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\bf
\centerline
{Proof of the Riemann Hypothesis, and some other Hitherto Undemonstrated
Propositions}
\rm
\bigskip
\centerline{ {\it Shalosh B. Ekhad}\footnote{$^1$}
{\eightrm  \raggedright
Department of Mathematics, Temple University,
Philadelphia, PA 19122, USA. 
%\break
e-mail:{\eighttt ekhad@math.temple.edu .}
www:{\eighttt http://www.math.temple.edu/$\tilde{\,\,}$ekhad   .}
April 1, 1995.
} 
}
 
{\bf Introduction}
 
One suggestion by Pierre Cartier (see Ruelle's
``Chance and Chaos'' p. 193) is that the axioms of
set theory are actually inconsistent, but that a proof of contradiction
would be so long that it could not be performed in our physical universe!
 
While the consistency of the axioms of set theory remains open,
in this note I announce, and sketch the proofs, of several hitherto
open problems. The proofs follow the celebrated WZ methodology, suitably
extended, that reduces the truth of statements to routinely verifiable
{\it certificates.} Fortunately, the lengths of these certificates
fit very comfortably, even in our very own planet. Unfortunately  
they far exceed the hard-disk allocation of 30MB of my ftp and
http directories. The application for enlarged allocation is still
pending. 
 
On the other hand, perhaps the actual certificates are besides the point.
John Littlewood even had a name for those pedants who insist on seeing the
certificates and carrying out the purely routine algebra: {\it obtuse.}
In fact, this is a good way to find out who is obtuse and who is not.
Once my application for extended disk-space is granted, I will take
a record of all those who try to download the certificates, and
this will be used to compile a {\it Directory of Obtuse Mathematicians},
that would be viewable for a modest fee, that will help defray the cost
of maintaining the ftp facility.
 
{\bf Two Proofs of the Riemann Hypothesis}
 
{\bf ``Elementary Proof''}
 
Let $\psi(x):=\sum_{p^m \leq x} \log p$. Tchebychev proved that,
for large $x$,
$A_1 x \leq \psi(x) \leq A_2 x$, by using the ridiculously simple
recurrence:
$$
(\log 2) \cdot x +O(log(x)) = \psi(x)-\psi(x/2)+\psi(x/3)-\psi(x/4) + 
\dots \quad ,
$$
that yields $A_1=\log 2$ and $A_2=2 \log 2$.
 
This was  considerably improved by 
Tchebychev himself and James Joseph Sylvester, who found
other, more complicated, but still linear, recurrences, that brought
$A_1$ up and $A_2$ down. 
 
The next step was realized by Erd\"os and Selberg, who combined
two simple recursive inequalities for $R(x):=\psi(x)-x$. The first one
is linear (in $\vert R(x)\vert$):
$$
\vert R(x) \vert \leq {{1} \over {\log x}} \sum_{n \leq x}
\vert R(x/n) \vert + O( {{x \log \log x} \over { \log x}}) \quad ,
$$
while the second one is quadratic:
$$
\sum_{n \leq x} {{\log n} \over {n}} R(n) =
- \sum_{n \leq x} {{1} \over {n}} R(n) R( x/n) + O(x) \quad,
$$
 
from which it follows that $\vert R(x)\vert\leq \sigma x$ 
for $x>x_\sigma$, for every $\sigma >0$.
 
Our enhancement of the Erd\"os-Selberg method is related to it in the
same way as it is related to the Tchebychev-Sylvester method.
 
By using {\it advanced sieving}, we derive, much longer, 
{\it super-holonomic} recursive inequalities that immediately imply
that $\pi(x)-li(x)=O(x^{1/2+ \varepsilon})$. 
These inequalities will be  available 
from  my ftp and http site mentioned above, in compressed,
packed, and gunzipped form, occupying $2000$ MB. \halmos
 
{\bf `Analytic' Proof of RH}
 
The first proof is always the hardest. After we found the above
proof, we have also found another, slightly shorter ($1000$ MB) proof.
The first step was undertaken by Jensen, who showed that
RH is equivalent to the following inequality:
 
$$
\Delta (x,y) :=
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}
\Phi(t) \Phi(s) e^{i(t+s)x} e^{(t-s)y} (t-s)^2 dt ds \geq 0 \quad ,
$$
where
$$
\Phi(t)= \sum_{n=1}^{\infty} (2 \pi^2 n^4 e^{2t} - 3 \pi n^2 e^{5t})
e^{-\pi n^2 e^{4t}} \quad .
$$
It follows from the super-WZ theory that $\Delta (x,y)$ is super-holonomic.
It follows from Hilearnie's $17^{th}$ problem that if it is indeed
non-negative, then it should be possible to write it as a sum of squares of
super-holonomic functions.  This indeed turned out to be the case.
We exhibit these explicitly, by giving
the super-holonomic recurrences each of these ($\leq 100000$)
functions satisfy. \halmos
 
{\bf P $\neq$ NP}
 
A.Razborov introduced {\it legitimate} lattices in order to obtain lower
bounds for the monotone circuit complexity of CLIQUE. By using very
naive `counting' (sometimes called, erroneously, `probabilistic') methods,
he found super-polynomial lower bounds. These were upgraded to
`exponential' (in $n^{1/6}$)
by Andreev, and N. Alon\footnote{$^2$}
{\eightrm N. Alon is a thinly disguised pseudonym of the brilliant
combinatorialist A. Nilli.}  and R.Boppana.
 
In order to tackle general circuits, we introduce the revolutionary
concept of {\it Kosher} lattice, and then exhibit a specific,
albeit very large, such lattice. By using {\it advanced sieving} once
again we get the desired super-polynomial lower  bounds for the
(generalized) distance of the Boolean function describing CLIQUE and
our Kosher lattice.
Both the description of our Kosher lattice, and the
rather long inequalities implying the bounds will be soon
available at the above-mentioned site. \halmos
 
{\it (Added in Proof:)} N. Alon and
A. Nilli have informed us that both our lattices and inequalities
are unnecessarily long, and that they are preparing a 1-page short
proof, using a slight modification of our Kosher lattices,
aptly named {\it Glatt Kosher lattices}. Their
proof is entirely self-contained,
and only uses polynomials over finite fields, and the trivial
fact that a polynomial can't have more zeros than its degree.
Due to this development, and as a concession to our system
administrator, we decided {\it not} to include the original
lattice and inequalities at our ftp site.
 
 
{\bf Lehmer's Conjecture : Ramanujan's $\tau (n)$ is Never Zero}
 
The key-idea was to realize that the statement
$\tau(n) \neq 0$ is equivalent to $\tau(n)^2 > 0$.
The proof now follows the same lines as the second proof of
RH. $\tau(n)$, and hence $\tau(n)^2$,
belongs to another generalization of the class of holonomic
sequences, called $\theta-$holonomic sequences.
As expected,  $\tau(n)^2$ is
exhibitable as a sum of squares of such sequences.
It turned out that $\leq 10000$ sequences, the description of each
requiring $\leq 3KB$, suffice.
Finally, it could be easily verified that they
cannot vanish simultaneously. \halmos
 
{\bf Wiles's Theorem}
 
Lest I would be accused of only solving open problems, I will
also outline a new proof of Wiles's theorem. Unlike
the original proof, that can only be understood and verified by
those that are referenced in Wiles's paper, the present proof can
be understood and verified by any obtuse mathematician.
 
Let $W(a,b,c,n):=(a^n+b^n-c^n)^2$. We have to prove that
when $a,b,c > 0$ and $n >2$, $W(a,b,c,n) >0$.
It is easy to find many recurrences (both linear and non-linear)
satisfied by $W(a,b,c,n)$. Amongst those, there is at least one,
of the form,
$$
W(a,b,c,n)= F( {n-3} \cup
\{ W(a-\alpha, b-\beta, c-\gamma, n-m) \vert 0<\alpha+\beta+\gamma +m \leq
10000 \}) \quad ,
$$
 
where $F$ is an explicit rational function of ${{10004} \choose {4}}-1$ 
variables, with
both numerator and denominator polynomials of (total)
degree $\leq 100000$, and
most importantly, with {\it positive} coefficients. 
Wiles's theorem now follows by induction. \halmos
 
 
\bye
 
 

