymath}
Thus we see that the notion of a quasi-fixed point is a direct
extension of the notion of a fixed point.
\end{re}
Quasi-fixed points have got a lot of nice algebraic properties:\\
(P1) $Q_{1}(f)$ is always a fixed point (i.e. $\hat{Q_{1}(f)} = Q_{1}(f)$)
and as we observed in the above remark each fixed point $f$ has the
form : $Q_{1}(f/2)$.\\
(P2) For each symmetric $f$  and arbitrary $\lambda$  , the sum :
$Q_{\lambda}(f)+\hat{Q_{\lambda}(f)}$ is a fixed point of ${\cal
F}$.\\
(P3) Each quasi-fixed point $F_{\lambda}$ is an {\bf idempotent} elemnt
of ${\cal F}$, i.e. $\hat{\hat{F_{\lambda}}} = F_{\lambda}$ .
By $Q\Omega$ we denote the set of all {\bf quasi-fixed points} of
${\cal F}$  and by $\Omega$ the set of all fixeed points which satisfies the
assumptions of the {\bf Poisson Summation Formula}
(PSF in short (cf.e.g.[Ch,II,Th.1], [Ma,Th.XVIII.26],[Na],[Fe])) 
\begin{displaymath}
(PSF)\;\;\;\;\sum_{n\in \LZ}f(n)\;=\;\sum_{n \in
\LZ}\hat{f}(n).
\end{displaymath}
\begin{re}
Let's consider the Banach space $l^{1}(\LR)$ of all absolutely
convergent real sequences with the norm $\vert\vert x
\vert\vert_{l^{1}}:= \sum_{n\in \LZ}\vert x_{n} \vert$ if $x = \{x_{n}\}
\in \l^{1}(\LR)$. Then on the cone of such $f$, that both $f$ and
$\hat{f}$  are real and positive -(PSF) asserts that ${\cal F}$ is an izometry
, i.e. 
\begin{displaymath}
\;\;\;\vert\vert \hat{f}\vert\vert \;\;=\;\;\vert\vert f \vert\vert. 
\end{displaymath}
Observe that the (PSF) is in fact a theorem on $\theta(f)$.
\end{re}
\begin{lem}(Functional equation for $\theta$).\\
For an arbitrary quasi-fixed point $F_{\lambda} \in Q\Omega$ and $x>0$ we have
\begin{equation}
(\theta.f.e.)\;\\;\;\;\theta(F_{\lambda})(x)\;=\;\frac{1}{2x}(2\theta(
\hat{F}_{\lambda}) (\frac{1}{x})+\hat{F}_{\lambda}(0))-\frac{F_{\lambda}}{2}.
\end{equation}
\end{lem}
{\bf Proof}. If we set $F_{\lambda}^{x}(y) = F_{\lambda}(xy); x>0,y\in \LR$, in
the PSF, and observe that
\begin{displaymath}
\hat{F_{\lambda}^{x}}(y) \;=\;\int_{\LR}e^{2\pi iyz}F_{\lambda}(xz)dz\;=\;
\{xz = w, dz=\frac{dw}{x}\}\;=
\end{displaymath}
\begin{displaymath}
=\frac{1}{x}\int_{\LR}e^{2\pi i\frac{yw}{x}}F_{\lambda}(w)dw\;=\;
\hat{F_{\lambda}} (\frac{y}{x})/x,
\end{displaymath}

then we have the $\theta$-relation
\begin{equation}
\sum_{k\in\LZ}F_{\lambda}(kx)\;=\;\sum_{k\in \LZ}\hat{F_{\lambda}^{x}}(k)
\;=\;\end{equation}
\begin{displaymath}
\;=\frac{1}{x}\sum_{k\in \LZ}F_{\lambda}(\frac{k}{x}).
\end{displaymath}
Since $f$ is {\bf symmetric}, then also $F_{\lambda}$ is so, and in
consequence also $\hat{F_{\lambda}}$ is symmetric.
Therefore if we observe that:
\begin{equation}
\;\;\theta(F_{\lambda})(x)\;:=\;\sum_{n=1}^{+\infty}F_{\lambda}(nx);\;\;x>0,
\end{equation}
and 
\begin{equation}
\;\;\theta(\hat{F_{\lambda}})(x)\;\sum_{n=1}^{+\infty}\hat{F_{\lambda}}(n
x)\;\;x>0,
\end{equation}
then we obtain the $(\theta.f.e)$.

In the case of a fixed point $\omega = Q_{1}(f)$ the result of the
Lemma 2 reduces to the famous {\bf classical $\theta$-equation} of the
form
\begin{displaymath}
\;\;\theta(\omega)(x)\;=\;\frac{1}{2x}(2\theta(\omega)(\frac{1}{x})+1)-
\frac{1}{2}.
\end{displaymath}
Let's write the {\bf Chandrasekharan's book [CH] form} of (R.f.e.)
using our notations from the previous section :
\begin{equation}
(CH.R.f.e)\;\;(\Gamma(G)\zeta)(s)\;=\;\frac{G(0)}{2s(s-1)}+\int_{1}^{+
\infty}(x^{s-1}+x^{-s})\theta(G)(x)dx;\;\;s\in \LC-\{0,1\}.
\end{equation}
It well-known that (CH.R.f.e.) is the immediate consequence of the (PSF)
and the identity : $\hat{G} = G$.

Replacing $G$ by an arbitrary fixed point $\hat{\omega} = \omega$ of
${\cal F}$ for which holds the above classical $\theta$-equation we
obtain the extension of (CH.R.f.e) to this case (cf. the proof of
Theorem in [CH]). But even the inflation (spliting) of (3.37) to the whole
class $\Omega$ fixed points of ${\cal F}$ is not suffiecient! The
reason is that - so called - (RH)-fixed points of ${\cal F}$ do not
exist! More exactly, the fundamental for the proof of (RH) - considered
below - the {\bf Fox integral equation} (i.e. some special {\bf
Fredholm} equation of second order) has only solutions on the level of
quasi-fixed points. These solutions $F_{\lambda}$ we call {\bf
(RH)-quasi-fixed points of ${\cal F}$} and exactly $F_{1}$ (being the
fixed point) is the {\bf singular} solution of the mentioned above Fox
equation.
Thus, fortunately for us - in the large class of quasi-fixed points of
${\cal F}$  denoted here by $Q\Omega$, there exist (RH)-quasi-fixed
points $F_{\lambda} = f+\lambda \hat{f}$.

The ($\theta$.f.e.) permits us to generalize the triplet functional
equation (1.2) to the whole class $\Omega_{0}$. In this sense the idea
of using such a generalization of (1.2) is very close to the {\bf Grothendieck's}
magnificent idea of {\bf etale cohomologies} on the category of algebraic varieties
(although our idea is obviously not so magnificent!). Grothendieck 
looked at the notion of an open set of a topological space as a morphism
of inclusion. In such a way he extended the notion of topology from a
set into a category.In particular Grothendieck topologies satisfy
all axioms of set topologies in the "cohomological sense". More exactly
, the idea of etale cohomologies is based on a beautiful notion of
"topology", which differs from a topological space in, that "open
sets" are not contained in a set , but satisfy fundamental properties, which
makes possible the construction of a necessary cohomology theory.

Most surprising is the role of fixed points in the general categorical sense in
the case of general Riemann hypothesis (Lefschetz fixed point formula) and
the role of the p-adic fixed points for the Sazonov topologies (cf.[M\c{a}1]) as well
, as in de Branges proof of (RH) [dB5].
Hence by $\Omega_{0}({\cal S})$ we denote the set of all {\bf
symmetric fixed points} of ${\cal F}$ from ${\cal S}(\LR)$.
\begin{th}({\bf Quasi-fixed points t.f.e})\\
For each $F_{\lambda} \in Q\Omega({\cal S})$ (for an arbitrary spectral
parameter $\lambda$) and for $s$ satisfying
$Re(s)>0$ the following f.e. holds
\begin{equation}
(Q.f.f.e.)\;\;\;\;\Gamma(\omega)(s)\zeta(s)=\frac{\omega(0)}{s(s-1)}+\int_{1}
^{+\infty}(x^{s-1}+x^{-s})\theta(\omega)(x)dx.
\end{equation}
\end{th}
{\bf Proof}. (According to Chandrasekharan's proof of the t.f.e. [Ch,
(2.23)]).\\
From the definition (2.25) of the generalized gamma-function
as an integral, we have 
\begin{displaymath}
\;\;\Gamma(F_{\lambda})(s)\;=\;\int_{0}^{+\infty}F_{\lambda}(x)x^{s-1}dx;
\;u=Re(s)>0.
\end{displaymath}
If $n$ is a positive integer, we have, on substituting $nx$ for $x$,
\begin{displaymath}
\Gamma(F_{\lambda})(s)=\int_{0}^{\infty}F_{\lambda}(nx)(nx)^{s-1}ndx=
n^{s}\int_{0}^{\infty}F_{\lambda}(nx)x^{s-1}dx,\;u>0,
\end{displaymath}
or
\begin{displaymath}
\;\;\;\frac{\Gamma(F_{\lambda})(s)}{n^{s}}\;=\;\int_{0}^{\infty}F_{\lambda}(nx)
x^{s-1}dx,\;\;u>0.
\end{displaymath}
If $u>1$, then we have
\begin{displaymath}
\;\;\;\Gamma(F_{\lambda})(s)(\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}})=
\sum_{n=1}^{\infty}\int_{0}^{\infty}(nx)
F_{\lambda}x^{s-1}dx,
\end{displaymath}
and since the series
\begin{displaymath}
\sum_{n=1}^{\infty}\int_{0}^{\infty}\mid F_{\lambda}omega(nx)x^{s-1}\mid dx
=\sum_{n=1}^{\infty} \int_{0}^{\infty}\mid F_{\lambda}(nx)\mid
x^{u-1}dx=\Gamma(\mid F_{\lambda} \mid)(u)\zeta(u)
\end{displaymath}
is convergent for $u$ with $u>1$, we can interchange the
order of summation and integration, obtaining
\begin{displaymath}
\;\;\;\Gamma(F_{\lambda})(s)
\zeta(s)\;=\;\int_{0}^{\infty}\theta(F_{\lambda})(x)x^{s-1}dx. 
\end{displaymath}
According to $(\theta.f.e.)$, this can be rewritten as
\begin{displaymath}
\Gamma(F_{\lambda})(s)\zeta(s)\;=\;\int_{0}^{1}x^{s-1}\theta(F_{\lambda})(x)dx
+\int_{1}^{\infty}x^{s-1}\theta(F_{\lambda})(x)dx=
\end{displaymath}
\begin{displaymath}
=
\int_{0}^{1}x^{s-1}(x^{-1}\theta(F_{\lambda})(\frac{1}{x})+
\frac{\hat{F_{\lambda}(0)}}{2x}-
\frac{F_{\lambda}(0)}{2})dx+\int_{1}^{\infty}x^{s-1}\theta(F_{\lambda})(x)dx=
\end{displaymath}
\begin{equation}
=\frac{\hat{F_{\lambda}}(0)}{2(s-1)}
-\frac{F_{\lambda}}{2s}+\int_{0}^{1}x^{s-2}\theta(F_{\lambda})(1/x)dx+\int_{1}
^{\infty}x^{s-1}\theta(F_{\lambda})(x)dx
\end{equation}
\begin{displaymath}
=\;\frac{\hat{F_{\lambda}}(0)}{2(s-1)}\;+\; \frac{F_{\lambda}(0)}{2s}
\int_{1}^{\infty}x^{-s}\theta(\hat{F}_{\lambda})(x)dx\;+\;\int_{1}^{
+\infty}x^{s-1} \theta(F_{\lambda})(x)dx.
\end{displaymath}
Applying the modular map $m(s)= 1-s$ to the both sides of (.) and
adding side to side we finally get
\begin{equation}
(\Gamma(F_{\lambda})\zeta)(s)+(\Gamma(F_{\lambda}\zeta)(1-s)=
\frac{(F_{\lambda}+\hat{F}_{\lambda})(0)}{2s(s-1)}+\int_{1}^{+\infty}(x
^{s-1}+x^{-s}) \theta(F_{\lambda}+\hat{F}_{\lambda})(x) dx.
\end{equation}
The integral on the right-hand side of (2.) converges uniformly for
$-\infty<a\le u\le b<+\infty$, since for $x\ge 1$ , we have
\begin{displaymath}
\;\;\;\mid x^{-s} +x^{s-1}\mid \le\; x^{-a}+x^{b-1},
\end{displaymath}
while
\begin{displaymath}
\;\;\;\theta(F_{\lambda})(x)<\sum_{n=1}^{+\infty}\mid F_{\lambda}(nx)\mid \le
\frac{p_{l,0}(F_{\lambda})\zeta(l)}{x^{l}};\;\;\;x>0,
\end{displaymath}
(for $l>1$ , $sup_{x\in \LR}\mid F_{\lambda}(nx) (nx)^{l} \mid =
p_{l,0}(F_{\lambda})<+\infty$)
Hence, the integral in (2.) represents an entire function of $s$. Since
obviously the left-hand side of (2.) is entire, the must be equality.

Beside the class of quasi-fixed points of ${\cal F}$ we also need a
quite different class of functions connected with ${\cal F}$.
Interesting is - similarly as in $(Q.f.f.e.)$-case -importance of an
imaginary part of a complex number.
We consider {\bf amplitudes} $A: \LR \longrightarrow \LC$, which are
${\bf {\cal F}-admissible}$ and {\bf antysymmetric} (i.e. $A(-x) =
A(x)$). Therefore they satisfy the functional equation:
\begin{equation}
\;\;\;{\cal F}(A)\;= \;2Im{\cal F}_{+}(A).
\end{equation}

In the sequel we consider only {\bf Continuous}, (Lebesgue)-{\bf
Integrable} (on $\LR$) and (strictly)-{\bf Decreasing amplitudes}
(CID-amplitudes in short) $A = A(x)$. Moreover, a word "amplitude"
always in this paper mean, that $A$ is {\bf positive}.
With such an amplitude $A$ is connected a {\bf "dissappear (fade) wave"}
: $A(x)sin(ax)$ (with a "phase" $a$) and {\bf ${\cal F}_{+}$-volume}
\begin{equation}
\;\;\;{\cal F}_{+}(AsinL_{a})\;:=\;\int_{0}^{+\infty}A(x)sin(ax)dx.
\end{equation}
Here $L_{a}(u) :=au, u\in \LR$ stands for the {\bf linear function}.
The below lemma is "intuitionally obvious", but we give its proof for
the sake of completness.It also underline some special place of
(CID)-amplitudes in the harmonic analysis.
\begin{lem}
For each CID-amplitude $A:\LR \longrightarrow \LR_{+}^{*}$ and phase
$a\in \LR^{*}$ holds
\begin{equation}
\;\;\;{\cal F}_{+}(AsinL_{a})\;>\;0.
\end{equation}
\end{lem}
{\bf Proof}. We have
\begin{displaymath}
{\cal F}_{+}(AsinL_{a}) \;=\;\int_{0}^{+\infty} A(x)sin(ax)dx
\;=\;\sum_{n=0}^{\infty}\int_{n\pi/a}^{(n+1)\pi/a}A(x)sin(ax)dx\;=\;(
\sum_{n=0}^{+\infty}(-1)^{n}A(x_{n}))P(a),
\end{displaymath}
where $P(a) = \int_{0}^{\pi/a}sin(ax)dx >0$ and a sequence $\{x_{n}\}$
with $x_{n}\in [n\pi/a,(n+1)\pi/a]$ is determined according to the {\bf
mean value theorem}.
Since $A$ is {\bf positive, continuous, integrable} and {\bf
decreasing}(strictly), then we obtain the thesis of the lemma.

We say that a quasi-fixed point $F_{\lambda} =(F_{\lambda}^{A})\in Q\Omega$
is a {\bf (RH)-quasi-fixed point of ${\cal F}$} iff there exists such a
(CID)-amplitude $A = A(x); x>0$ that 
\begin{equation}
(RH.Q.f.)\;\;(F_{\lambda}^{A}\;-\;G)(x)\;\;=\;\;A(x)\;for\;all\;x\ge 1.
\end{equation}

The problem of the existence of (RH)-fixed points immediately leads to
the problem of the solution of the {\bf second order} (non-homogeneous)
{\bf Fredholm integral equation} of the form
\begin{displaymath}
(FE)\;\;f(x)=\lambda \int_{a}^{b}K(x,y)f(y)dy\;=\;g(x)\;\;;x\in [a,b],
\end{displaymath}
with a {\bf symmetric singular}(i.e. on $\LR_{+}$) and {\bf Fox
kernel}(i.e.)
\begin{displaymath}
(FK)\;\;\;\;\;K(x,y)\;=\;K(xy) \;\;(or \;K(x/y)).
\end{displaymath}

The Fredholm integral equations which kernels satisfies (FK)-condition
are called the {\bf Fox integral equations} (cf.e.g.[KKM, Sect.II.23])
in oppositeto the more popular {\bf Wiener-Hopf integral equations}
which kernels satisfies "arithmetically dual" condition to the (FK):
\begin{displaymath}
(WH)\;\;\;\;K(x,y)\;=\;K(x-y).
\end{displaymath}
\begin{re}
It is very surprising that the theory of integral equations -which
comparing with the theory of partial differential equations - is not
such popular. (The theory of integral equations seems to be very
"modest" if we compare it with the theory of differential equations,
but in this case it is deciding).

Observe that according to the formal resemblense of a Fredholm integral
equation of the second order
\begin{displaymath}
\lambda \int_{a}^{b}K(x,y)f(y)dy\; + \;f(x)\;=\;g(x),
\end{displaymath}
to a non-homogeneous differential equation of the first order 
\begin{displaymath}
\lambda(x) f^{\prime}(x)\;\;\;+\;\;f(x)=\;g(x),
\end{displaymath}
the modern name should be a " non-homogeneous integral equation of the
first order (instead a historical one).
Obviously - beside the same {\bf formal structure} of above integral
and differential equation - the {\bf structure of solutions} is quite
different. For example - integral operatorors are global, whereas
differential operators have got a totally local structure.
Probably the most importantant examples of integral operators are the
Fourier, Laplace and Mellin transforms.

The Fox integral equations can be solved {\bf algebraically} by using
the {\bf Mellin transform}. It is very detailed explained in [KKM] ,
what is a little bit amazing coincidence, since in [KKM, Sect.II.23] 
is exactly solved such a Fox equation what we need.
The {\bf Mellin transform} is a specific functor, which reduces a
rather difficult analytic problem of the solution of the integral Fox
equation to a relatively easy {\bf algebraic} problem of the solution
of an operator equation. (It is well-known that in the theory of ordinary
linear differential equations of a finite order, the same role is
played by the {\bf Laplace transform}).
Obviously, the full algebraization of a Fox equation is possible
according to ( a deeper property of Mellin transform): the theorem on the
{\bf convolution} for Mellin transform:
\begin{equation}
\;M[\int_{0}^{+\infty}
f(t)\phi(\frac{x}{t})\frac{dt}{t}]\;=\;[(Mf)\cdot (M\phi)] (s).
\end{equation}
Finally, let's remark that the Riemann hypothesis (RH) is then very
strictly associated with the problem of the solution of the Fox
equation. (Blisko pracy (Calkowalnosc to (RH)?)).
\end{re}

We are ready now to prove a following preliminary technical result :
\begin{pr}({\bf On the existence of (RH)-quasi-fixed points})
For each positive CID-amplitude $A = A(x)$ from ${\cal S}(\LR)$ and a spectral
parameter $\lambda\ne \pm 1$ there exists {\bf (RH)-quasi-fixed point}
$F_{\lambda}$ associated with $A$, i.e.
\begin{equation}
\;\;\;(F_{\lambda}\; -\; \alpha G(x)) \;=\;A(\mid x \mid);\;x\in \LR.
\end{equation}
Moreover $F_{\lambda} \in {\cal S}(\LR)$.
\end{pr}
{\bf Proof}. We search $F_{\lambda}$ in the form : $F_{\lambda}(x) =
f(x)+ \lambda \hat{f}(x)$ , for an unknown and symmetric $f :\LR
\longrightarrow \LR$.

According symmetricity of $f$, we see that the equality (3. ) reduces
to the problem of the solution of the following {\bf Fox integral
equation} (with the kernel $K(xy) :=cos(2\pi xy)$) :
\begin{equation}
\;f(x)\;+\;2\lambda \int_{0}^{\infty}cos(2\pi xy)f(y)dy\;=\;A(x)+\alpha
G(x)(=:g_{A}(x))\;\;;x\in \LR_{+}.
\end{equation}
Multiplying both sides of (.) $x^{s-1}$ , integrating w.r.t. the
Lebesgue measure $dx$ on $\LR_{+}$ and applying {\bf Fubini theorem} we get
\begin{equation}
\int_{0}^{\infty}g_{A}(x)x^{s-1}dx\;=\;\int_{0}^{\infty}f(x)x^{s-1}dx+
\lambda\int_{0}^{\infty}f(y)dy \int_{0}^{\infty} cos(2 \pi xy) x^{s-1}
dx.
\end{equation}
Denoting by $\tilde{K}(s)$ the {\bf Mellin transform} of $K(x) = cos(2
\pi x)$ and applying {\bf convolution theorem for $M$} (.) , we can
write (.) of the form
\begin{equation}
\;\;(Mg_{A})(s)\;=\;(Mf)(s)\;+\;\tilde{K}(s)\int_{0}^{\infty}f(y)y^{-s}
dy.
\end{equation}
It is easy to check that  $\int_{0}^{\infty} f(y)y^{-s}dy = (Mf)(1-s)$.
Therefore the equality (.) we can write of the {\bf operator form}:
\begin{equation}
\;\;(Mg_{A})(s)\;=\;(Mf)(s)\;+\;\tilde{K}(s)(Mf)(1-s).
\end{equation}
Replacing in (.) $s$ by $1-s$ (i.e. acting the generator $m$ of
$Mod(\LC)$) we obtain
\begin{equation}
\;\;(Mg_{A})(1-s)\;=\;(Mf)(1-s)\;+\;\tilde{K}(1-s) (Mf)(s).
\end{equation}
From (.) and (.) follows that
\begin{equation}
\;(Mf)(s)\;=\;\frac{(Mg_{A})(s)-2 \lambda
(Mg_{A})(1-s)\tilde{K}(s)}{1-4\lambda^{2}\tilde{K}(s)\tilde{K}(1-s)}.
\end{equation}
The formula (.) gives an operator solution of the Fox equation (.).
According to the formula concerning the {\bf inversion} of the Mellin
transform, we find the solution of the Fox equation (.) as 
\begin{equation}
\;\;f(x)\;=\;\frac{1}{2\pi i} \int _{\sigma-i \infty}^{\sigma+ i \infty}
\frac{(Mg_{A})(s)-2 \lambda (Mg_{A})(1-s)\tilde{K}(s)}{1-4
\lambda^{2}\tilde{K}(s)\tilde{K}(1-s)}x^{-s} ds.
\end{equation}

At this place we repeat the calculations (practically without changes)
given at the last example of the Section II.23 from [KKM].

We have
\begin{equation}
\;\;\tilde{K}(s)\;=\;2\lambda \int_{0}^{\infty}x^{s-1}cos(2 \pi x)dx.
\end{equation}
To calculate the above integral we recall the first the definition of
the classical {\bf Gamma}:
\begin{equation}
\;\;\int_{0}^{\infty}e^{-x}x^{z-1}dx \;=\;\Gamma(z).
\end{equation}
If in (.) we make a turn of the radius of integration at $\pi/2$ ( it
is possible according to the {\bf Jordan lemma} if $z \in (0,1)$ , i.e.
we formally make the substitution :$x = 2\pi y i$) we
obtain the formula:
\begin{equation}
\;\;\int_{0}^{\infty}e^{-2\pi iy} (2\pi i y)^{z-1}2\pi i
dy\;=\;\Gamma(z),
\end{equation}
or
\begin{equation}
\;\;\int_{0}^{\infty}e^{-2\pi iy}y^{z-1}dy\;=\;(2\pi)^{-z}e
^{-\frac{\pi zi}{2}}\Gamma(z).
\end{equation}
The comparison of the real and imaginary part of the both sides of (.)
gives
\begin{equation}
\tilde{K}(s)\;=\;int_{0}^{\infty}x^{s-1} cos(2\pi
x)dx\;=\;(2\pi)^{-s}cos(\frac{\pi s}{2})\Gamma(s),
\end{equation}
and \begin{equation}
\;\;\;\int_{0}^{\infty}x^{s-1}sin(2\pi
x)dx\;=\;(2\pi)^{-s}sin(\frac{\pi s}{2})\Gamma(s).
\end{equation}
From (.) and (.) we obtain that
\begin{equation}
\;\;\;\tilde{K}(s)\;\;=\;\;(2 \pi)^{-s} \Gamma(s)cos(\frac{\pi s}{2}).
\end{equation}
Moreover:
\begin{equation}
\tilde{K}(s)\tilde{K}(1-s) =(2 \pi)^{-s}\Gamma(s) cos\frac{\pi s}{2}
(2\pi)^{s-1}\Gamma(1-s)sin\frac{\pi s}{2}=
\end{equation}
\begin{displaymath}
=\frac{1}{4\pi}2sin\frac{\pi s}{2}cos\frac{\pi
s}{2}\Gamma(s)\Gamma(1-s)=\frac{1}{4},
\end{displaymath}
since $\Gamma(s)\Gamma(1-s) = \pi/sin \pi s$ (according to the
well-known {\bf Euler formula}).
Therefore, following (.) , if $\mid \lambda \mid \ne 1$  we have 
\begin{displaymath}
(Mf)(s)\;=\;\frac{(Mg_{A})(s)- 2\lambda (Mg_{A})(1-s)(2
\pi)^{-s}cos\frac{\pi s}{2}\Gamma(s)}{1-\lambda^{2}}.
\end{displaymath} 
Hence
\begin{equation}
f(x)=\frac{1}{2\pi(1-\lambda^{2})} \int_{\sigma - i\infty}^{\sigma
+i\infty}(Mg_{A}(s)- 2 \lambda (2 \pi)^{-s}cos\frac{\pi
s}{2}\Gamma(s))x^{-s}ds=
\end{equation}
\begin{displaymath}
=\frac{1}{1-\lambda^{2}}\cdot \frac{1}{2\pi i}\int_{\sigma
-i\infty}^{\sigma +i
\infty}(Mg_{A})(s)x^{-s}ds+\frac{2\lambda}{1-\lambda^{2}}\cdot
\frac{1}{2\pi i}\int_{\sigma-i\infty}^{\sigma+i\infty}
(Mg_{A}(1-s)(2\pi)^{-s}cos\frac{\pi s}{2}\Gamma(s)x^{-s}ds).
\end{displaymath}
In the right-hand side of (.), let's replace in the second integral
$Mg_{A}(1-s)$ through $\int_{0}^{\infty}g_{A}(x)x^{-s}dx$ and
observe that
\begin{displaymath}
\;\;\;\frac{1}{2\pi i}\int_{\sigma-i \infty}^{\sigma+i
\infty}(Mg_{A})(s)x^{-s}ds\;=\;g_{A}(x).
\end{displaymath}
The formula (.) obtain now the following form:
\begin{equation}
f(x)=\frac{g_{A}(x)}{1-\lambda^{2}}+\frac{2\lambda}{1-\lambda^{2}}\cdot
\frac{1}{2 \pi i}\int_{\sigma-i\infty}^{\sigma+i\infty} (2
\pi)^{-s}cos\frac{\pi s}{2}\Gamma(s)(xy)^{-s}\int_{0}^{\infty}f(y)dy.
\end{equation}
According to the inversion formula for the Mellin transform we get
\begin{displaymath}
\;\;\;\frac{1}{2\pi i}\int_{\sigma-i \infty}^{\sigma+i
\infty}(2\pi)^{-s}\Gamma(s)cos\frac{\pi s}{2}(xy)^{-s}ds\;=\;cos(2\pi
xy),
\end{displaymath}.
Therefore , finally we obtain
\begin{equation}
\;f(x)\;\;=\;\;\frac{g_{A}(x)}{1-\lambda^{2}}+\frac{2\lambda}{1-\lambda^{2}}
\int_{0}^{\infty}g_{A}(y)cos(2\pi xy)dy.
\end{equation}

\section{From (RH)-quasi-fixed to (RH)-fixed points of ${\cal F}$}

According to the Proposition 3.1, for each CID , {\bf bimodal} amplitude
$A$ and real $\lambda$ with $\mid \lambda \mid \ne 1$, there exists a (RH)
-quasi-fixed point $F_{\lambda} = f_{\lambda}+ \lambda \hat{f}_{\lambda}$ 
of ${\cal F}$ with a {\bf symmetric} $f_{\lambda}$ from ${\cal S}(\LR)$,
and such that it solves the Fox equation:
\begin{equation}
F_{\lambda}(x)=f_{\lambda}(x)+\lambda \hat{f}_{\lambda}(x)
\end{equation}
\begin{displaymath}
=G(x)+A(x) \;=:\;g_{A}(x),\;x \in \LR_{+}.
\end{displaymath}
and
\begin{equation}
\;\;F_{\lambda}(0)\;=\;G(0).
\end{equation}

Thus, there exists a {\bf whole family}
\begin{displaymath}
\;\;F(x)\;:=\;\{F_{\lambda}(x)\;:\;\mid \lambda \mid \ne 1\}
\end{displaymath}
, of quasi-fixed points $F_{\lambda} = f_{\lambda} +
\lambda\hat{f}_{\lambda}$, which constitutes a material (stuff) for
obtaining a $(RH)$-fixed point of ${\cal F}$ - let's write - $F_{1}$ -
the main technical object of this paper.
Let's observe also that is very important the exact structure of $f_{\lambda}$
(cf.(3.64)
\begin{equation}
f_{\lambda}(x)\;=\;\frac{g_{A}(x)}{1-\lambda^{2}}
+\frac{\lambda}{1-\lambda^{2}}({\cal F}g_{A})(x).
\end{equation}
From (4.67) we immediately get 
\begin{equation}
\;\;f_{\lambda}(x)+f_{1/\lambda}(x)\;=\;g_{a}(x) \;;\;x\in \LR_{+},\mid
\lambda \mid \ne 1.
\end{equation}
In the family of the equations (4.67) are also written the important {\bf
algebraic} expression  on $\lambda$ . Let $I_{\LR}(\lambda) = 1/\lambda$ be
the {\bf inverse} (hiperbolic) map on $\LR^{*}$. Observe that the function
\begin{equation}
\;\;\;{\cal I}_{\LR}(\lambda) \;=\;\frac{\lambda}{\mid 1-\lambda^{2} \mid}
\end{equation} 
is {\bf $I_{\LR}$-invariant} on $\LR_{+}^{*}-\{1\}$, i.e. ${\cal
I}_{\LR}(I_{\LR}(\lambda)) = {\cal I}_{\LR}(\lambda)$.
Moreover, the function $\mid {\cal I}_{\LR}(\lambda)\mid/\lambda$ is the
{\bf density} of a {\bf finite Borel measure} $\mu_{I}$ on $\LR_{+}^{*}$ given
by the formula
\begin{equation}
\;\;\mu_{I}(B)\;:=\;V.p.\int_{B}\frac{d\lambda}{\mid 1- \lambda^{2} \mid} ,
\end{equation}
where the integral exists in the sense of the Cauchy's {\bf "Valeur
principale"} in 1 (cf.e.g.[Fi,XIII.484]) , and $B$ is a Borel set in
$\LR_{+}^{*}$.

The measure $\mu_{I}$ is the {\bf invariant measure} of the {\bf
topologically-dynamical system} $(\LR_{+}^{*},\{I_{\LR}^{n} : n\in
\LN\})$ , i.e.
\begin{equation}
\;\;\;\;\;\mu_{I}((I_{\LR}^{n})^{-1}(B))\;=\;\mu_{I}(B),
\end{equation}
for each $n\in \LN$ and a Borel set B.

According to the above notations , the component $f_{\lambda}$ of the fixed point
$F_{\lambda}$ we can write in the form:
\begin{equation}
\;f_{\lambda}(x)\;=\;\frac{d\mu_{I}(\lambda)}{d\lambda}g_{A}(x)+{\cal
I}_{\LR}(\lambda)\hat{g_{A}}(x) .
\end{equation}

\begin{re}
According to the classical {\bf Bogoluboff-Kriloff theorem} (cf.[BK])
every topological, dynamical system $\{T^{n}\}$ generated by a map $T:X
\longrightarrow X$ defined on a compact metric space $X$, admits an
{\bf invariant measure } $\mu_{T}$. In the sequel the measure $\mu_{T}$
is called the {\bf Bogoluboff-Kriloff measure} of a (topologically)
dynamical system $(X,T)$.
\end{re}

The constructed above system $\{F_{\lambda}\}$ of (RH)-quasi-fixed
points , commonly with the {\bf Bogoluboff-Kriloff measure} $\mu_{H}$ of 
$I_{H}(\lambda)= \frac{1}{\lambda} , \lambda \in H$ (where $H$ is the
field of {\bf quaternions}),permits us to prove the main technical result
of this paper : the existence of (RH)-fixed points of ${\cal F}$.

We start from some preliminary results concerning {\bf integration} in
the area of {\bf local fields}. (Let recall that our Brouwer logic proof
of (RH) given in [M\c{a}2] is strongly based - as it was observed by W.
Narkiewicz - on the integration in the infinite-dimensional function
spaces).
The idea of using local fields in the context of the Riemann hypothesis is not
new. In fact {\bf de Branges} in the mentioned in the introduction - series
of papers, showed that {\bf $p$-adic analysis} is strictly connected with $(RH)$.

The main idea developes here below, is based on the observation, that
suitable results concerning the {\bf transition} from the family
$\{f_{\lambda}:\lambda >0, \lambda \ne 1\}$ to $f_{1}$, which normally
needs analysis on $\LR = :\LQ_{\infty}$, can be {\bf algebraically}
obtained.

Let $K$ be an arbitrary {\bf local field} , i.e. in the Weil's {\bf Basic
Number Theory} terminology, it is an arbitrary non-discrete Locally
Compact (LC in short) field (not necessary {\bf commutative})(cf.e.g. [We]).

Let $\mid . \mid_{K}$ denotes the canonica non-trivial {\bf valution}
(absolute value or point of $K$), which can be {\bf archimedean} (in the case of 
{\bf connected} fields) or {\bf non-archimedean} (in the case of {\bf
totally-disconnected} fields).

Let $H_{K}$ be the {\bf Haar measure} on $(K,+)$ ( the {\bf left Haar measure} in
the case when $K$ is non-commutative). Let's denote :  $R_{K} := \{k\in K: 
\mid k \mid_{K}\le 1\}$. It is very convenient to normalize $H_{K}$ in
the following standard way :\\
(i) if $K$ is non-archimedean then $R_{K}$ is the maximal compact subring of
$K$ (the ring of integers of K). We assume that $H_{K}(R_{K}) =1$.\\
(ii) in the archimedean case we obviously have: $H_{\LR}(R_{\LR}) = 2,
H_{\LC}(R_{\LC}) = \pi$ and $H_{H}(R_{})=7 \pi/4$.

Each {\bf automorphism} $\lambda$ of $(K,+)$ change the Haar measure $H_{K}$
into $c H_{K}$ with $c\in \LR_{+}^{*}$ (since obviously according to the
Neumann-Weil theorem a Haar measure is unique up to the constant). The
number $c$ do not depend on the choice of a Haar measure and we denote it by
$\Delta_{K}(\lambda)$ or $mod_{K}(\lambda)$. In the other words, it is
defined by the one of the below (equivalent) formulas (cf.[We,I.2]) :
\begin{displaymath}
\;\;\;\;H_{K}(\lambda(B))\;=\;\Delta_{K}(\lambda)H_{K}(B),
\end{displaymath}
or
\begin{displaymath}
\;\;\int f(\lambda^{-1}(x))dH_{K}(x)\;=\;\Delta_{K}(\lambda)\int
f(x)dH_{K}(x),
\end{displaymath}
where $B$ is any Borel set and $f$ is an integrable function with $\int
f dH_{K}\ne 0$.
The second formula we can symbolically write in the form :
$dH_{K}(\lambda(x))= \Delta_{K}(x)(\lambda)dH_{K}(x)$.

Let $L_{a}(x) = a\cdot x; a\ne 0, x\in K$ be the linear multiplication (by $a$)
automorphism of $K$. We put
\begin{displaymath}
\;\;\;\Delta_{K}(a)\;\;:= \;\;\Delta_{K}(L_{a});\;\;\;a\in K^{*} .
\end{displaymath}
By $I_{K}(\lambda) = \lambda^{-1}; \lambda \in K^{*}$ we denote the
the {\bf inversion of $K^{*}$}, i.e. the {\bf crucial}
(topologically-algebraic) automorphism of $K^{*}$ of order $2 :
I_{K}^{2}\;=\;id_{K^{*}}$.
It is well-known (cf.[We]) that
$\frac{dH_{K}(\lambda)}{\Delta_{K}(\lambda)}$ is a {\bf Haar measure} of
$(K^{*},\cdot)$ (cf.e.g. [We,VII.4, Lemma5]).

Let's also mention at this place, that in the most important cases $\mid \cdot
\mid_{K}$ and $\Delta_{K}$ coincides. In particular, $\Delta_{\LR} = \mid .
\mid_{\LR} = \mid . \mid_{\infty}$ and for all p-adic number fields
$\LQ_{p}$ we have: $\Delta_{\LQ_{p}} = \mid \cdot \mid_{p}$ , $p$ is a
prime. Moreover $\mid z \mid^{2}_{\LC} = \Delta_{\LC}(z))$ , $\mid h
\mid_{H}^{4} = \Delta_{H}(h)$ and generally : $\Delta_{K}(k) = \mid
k \mid_{K}^{d}, d\ge 1, d\in \LN $ (cf.[We,I.2 and Corollary 2]).

In the sequel we rather prefer to work with $\Delta_{K}(k)$ instead of $\mid k
\mid_{K}$.

We start from the following simple algebraically-measure formulas
for the Haar measure $H_{K}$, (where only algebraic and measure structure are
important). That is an short algebraic calculus.
\begin{lem}
Let $K$ be a local field. Then for each {\bf integrable} function $f$
w.r.t $H_{K^{*}}$ on $K^{*}$ we have:
\begin{equation}
(i)\;\int_{K^{*}}f(\lambda^{-1})\frac{dH_{K}(\lambda)}{\Delta_{K}(\lambda)}\
\;=\;\int_{K^{*}} f(\lambda)\frac{dH_{K}(\lambda)}{\Delta_{K}(\lambda)},
\end{equation}
, i.e. $\frac{dH_{K}(\lambda)}{\Delta_{K}(\lambda)}$ is the {\bf
Bogoluboff-Kriloff measure} of $I_{K}$, and
\begin{equation}
(ii) \;\int_{K^{*}}
f(\lambda)dH_{K}(\lambda)\;=\;\int_{K^{*}}\frac{f(\lambda)}{\Delta_{K}^{2}(
\lambda)}dH_{K}(\lambda).
\end{equation}
\end{lem}
{\bf Proof}. Let's consider the Haar measure
$\frac{dH_{K}(\lambda)}{\Delta_{K}(\lambda)}$ and the second order
automorphism $I_{K}$ with $I_{K^{*}}^{2} = id_{K^{*}}$. Since, obviously
$I_{K}^{*}(\frac{dH_{K}(\lambda)}{\Delta_{K}(\lambda)})$ is a Haar
measure on $K^{*}$, then according to the von Neumann-Weil theorem
there  exists unique constant $c = mod_{K^{*}}(I)$ with the property
\begin{displaymath}
\;\;I^{*}(\frac{dH_{K}(\lambda)}{\Delta_{K}(\lambda)})\;=\;mod_{K^{*}}(I)
\frac{dH_{K}(\lambda)}{\Delta_{K}(\lambda)}.
\end{displaymath}
But 
\begin{displaymath}
c^{2}\;=\;mod_{K^{*}}^{2}(I)\;=\;mod_{K^{*}}(I^{2})\;=\;mod_{K^{*}}(id_{K
^{*}})\;=\;1.
\end{displaymath}
Therefore
\begin{displaymath}
\int_{K^{*}}f(\lambda^{-1})\frac{dH_{K}(\lambda)}{\Delta_{K}(\lambda)}=
\int_{K^{*}}
f(\lambda)dI^{*}(\frac{dH_{K}(\lambda)}{\Delta_{K}(\lambda)})=
\end{displaymath}
\begin{displaymath}
= \int_{K^{*}}f(\lambda)\frac{dH_{K}(\lambda)}{\Delta_{K}(\lambda)}.
\end{displaymath}
(In fact more general fact is true: each left (right) Haar measure
$H_{G}$ on a unimodular group $G$ is simultaneously the
Bogoluboff-Kriloff measure of the inversion of that group, cf.e.g. [BR,
II.3, Proposition 3]).

The whole above calculus is written in fact on the page 174 in the 1972
Russian edition of the Weil's book [We] (in the case of idele group of
a global field).

To obtain the second formula it suffices to apply the first formula for
the "new" function $\phi(\lambda) = f(\lambda^{-1})/\Delta_{K}(\lambda)$.

\begin{re}
Let's observe that the formulas (4.73) and (4.74) in the case of $K = \LR$
are usually analyticaly proven , by using 1-dimensional Jacobi formula. 
\end{re}

Let's consider now {\bf $K$-segment (ring)}  $R_{K}(M,N):=\{k\in K: M \le
\mid k \le N\}, N>0$. Obviously $R_{K} = R_{K}(0,1)$ Then we can consider a
topologically-dynamical system $D_{K}(M,N) :=
(R_{K}(M,N)-R_{K}(N^{-1},M^{-1}), I_{K}), N>M>0$.
\begin{lem}
Let $K$ be a local field and $H_{K}$ be a left Haar measure of $(K,+)$.
Then the formula
\begin{equation}
\;\;\mu_{BK}^{K}(B)\;:=\;\int_{B}\frac{dH_{K}(\lambda)}{\Delta_{K}(1
-\lambda^{2})},
\end{equation}
, where $B$ is a Borel set in $X_{K}(M,N):=R_{K}(N,M)- R_{K}(M^{-1},N^{-1})$, gives a
Bogoluboff-Kriloff measure of the dynamical systewm $D_{K}(M,N)$.
\end{lem}
{\bf Proof}. It suffices to apply the formula (4.74) for the function
\begin{displaymath}
\;\;\;f(\lambda)\;=\;\Delta_{K}((1-\lambda^{2})^{-1})\chi_{X_{K}(M,N)}(
\lambda)\;;\lambda \in  K^{*}.
\end{displaymath}
\begin{re}
Let's observe that Bogoluboff-Kriloff measures are generally not
uniqne. For example the dynamical system $D_{K}(M,N)$ has got at least
two invariant measures :
$\frac{dH_{K}(\lambda)}{\Delta_{K}(1-\lambda^{2})}$ and
$\frac{dH_{K}(\lambda)}{\Delta_{K}(\lambda)}$.
For theorems which approximate the number of invariant measures of a
dynamical system ,cf. [FKS,V.2, Th.1] .

It is also convenient to call the density $\frac{d\mu^{K}_{BK}}{dH_{K}}$
the {\bf $K$-Poincare metric}, since that function is modeled on the
true Poincare metric 
\begin{displaymath}
\;P(z)\;=\;\frac{1}{\mid 1-\mid z mid^{2}\mid^{2}}
\end{displaymath}
of $R_{\LC}$. In the number field $\LC$ case additionally appears the
module (cf. [Kr, I.1.4])
\end{re}

A quite unexpected for us was a help from the huge {\bf potential
theory}. According to one of the famous specialists in the potential
theory - W.K. Hayman (cf.[HK, An Introduction]) it is in fact, a theory
of subharmonic functions, with its deep and difficult results:\\
(1) {\bf Riesz theorem} - on the integral representation of subharmonic
functions ( i.e. existence of an {\bf Abstract Hodge Decomposition} of
a subharmonic function $s(x)$ in a domain from $\LR^{m}$ with respect
to the {\bf Poisson type} kernel $K_{m}(x-y)$, which is moreover
Wiener-Hopf type and {\bf algebraic} ): there exists a {\bf Riesz
measure} $dR = \bigtriangledown^{2}s dx$ and {\bf harmonic} $h(x)$
outside a compct set $E$ function with
\begin{equation}
(RT)\;\;\;s(x)\;=\;\int_{E}K_{m}(x-y)dR(y)\;+\;h(x),
\end{equation}
(cf.[HK,3.5, Theorem 3.9]). Obviously the Riesz measure plays the role
of the Hodge measure in this Abstract Hodge Decomposition.
The role of {\bf Green functions} $K_{m}(x)$ , which are equal to
$log\mid\mid x \mid\mid_{2}$ if $m=2$  and  $\mid\mid x
\mid\mid^{2-m}_{m}$ if $m\ge3$ , where $\mid\mid x \mid\mid_{m}:=
\sqrt{\sum_{j=1}^{m}x_{j}^{2}}$ is m-euklidean norm in $\LR^{m}$ is
not to overestimate.

First of all they are {\bf fundamental solutions} of the {\bf Laplace
operator} $\Delta = \bigtriangledown^{2}$  in $\LR^{m}$:
\begin{displaymath}
\;\;\;\;\Delta K_{m}(x)\;=\;\delta_{0}(x),
\end{displaymath}
 i.e. they are also {\bf harmonic functions} in the domain
$\LR^{m}-\{0\}$.
However, for our purposes , more important is the below {\bf Brelot's
theorem}(cf.[HK,3.6, Theorem 3.10]) on there existence of the {\bf
harmonic measure}.
Let $D$ be a {\bf regular} and bounded domain of $\LR^{m}$ with the border
$\partial D = F$. Then for each $x\in D$ and arbitrary Borel set $B$ in
$F$ , there exists a {\bf unique} function of two variables :
$\omega(x,B;D)$ with the following three properties :\\
(i) for each $x\in D ,  \omega(x,.;D) $ is a {\bf probability} Borel
measure on $\partial D$ , i.e. $\omega(x,\partial D;D) =1$.\\
(ii)For each Borel set $B\subset \partial D,  \omega(.,B;D)$ is a {\bf
harmonic function} on $D$.\\
(iii) (Abstract Hodge Decomposition of harmonic extension).\\
If $f(\xi)$ is a {\bf semi-continuous} function on $\partial D$, then
the formula
\begin{equation}
\;\tilde{f}(x)\;=\;\int_{\partial D}f(\xi)d\omega(x,\xi;D)\;;\;\xi\in
D-F,
\end{equation}
gives the harmonic extension of $f$ from $F$ to $D$.\\
The measure $\omega(x,.;D)$ is called the {\bf harmonic measure} in $x$
w.r.t. $D$ and its existence follows from the well-known Riesz theorem
on the representations of linear functionals as measures.
\begin{re}
The family $\omega(D):=\{\omega (x,.;D):x\in \partial D\}$ of harmonic measures on
$D$ solves the famous {\bf Dirichlet problem} for a pair $(D,\partial
D)$: find the {\bf harmonic} function $u(x)$ on $D$ and {\bf continuous}
on $D\cap \partial D$ which takes given values on $F$. It is well-known
that the solution of a Dirichlet problem is {\bf unique}, under the
assumption that one exists.
\end{re}

The {\bf Riesz measure} $R$ and family $\omega(D)$ of {\bf harmonic
measures} commonly appear in the famous {\bf Poisson-Jensen formula} (
a double Abstract Hodge Decomposition of a {\bf subharmonic} function
$s(x)$ on a domain $D$, cf.[HK, Sec.3.7,Th.3.14]):
\begin{displaymath}
s(x)\;=\;\int_{\partial
D}s(\xi)d\omega(x,\xi;D)-\int_{D}g(x,\xi;D)dR(\xi).
\end{displaymath}
The second integral is often called the {\bf Riesz potential} from the
{\bf Green function} $g$.

From the above mentioned machinery of the potential theory we need only
some {\bf simple} result concerning the Green function $K_{6}(x) =
\mid\mid x \mid\mid_{6}^{-4}, x\ne 0$ , i.e. the case $m=3$.

Before we write the main needed formulas we recall the very important
notion of the {\bf Hilbert transform} on a local field .
Thus, let $K$ be a local field (not necessary commutative). For some
class of admissible functions $f$ defined on a Borel subset $B$ of $K$
we define {\bf $K$-Hilbert transform} $({\cal H}_{K}f)$ of $f$ as the
integral
\begin{equation}
\;({\cal H}_{K}f)(x)\;:=\;\int_{B}\frac{f(y)}{\Delta_{K}(x-y)}
dH_{K}(y);\;x\in B.
\end{equation}
If $B$ is pre-compact , then we say on a {\bf compact $K$-Hilbert
transform}. 
More generally, iff $\mu$ is an admissible measure on $K$ , then we can
define $K$-Hilbert transform $({\cal H}\mu)$ of a measure $\mu$:
\begin{equation}
\;\;\;({\cal
H}\mu)(x)\;:=\;\int_{K}\frac{d\mu(y)}{\Delta_{K}(x-y)}\;;\;x\in K.
\end{equation}

Observe that kernel of $K$-Hilbert transform is {\bf algebraic} and
{\bf Wiener-Hopf type}.

In the most important case of $K = \LR$ of reals , the "module" does not
appear, i.e. (cf.e.g. [Pi, Sections 5.3 and 5.4]) and the "true"
Hilbert transform is given as
\begin{displaymath}
\;({\cal
H}_{\LR}f)(x)\;=\;\frac{1}{\pi}\int_{-\infty}^{+\infty}\frac{f(y)dy}{y-x},
\end{displaymath}
whereas the {\bf bounded} Hilbert transform is defined as
\begin{displaymath}
\;\;(\tilde{\cal
H}_{\LR}f)(x)\;=\;\frac{1}{\pi}\int_{-1}^{+1}\frac{f(y)dy}{y-x},
\end{displaymath}
and leads to the theory of {\bf singular integrals} (cf.e.g. [St]).

The importance of $\tilde{{\cal H}}_{\LR}$ follows from the fact that
it solves a very important Fredholm equation of the first order:
\begin{equation}
\;\;\int_{-1}^{+1}\frac{f(y)dy}{y-x}\;=\;g(x),
\end{equation}
which is called the integral equation of the {\bf profile of a wing of
a plane} (cf.e.g. [Pi, Sect.5.4]), whereas ${\cal H}_{\LR}$ solves a
very important Fredholm equation of the second order of this kind- so called 
{\bf Carleman integral equation} (cf.[Pi,Sect.5.5]).
\begin{re}
Observe additionally, that (4.80) is an analogue of the family of the
important {\bf Abel integral equations} (in the class of the {\bf
Voltera integral eqations} of the first order):
\begin{equation}
\;\;\int_{0}^{x}\frac{f(y)dy}{(x-y)^{m}}\;=\;g(x).
\end{equation}
For $m$ with $0<m<1$ , the solution of (4.81)is given by a {\bf
self-similar} formula:
\begin{displaymath}
\;f(x)\;=\;\frac{1}{\Gamma(1-m)\Gamma(x)}\frac{d}{dx}\int_{0}^{x}\frac{g
(y)dy}{(x-y)^{(1-m)}}.
\end{displaymath}
It can be find in the Titchmarsh paper [Ti].
\end{re}
\begin{re}
In the case of $p$-adic fields $\LQ_{p}$, the $\LQ_{p}$-Hilbert transforms
probably first were considered in the {\bf Vladimirov} et al. paper'[VWZ]
as the $\gamma$-order {\bf derivative} $D^{\gamma}f$ of a locally
constant function $f$. It is describable by pseudo-differential
operator and explicitly written as 
\begin{equation}
D^{\gamma}f(x)\;=\;\int_{\LQ_{p}}\mid \xi
\mid_{p}^{\gamma}\hat{f}(\xi)\chi_{p}(-\xi x)H_{p}(d\xi)\;=\;=\frac{p
^{\gamma}-1}{1-p^{-\gamma-1}}\int_{\LQ_{p}}\frac{f(x)-f(y)}{\mid
x-y\mid_{p}^{\gamma+1}}H_{p}(dy),
\end{equation}
where $\chi _{p}$ is the additive character of $\LQ_{p}$ and $\hat{f}(\xi)$
stand for the Fourier transformation $\int_{\LQ_{p}}\chi_{p}(\xi
x)f(x)H_{p}(dx)$ of the function $f$.
The deeper analysis of p-adic fractional differentiation $D^{\gamma}$
is given in the Kochubei' paper [Ka] , where using Madrecki's theorem
he established the existence of a {\bf Kachubei-Gauss measure} $\mu$ over
infinite-dimensional field extensions $\Omega_{p}$ of $\LQ_{p}$ , which is a 
{\bf harmonic measure} for $D^{\gamma}$ and solved p-adic integral equations
of a profile of wing of a plane.(cf.[Ka,Prop.6]).

In the sequel, it will be very convenient to use the {\bf language of
Hamilton's quaternions} $H$ (cf.e.g. [Si,])- probably the most important example of a
local and non-commutative field. 
From the point of view of the  theory of vector spaces over $\LR$, the field
of quaternions $H$ is only the 4-dimensional euclidean-space
$(\LR^{4},\mid\mid\mid .\mid\mid_{4})$.

From the physical point of view, it is only a model of the {\bf
Einstein-Minkowski time -space}: $H = T\times \LR^{3}$, i.e. the
quaternions $h \in H$ can be considered as a position $r = (x,y,z)\in
\LR^{3}$ of a {\bf particle} at a {\bf momentum} $t\in T=\LR$ , i.e.
\begin{displaymath}
 h \;=\;(t,r)\;=\;(t,x,y,z)\in  H.
\end{displaymath}

Finally, from the {\bf algebraic} point of view, it is very convenient
to consider $H$ as a product of the complex planes : $H = \LC\times \LC$.
It means that 
\begin{displaymath}
\;\;\;H\in h\;=\;(z_{1},z_{2}) \;\in\;\LC \times \LC.
\end{displaymath}

Obviously, for all $h = (z_{1},z_{2}) = (t,x,y,z)$  we have
\begin{equation}
\;\mid h \mid_{H}\;=\;\sqrt{\mid z_{1}\mid^{2}_{\LC}+\mid
z_{2}\mid^{2}_{\LC}} \;=\;\sqrt{\mid t \mid_{\LR}^{2}+\mid x
\mid_{\LR}^{2}+\mid y \mid_{\LR}^{2}+\mid z\mid_{\LR}^{2}},
\end{equation}
and (cf.e.g. [We,p.11?])
\begin{equation}
\;\;\Delta_{H}(h)\;\;=\;\;\mid h\mid^{4}_{\LH}.
\end{equation}
\end{re}
For each $r_{1},r_{2}>0, r_{1} <r_{2}$ , let's consider a {\bf ring} of
$\LR^{m}$ : $R^{m}(r_{1},r_{2})\;:=\;\{x\in \LR^{m} : r_{1} \le \mid\mid x
\mid\mid_{3} \le r_{2}\}$ (with the radiuses :inner $r_{1}$ and outer
$r_{2}$). By $S^{m-1}_{\infty}(r):= \{x\in \LR^{m}:\mid\mid x \mid\mid_{m} =
r\} = R^{m}(r,r)$ we denote the $(m-1)$-dimensional {\bf sphere} of radius $r$.
Moreover we write $S^{m}_{\infty}$ instead of $S^{m}_{\infty}(1)$.
\begin{pr} ({\bf The existence of the $\LH$-profile of a wing of a plane
equation in measures})\\
There exists such a probability Borel measure $R$ on $S^{3}$ that for each
$r\in X_{\LH}(M,N)$ with $N>M>0$  the following Abstract Hodge
Decomposition holds
\begin{equation}
\;\Delta_{\LH}^{-1}(l^{2})\;=\;\int_{S^{3}}\frac{dR(y)}{\Delta(r^{2}-y^{2})}.
\end{equation}
\end{pr}
{\bf Proof}.(1) First we give a proof of this proposition on the "intuive
level" based on the Riesz theorem.
Let $\epsilon_{n}>0$ be arbitrary sequence which converges to zero.
Then the function $\vert\vert \cdot \vert\vert_{6}^{-(4+\epsilon_{n})}$
is {\bf subharmonic} as a suitable power of {\bf harmonic} $K_{6}$
(cf.e.g. [HK,Sect.2.3, Corollary 2]) and obviously is not harmonic!
Therefore, according to the {\bf Riesz theorem}, there exists a
sequence of {\bf Riesz measures} $\{R_{n}\}$ and a sequence $\{h_{n}\}$
of harmonic functons inside of $S^{3}$ with the property
\begin{equation}
\;\mid\mid r \mid\mid_{6}^{-(4+\epsilon_{n})}\;=\;\int_{S^{3}(=\{h\in
H: \mid h \mid_{H}=1\})} \frac{dR_{n}(x)}{\mid\mid r-x
\mid\mid_{6}^{4}}\;+\;h_{n}(r).
\end{equation}

Since $dR_{n}(x) = \bigtriangledown(\mid\mid
x\mid\mid_{6}^{-(4+\epsilon_{n})})dx$ (cf.[HK,Section 3.5]). Therefore
the sequence $\{R_{n}(S^{3})\}$ is {\bf bounded}, i.e. $R_{n}(S^{3})\le
A$ , for some $A>0$ and all $n\in \LN$.

According to the {\bf Frostman's theorem}(cf.[HK, Theorem 5.3]), we can
choose a subsequence $\{R_{n_{p}}\}$ which is {\bf weakly convergent}
to a limit measure $R_{\infty}$ on $S^{3}$, i.e. $R_{\infty} :=
(w)lim_{p\longrightarrow \infty} R_{n_{p}}$.
In particular, we obtain that there exists $h(x) :=\lim_{p\rightarrow
+\infty}h_{n_{p}}(x)$ , which is harmonic inside $S^{3}$(Harnak?) and
\begin{equation}
\;\mid\mid r
\mid\mid_{6}^{-4}\;=\;\int_{S^{3}}\frac{dR_{\infty}(x)}{\mid\mid r-x\mid\mid
_{6}^{4}}\;+\;h(x).
\end{equation}
Since $int(S^{3}) = \empty$ then without loss of generality we can
assume that $h\equiv 0$.

On the other hand, according to the {\bf Brelot's
theorem}(cf.[HK,Theorem 3.10]) applied to the triplet
$(D_{6}(r),0,S^{5}(r))$- there exists the {\bf harmonic measure}
$\omega_{r}(\cdot) := \omega_{r}(\cdot,0;D_{6}(r))$ na $S^{5}(r)$ with
the property (cf.[HK,Lemma 3.1])
\begin{equation}
\;\;\mid\mid l
\mid\mid_{6}^{-4}\;=\;\int_{S^{5}(r)}\frac{d\omega_{r}(y)}{\mid\mid
l-y \mid\mid_{6}^{4}};\;\;\;l\in D_{6}(r).
\end{equation}
Consider the map of measure spaces $j_{3}: (S^{3},R_{\infty})\longrightarrow
(S^{5}(r),j_{3}^{*}(R_{\infty}))$ , where $j_{3}(x_{1},x_{2},x_{3}) =
(rx_{1},rx_{2},rx_{3},0,0,0)$ .Then 
\begin{equation}
\;\;\int_{S^{3}}\frac{dR_{\infty}}{\mid\mid rx - l \mid\mid_{6}^{4}}
\;=\;\int_{S^{5}(r)}\frac{d(j_{3}^{*}R_{\infty})(y)}{\mid\mid y-l
\mid\mid_{6}^{4}}.
\end{equation}

We know (it is well-know), as a huge consequence has got (pociaga za
soba) fact, that a Haar measure on LC group is uniquely determined (up
to the constant multiple)- there existence of $\Delta$ and its main
properties.

Now, we explore the strong facts that the {\bf Riesz} and {\bf harmonic}
measures are unique (in the well-know sense) and that the solution of
the {\bf Dirichlet problem} has always unique solution. 
Thus, from the one-hand side we have 
\begin{equation}
\;\mid\mid l
\mid\mid_{6}^{-4}\;=\;\int_{S^{5}(r)}\frac{d(j_{3}^{*}R_{\infty})(y)}{
\mid\mid l-y \mid\mid_{6}^{4} \;+\;h(l).
\end{equation}
On the second-hand we have
\begin{equation}
\;\mid\mid l
\mid\mid_{6}^{-4}\;=\;\int_{S^{5}(r)}{d\omega_{5}^{r}}\mid\mid l-y
\mid\mid_{6}^{4}.
\end{equation}
Let's consider the function $f_{l_{0}}(y) := \mid\mid l_{0} - y
\mid_{6}^{-4} ; y\in S^{5}(r)$ on $S^{5}(r)$ under fixed $l_{0}\in
D_{6}(r)- S^{5}$.
All this gives the assumpt of the following hipothetical statements :
$j_{3}R_{\infty} = \omega$ and $h\equiv 0$.\\
({\bf Non-spectacular - based on the Brelot's theorem}).\\
Let's consider the sequence of {\bf regular} domains (in the sense of
Th.3.1 of [HK]):
\begin{displaymath}
\;\;D_{n}^{6}\;:=\;B^{4}\times [1-1/n,1+1/n]^{2}\cup B^{6}(1/n) 
\end{displaymath}
, of $\LR^{6}$. Then according to the Lemma 3.9 of [HK] applied for the
pair $(D_{n}^{6},K_{6})$ and $x=0, r=\eta$ we get
\begin{equation}
\int_{F_{n} = \partial D_{n}^{6}}\frac{d\omega(0,\xi;D_{n}^{6}))}
{\mid\mid  \xi - r \mid\mid_{6}^{4}}\;=\;\frac{1}{\mid\mid r
\mid\mid_{6}^{4}},
\end{equation}
, if $r\nin D_{n}^{6}$, \$where $\omega(0,.;D_{n}^{6})$ is the {\bf harmonic
measure} in $0$ w.r.t. $D_{n}^{6}$.

We extend each measure $\omega_{n}$ to the measure $\tilde{\omega_{n}}$
on $D_{1}^{6}\{0\}$ by the restriction procedure:
\begin{displaymath}
\;\;\tilde{\omega _{n}}(B)\;:=\;\omega_{n}(B\cap(D_{n}^{6}-\{0\})),
\end{displaymath}
if $B$ is a Borel set in {\bf precompact $D_{n}^{6}-\{0\}$}.

Since $\omega_{n}(\partial D_{n}^{6}) = 1$  for each $n$, the sequence
$\{\tilde{\omega_{n}}\}$ of measures on $D_{1}^{6}-\{0\}$ is {\bf
bounded}. According to the {\bf Frostman theorem}(cf.[HK,Th.5.3.]) (in
fact its easy modification for {\bf precompacts} and {\bf continuous
bounded} functions), we can assume that $\{\tilde{\omega_{n}}\}$ is
{\bf weakly convergent} to a measure $\tilde{\omega}$ on $D_{1}^{6}-\{0\}$
(after a choice of a subsequence), i.e.
\begin{displaymath}
\;\;\;\tilde{\omega}\;=\;(w)\lim_{n\longrightarrow \infty}
\tilde{\omega_{n}}.
\end{displaymath}

According to the "cobordysm theory"
\begin{displaymath}
\partial D_{n}^{6}\;=\;\partial(B^{4}\times [1-1/n,1+1/n])\cap \partial
B^{6}(1/n)\;=\;
\end{displaymath}
\begin{displaymath}
\partial B^{4}\times [1-1/n,1+1/n]^{2}\cup B^{4}\times \partial
[1-1/n,1+1/n]^{2}\cup S^{5}(1/n). 
\end{displaymath}
Thus, we see that there exists a natural {\bf projective system
(sequence)} : $\{\partial D_{n}^{6}, p_{mn}\}$, where 
\begin{displaymath}
p_{mn}(x)\;=\;
\end{displaymath}
which {projective limit} is homeomorphic with the $3$-dimensional
sphere $S^{3}\cap \{0\}$.

Moreover, with the above projective sequence of compact topological spaces
is associated the {\bf projective seqence of measures} :
$\{\omega_{n},p_{mn}\}$ . According to the  {\bf Prohorov-Kisynski
theorem} (cf.e.g. [Mau]), there exists the {\bf projective limit of measures}
$\tilde{\omega}$.

Let's consider a sequence of functions $\{f_{n}^{m}\}$ on $D_{1}^{6}-\{0\}$
with the following three properties:\\
(1). $f_{n}^{m}(x) = 1$ if $x \in \partial D_{n}^{6}$,\\
(2). $0\le f_{n}^{m} \le 1$ on $D_{1}^{6}-\{0\}$, and  $f_{n}^{m}$ is
continuous,\\
(3). $,\lim_{m}f_{n}^{m}=\chi_{\partial D_{n}}\lim_{n,m}f_{n}^{m}(x) = \chi_{S^{3}}(x), x\in D_{1}^{6}^{*}$ , where $\chi_{S^{3}}$ is the
characteristic function of $S^{3}$.\\ 
Shortly,  $f_{n}(x)$ is a "continuous modification" of $\chi_{\partial
D_{n}^{6}}(x)$.

We show that
\begin{equation}
\lim_{n\longrightarrow +\infty}\int_{D_{1}^{6*}}\chi_{\partial D_{n}^{6}}(x)
d\tilde{\omega_{n}}\;=\;\int_{D_{1}^{6*}}\chi_{S^{3}}(x).
\end{equation}
Reely, for each $n$ we can find such $m$ that
\begin{displaymath}
\mid \int \chi_{\partial D_{n}} d\tilde{\omega_{n}} -\int
\chi_{S^{3}}d\tilde{\omega}\mid\;\le\;\mid \chi_{\partial D_{n}}d
\tilde{\omega_{n}} -\int f_{m} d\tilde{\omega_{n}}\mid +
\end{displaymath}
\begin{displaymath}
+ \mid \int f_{m}d(\tilde{\omega_{n}} - tilde{\omega})\mid \;+\;\int
\mid f_{m} - \chi_{S^{3}}\mid d\tilde{\omega}.
\end{displaymath}
Reely, let $\epsilon>0$ be arbitrary. Then - basing on the construction
of $f_{m}^{n}$ - for each $n$, there exists such $N(n,\epsilon)$ , that
for all $m \ge N(m,\epsilon)$, 
\begin{displaymath}
\mid \int \chi_{\partial D_{n}} d\tilde{\omega_{n}} - \int
f_{n}^{m}d\tilde{\omega_{n}} \mid\;< \epsilon,
\end{displaymath} 
(the Lebesgue dominated convergence theorem).

Thus, according to (.) we finally obtained
\begin{equation}
\frac{1}{\mid\mid r \mid\mid_{6}^{4}}\;=\;\lim_{n\longrightarrow 
\infty}\int_{\partial D_{n}^{6}} \frac{d \omega _{n}(y)}{\mid\mid
r-y\mid\mid_{6}^{4}}
\;=\;\int_{S^{3}}\frac{d\tilde{\omega}(y)}{\mid\mid r-y
\mid\mid^{4}_{6}}; \;r\in D_{1}^{6} - B^{6}(1/2).
\end{equation}

In the sequel we write $R_{\infty}$ instead of $\tilde{\omega}$.

So,if we restrict ourselves in () for $h\in \LH$ , we finally get
\begin{equation}
\;\;\Delta_{\LH}(h)^{-1} \;=\;\int_{S^{3}}\frac{dR(x)}{\Delta (
h-x )}\;\;h\in R_{4}(r,r^{-1}) (r\in (0,1)).
\end{equation}
Let's consider a {\bf branch of the hamiltonian square} $\sqrt{\cdot}$
and the induced map of measure spaces : $\sqrt :
(S^{3},R)\longrightarrow (S^{3}, \sqrt{\cdot}^{*}R)$. Putting in (.)
$h^{2}$ instead of $h$ we finally get:
\begin{equation}
\;\;\Delta_{\LH}(h^{2})\;=\;\int_{S^{3}}\frac{d(\sqrt{\cdot}R_{\infty})(y)}
{\Delta_{\LH}(h^{2}-y^{2})}\;=\;{\cal
H}_{\LH}(\sqrt{\cdot}R_{\infty})(h^{2}),
\end{equation}
where $h \in R_{H}(r^{-1},r)$.

\begin{th}({\bf On the existence of (RH)-fixed points of ${\cal F}$)\\
Let $A = A(x)$ be a positive CID-amplitude with $A(0) = 0$ and $A(x)
\searrow 0$ for $\mid x \mid \ge 1$.
Then , there exists such a {\bf fixed point} $\omega$  of ${\cal F}$
(from ${\cal S}(\LR)$) - the (RH)-fixed point, that
\begin{equation}
\;\;\;(\omega\;-\;G)(x)\;=\;A(x)\;;\;x\in \LR_{+}.
\end{equation}
In particular
\begin{equation}
\;\;\;\omega(0)\;=\;G(0)\;=\;1.
\end{equation}
\end{th}
{\bf Proof}. We start from the recalling of the fact, that we have got
to the desposition? the whole family 
\begin{equation}
\;\;F(g_{A})(x)\;:=\;\{f_{\lambda}(x) : \mid \lambda \mid\}
\end{equation}
of solutions of the family of the Fox equations:
\begin{equation}
\;f_{\lambda}(x)\;+\;\lambda
\hat{f}_{\lambda}(x)\;=\;g_{A}(x):=G(x)+A(x).
\end{equation}

Since the consideration of the pair $(\LR, the Lebesgue measure)$ seems
to be {\bf unsufficient}, then we work in this moment with an arbitrary
local field $K$.

We make in (.) the substitution : $\lambda = \Delta_{K}^{2}(k) ; k\in K$
, with $\Delta_{K}(k) \ne 1$ to obtain
\begin{equation}
\frac{f_{\Delta^{2}_{K}(k)}(x)}{\Delta_{K}^{2}(k)}\;+\;\hat{f}_{\Delta^{2}_{K}}
(k)(x)\;=\;\frac{g_{A}(x)}{\Delta_{K}^{2}(k)},
\end{equation}
where $x \in \LR$ and $k \in K$.
Integrating the both sides of (.) with respect to the left {\bf Haar
measure} $H_{K}$ of $(K,+)$ and applying the formula (4.74) we obtain
\begin{equation}
\int{X_{N}(K)}\Delta^{-2}_{K}(k)
f_{\Delta^{2}_{K}(k)}(x)dH_{K}(k)\;+\;\int{X_{N}(K)}\Delta_{K}^{-2}(k)
\hat{f}_{\Delta^{-2}_{K}(k)}(x)dH_{K}(k)
\;=\;g_{A}(x)\int_{X_{N}(K)}\Delta^{-2}_{K}(k)dH_{K}(k) =:M_{-2}(N,K)).
\end{equation}
In the sequel it is convenient to call $M_{-2}(N,K)$  the {\bf $-2-K$-moment} of
the Haar measure $H_{K}$ on the $X_{N}(K)$.

Now, it is the time to use {\bf compact-  $K$ -Hilbert transform}
${\cal H}_{K}$ (in the case of $K = \LH$) to the expressions:
\begin{equation}
\;\;\;\int_{X_{N}(K)}\frac{f_{\Delta_{K}^{\pm 2}(k)}(x)
dH_{K}(k)}{\Delta_{K}^{2}(k)}.
\end{equation}
According to the Proposition 6.1? we have
\begin{equation}
\;\frac{1}{\Delta_{\LH}(h^{2})}\;=\;\int_{S^{3}}\frac{dR(y)}{\Delta_{H}(h
^{2}-y^{2}})\;=\;{\cal H}_{H}(R_{\infty})(h^{2}),
\end{equation}
if $h \in X_{N}(H)$.

Applying the {\bf Fubini theorem} we obtain:
\begin{equation}
\int_{X_{N}(H)}\frac{f_{\Delta^{\pm
2}_{H}(h)}(x)dH_{H}(h)}{\Delta_{H}(h^{2})}\;\;=\;\;\int_{X_{N}(H)}f
_{\Delta^{\pm 2}_{H}(h)(x)} (\int_{S^{3}}\frac{dR(y)}{\Delta_{H}(h^{2}-
y^{2})}dH_{H}(h))=
\end{equation}
\begin{displaymath}
=\;\;\int_{S^{3}}dR(y)(\int_{X_{N}(H)}\frac{f_{\Delta^{\pm 2}_{H}(h)}(x)
dH_{\LH}(h)}{\Delta_{\LH}(h^{2}-y^{2})}).
\end{displaymath}

But, the second inner integral in the above iterated integral, we can
write in the form:
\begin{equation}
\int_{X_{N}(\LH)}\frac{f_{\Delta^{\pm
2}_{H}(\frac{h}{y})}(x)dH_{H}(h)}{\Delta^{2}_{\LH}(y)\Delta_{H}(1-
\frac{h}{y})} =\int_{X_{N}(H)} f_{\Delta_{H}^{\pm
2}(h)(x)dH_{\LH}(h)}{\Delta_{\LH}(1-h^{2})},
\end{equation}
according to the formula (W).

But $\frac{dH_{\LH}(h)}{\Delta_{H}(1-h^{2})} =: \mu_{BK}$ is a {\bf
Bogoluboff-Kriloff measure} of $I_{\LH^{*}}$, i.e.
\begin{equation}
\;\int_{X_{N}(H)}\frac{f_{\Delta^{2}(h)}(x)d
H_{\LH}(h)}{\Delta_{H}(1-h^{2})}\;=\;\int_{X_{N}(H)}f_{\Delta^{-2}_{\LH}(x)d
H_{\LH}(h)}{\Delta_{H}(1-h^{2})},
\end{equation}
since, for each integrable function $\phi$ the following calculus is
true
\begin{displaymath}
\int_{X_{N}(H)}\phi(I_{H^{*}}(h))d\mu_{BK}(h)\;=\;\int_{X_{N}(H)}\phi(h)d
(I^{*}_{H^{*}}\mu_{BK})(dh)\;=
\end{displaymath}
\begin{displaymath}
\;=\;\int_{X_{N}(\LH)}\phi(h)d\mu_{BK}(h).
\end{displaymath}

Now, observe that $f_{1}$ wasn't defined yet. Therefore we define
\begin{equation}
\;\;\;f_{1}(x)\;:=\;R(S^{3})\int_{X_{N}(H)}\frac{f_{\Delta_{\LH}^{2}(h)}
(x)d H_{H}(h)}{\Delta_{\LH}(1-h^{2})}.
\end{equation}

Applying the {\bf Fubini theorem} we finally get, that there {\bf exists
a singular solution} $f_{1}(x)$ of the Fox equation (.) :
\begin{equation}
\;\;\;f_{1}(x)\;\;+\;\;\hat{f_{1}}(x)\;\;=\;I(-2)g_{A}(x).
\end{equation}
Finally putting
\begin{equation}
\;\omega(x)\;:= f_{1}(x)\;+\;\hat{f_{1}}(x)\;
\end{equation}
we obtain the required (RH)-fixed point.

\begin{pr}({\bf On the positivity of the (Rhfe)-trace}).\\
For each (RH)-fixed point $\omega_{A}$ of ${\cal F}$ associated with
a CID-amplitude $A$ as in the Th.2 ,for each $a>0$ and complex $s=u+iv$
with $u\in [0,1/2]$ and $v>0$, the following inequality
holds:
\begin{equation}
Tr_{an}(\zeta,A)(s):=\int_{1}^{+\infty}(x^{u-1}-x^{-u})\theta(A)(x)
sin(2\pi v ln x)dx\;=\;
\end{equation}
\begin{displaymath}
={\cal F}_{+}(\theta(A(exp)))\cdot Exp(u) \cdot sin(2\pi L_{v})\;>\;0,
\end{displaymath}
,where by $Exp(u)$ we denoted the function $e^{ux}-e^{(1-u)x}$ of the
variable $x$.
\end{pr}
{\bf Proof}. We have
\begin{equation}
Tr_{an}(\zeta,A)(s)
\;=\;\sum_{n=1}^{\infty}\int_{1}^{+\infty}(x^{u-1}-x^{-u})A(nx)sin(2\pi
v lnx)dx=
\end{equation}
(after the substitution $x = e^{r}$)
\begin{displaymath}
=\sum_{n=1}^{+\infty}\sum_{n=1}^{+\infty}(e^{ru}-e^{r(1-u)})A(ne^{r})sin
(2\pi vr) dr \;=\;
\end{displaymath}
\begin{displaymath}
:=\sum_{n=1}^{+\infty}{\cal F}_{+}(EXp(u)\cdot A(nexp)\cdot sin(2\pi
L_{v})).
\end{displaymath}
Consider the {\bf amplitudes} ${\cal A}_{n}(r):=
e^{ru}(1-e^{r(1-2u)})\cdot A(ne^{r})$. Since
$\frac{d}{dr}(1-e^{r(1-2u)})= (2u-1)e^{r(1-2u)}<0$ for $u \in (1/2,1]$,
then  ${\cal A}_{n}(r)$ is strictly positive CID-amplitude (with ${\cal
A}_{n}(0) = 0$)

\section{The Proof of the Riemann Hypothesis}
\begin{th}
There exists such functions $F_{1}(s)$  and $F_{2}(s)$ ({\bf $\Gamma \theta
sinlog$-factors}) and real valued $f_{1}(s)$ and $f_{2}(s)$
({\bf $\theta$ sinlog-factors}) with $f_{1}(s)-f_{2}(s)\ne 0$ that the following
{\bf Riemann hypothesis functional equation} (with a {\bf rational
term} $I$ and the {\bf action of $Gal(\LC/\LR)$}) (R.h.f.e. in short) holds for 
$Re(s)\in (0,1/2)$ :
\begin{equation}
(Rhfe)\;\;\;\;Im((F_{1}\zeta)(s)+(F_{2}\zeta)(c(s)))\;=\;\frac{(f_{1}(s)-
f_{2}(s))}{\mid s(s-1)\mid^{2}} I(s).
\end{equation}
\end{th}
{\bf Proof}. Let $a_{1}>a_{2}>0$ be aritrary ({\bf artificially chosen
$\zeta-Cramer$ initial conditions}) and let $s = u+iv =Re(s)+iIm(s)$ be a fixed 
complex number. Let's consider the simple {\bf non-homogeneous system} of two 
linear equations with two variables $p_{1}$ and $p_{2}$:
\begin{equation}
\;\;\;\;p_{1}v(u-1)\;+\;p_{2}vu\;=\;a_{1}-a_{2}
\end{equation}
\begin{displaymath}
\;\;\;\;p_{1}v(u-1)\;+\;p_{2}vu\;=\;a_{2}-a_{1}
\end{displaymath}
The system (3.39) is the {\bf Cramer system} iff $s = (u,v)$ belongs to the
{\bf algebraic $\LR$-variete} $I(\LC)$ . The main determinant of (3.38) is
$I(s)$ and its solution is given by
\begin{equation}
p_{1}\;=\;p_{1}(Im(s);a_{1}-a_{2})\;=\;\frac{(a_{1}-a_{2})I(s)}{3vI(s)}=
\frac{(a_{1}-a_{2})}{3v}>0
\end{equation}
and
\begin{equation}
p_{2}\;=\;p_{2}(Im(s);a_{1}-a_{2})\;=\;\frac{(a_{2}-a_{1})T(s)}{3vI(s)}=
\frac{(a_{2}-a_{1})}{3v} =- p_{1}<0.
\end{equation}
Let's denote by $J_{1}(s)$ the integral in the right-hand side of
$Im(F.p.t.f.e)$ for $\omega_{A}$  and by $J_{2}(s)$ the respectible
integral in $Im(Ftfe)$ for $\omega = G_{2}$.
The main observation on $J_{1}$ and $J_{2}$
is that they are {\bf "quasi-invariant"} under the changes of variables
w.r.t. the {\bf algebraic} substitutions : $x = y ^{r}$ for positive $r$.
For $p_{1}>0$ given by (3.38) and for $v>0$ let's change  variables in
$J_{1}$ by the substitution : $x = y^{p_{1}v}$. Then
\begin{equation}
J_{1}(s)=p_{1}v\int_{1}^{\infty}(y^{p_{1}v(u-1)}-y^{-p_{1}vu})
sin(logy^{p_{1}v})
\theta(G_{1})(y^{p_{1}v})y^{(p_{1}v-1)}dy=:J_{1}^{r}(s).
\end{equation}
Analogously, for $p_{2}<0$ and $v>0$ we change variables in $J_{2}$
according to the substitution : $ x = y^{-p_{2}v}$. Then
\begin{equation}
J_{2}(s)=-p_{2}v\int_{1}^{+\infty}(y^{-p_{2}v(u-1)}-y^{p_{2}vu})
sin(log y^{-p_{2}v})\theta(G_{\sigma_{0}})(y^{-p_{2}v})y^{-(p_{2}v
+1)}dy=:J_{2}^{r}(s) ,
\end{equation}
($r$ means right).

Let's observe that although the integrals $J_{i}$ and $J_{i}^{r}$ are
equal however the subintegral functions are different.

Now is the unique non-algebraic (purely analytic) part of considerations.

According (3.40) and (3.41) the equalities :$J_{i}(s) = J_{i}^{r} ;
i=1,2$ hold on the {\bf Poincare half-plane} ${\cal P} :=\{s\in \LC:
Im(s)> 0\}$. But, without loss of generality, we can assume that those equalities 
of integrals hold on $Im(s) = 0$. But obviously they are imaginary parts of 
{\bf holomorphic} $\Gamma(G_{i})\zeta-1/W$ defined on the whole {\bf
connected} domain $\LC-\{0,1\}$. Thus,  they must be equal on the whole complex
domain $\LC-\{0,1\}$(cf.e.g. [Ma,Th.XV.2]). Thus the equality of integrals is
{\bf preserved} after the action of the {\bf complex conjugation} $c$ to the 
both sides of equality : $J_{2} = J_{2}^{r}$, i.e. we get
\begin{equation}
J_{2}(c(s))\;=\;p_{2}v
\int_{1}^{\infty}(y^{p_{2}v(u-1)}-y^{-p_{2}vu})sin(logy^{p_{2}v})
\theta(G_{2})(y^{p_{2}v})y^{(p_{2}v-1)} dy.
\end{equation}
Now, if we apply the elementary {\bf mean value theorem} for non-proper
integrals $J_{i}^{r}$ (cf.e.g.[Fi]) we obtain that there exists such
$y_{i} = y_{i}(s,a)$ from $[1,+\infty)$ that
\begin{equation}
J_{i}(c_{i}(s))=p_{i}v
sin(logy_{G}^{p_{i}v^{2}})\theta(G_{i})(y_{i}^{p_{i}v})y_{i}^{p_{i}v-1}\cdot
\int_{1}^{\infty}(y^{p_{i}v(u-1)-a_{i}}-y^{-p_{i}vu-a_{i}})dy
\end{equation}
where by $c_{1}$ we denoted the identity map $id_{\LC}$ and by $c_{2}$
the complex conjugation $c$.
(May be even these elementary analytic facts we would not recall and
write but we enclosed them for the sake of completness).

Obviously we can always take artifficial $a_{i}$ in (3.38) in such a
manner that integrals in the right-hand side of (3.44) are {\bf
convergent}.
Let's define {\bf $\theta sinlog$-factors} as
\begin{equation}
f_{i}(s)=f_{i}(Im(s)) =p_{i}Im(s)sin(logy_{i}^{p_{i}Im^{2}(s)})
\theta(G_{i})(y_{i}^{p_{i}Im(s)})y_{i}^{p_{i}Im(s)-1} ;
\end{equation}
$i=1,2$.
If now we write down (Fptfe) (2.30) in the case of fixed points $\omega = 
G_{i}$ , change variables as above and use the above factors we obtain
\begin{equation}
Im((\Gamma(G_{i}) \zeta)(s))=\frac{I(s)}{2\mid
s(s-1)\mid^{2}}+f_{i}(s)\int_{1}^{+\infty}(y^{p_{i}v(u-1)-a_{i}}-y
^{p_{i}vu-a_{i}})dy.
\end{equation}
Hence we obtain the following system of two eqations
\begin{equation}
Im((\Gamma(G_{1})f_{2}\zeta)(s))=\frac{f_{2}(s)I(s)}{2\mid
s(s-1)\mid^{2}}+
\end{equation}
\begin{displaymath}
+(f_{1}f_{2})(Im(s))\int_{1}^{+\infty}(y
^{p_{1}Im(s)(Re(s)-1)-a_{1}}-y^{-p_{1}Im(s)Re(s)-a_{1}})dy
\end{displaymath}
\begin{displaymath}
Im((\Gamma(G_{2})f_{1}\zeta)(c(s)))=
\frac{-f_{1}(s)I(s)}{2\mid s(s-1)\mid^{2}}+
\end{displaymath}
\begin{displaymath}
+(f_{1}f_{2})(Im(s))\int_{1}^{+\infty}(y^{p_{2}Im(s)(Re(s)-1)-a_{2}}-y
^{p_{2}Im(s)Re(s)-a_{2}}dy.
\end{displaymath}
Let, s introduce the {\bf $\Gamma \theta sinlog$-factors} by the
formula
\begin{equation}
\;\;F_{1}(s)\;:=\;(\Gamma(G_{1})f_{2})(s)\;and\;F_{2}(s)\;:=\;(\Gamma(G_{2})
f_{1})(s).
\end{equation}
If we add the equations in (3.47) - side to side -, then according to
(3.47) we get $(Rhfe_{G})$. 
Finally, according to the Lemma 3,  $(J_{1}-J_{2})(s)>0$ if $Re(s)\in
(0,1/2)$. Since obviously integrals in the right-hand sides (3.44) and
(3.39) of $J_{i}$ are {\bf equal} each other then
\begin{displaymath}
\;\;\;f_{1}(Im(s))-f_{2}(Im(s))\;\ne 0\;\;for\;Re(s)\in (0,1/2),
\end{displaymath}
which proves Theorem 2.
\begin{re}
The following natural question arises : why the important is the
imaginary part of t.f.e but the real one is not ( so it is a strange
{\bf violation (breaking) of symmetry} in the Riemann hypothesis
problem). May be the explanation is the following : in the integral
$Im(J(s,\omega)) = Im(\int_{\LR}(x^{s-1}+x^{-s}))\theta(\omega)(x)dx$
the exponential factor has the form $(x^{u-1} -x^{-u})$, whereas in
$Re(J)$ the exponential factor is $(x^{u-1}+x^{-u})$. So, in the
imaginary part appears the arithmetic operation $"-"$, whereas in the
real part $"+"$. Now, let's observe that the formula : $l([a,b]) :=
(b-a)$ gives the very important (and fundamental) notion of the {\bf
Lebesgue measure} on $\LR$, whereas the formula : $c([a,b]) :=
\frac{(b+a)}{2}$ gives less important notion of the {\bf centre} of interval 
. Moreover the fade wave volume of the form : $
A_{s} := \int_{0}^{+\infty}A(x)sin(ax)dx$ for a non-increasing function
$A(x)$ is always {\bf non-vanishing} , whereas non-vanishing of the
integral $A_{c} := \int_{0}^{\infty}A(x)cos(ax)$ needs some additional
conditions on $A$.
\end{re}
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\end{document}