As a hobbyist mathematician, my interest in RH stems from a
seminal paper presented by Professor Berry, at the Royal Society in the late
80’s, in which he shows an amazing agreement between the statistics of the
spacings distributions of the roots of zeta on the Critical Lines (re= ½) and
chaotic billiards.

He
infers that there is a Quantum Mechanical operator (a ‘vibrating drum’) plus ‘Chaotic
quantum billiards’ that underlies the RH (cf Signal + Noise), and Physics and
number theory may share this common mechanism.

The
‘level spacings’ statistics i.e. the distribution of separation distances
between points on the Critical Line (re(z)= ½ ) (M.V. Berry, The Royal Society
Bakerian Lecture 1987) have been shown to exactly match the chaotic billiard
domain using results through simulated ray tracing. They are the Complex
Hermitian statistics, whose origin lies in Atomic Physics.

I was
intrigued to notice that many of the distributions and statistics presented,
also appear in Sonar, which I’ve spent many times measuring these at sea.
Furthermore, they appeared to be the noise ‘footprints’ of the Fourier spectrum
often known as Boltzmann noise or Thermal noise ‘floor’ by practitioners of
Sonar and Radar and have a random origin from Central Limit Theorems (CLT)
based on the summing of independent sound sources in the ocean. From these
curves (called Receiver Operator Characteristics), the Signal-to-Noise (SNR)
ratio of sonars may be calibrated to give measures of effectiveness viz.
probabilities of detection and false alarms.

In
the Platonic world, the solutions are pure or signal like whereas the physical
world introduces an element of ‘noise’. In most problems encountered, the maths
provides good approximation to physical reality because the noise is
manageable. It is this ‘noise’ that characterises the behaviour of the root
finding in the case of the Zeta function.

I
took the view that computed zeros must ultimately have a physical explanation,
and could be modelled physically. The many mathematical theorems, facets and
approaches must be consistent between themselves and have an underlying cause,
even though the connection may not appear to be obvious and may fall within
differing subject domains and different experts. If it is True in one domain it
must be True in another.

I
then decided to look at the effect of adding arbitrary small numerical ’noise’
in the (analytically continued) zeta function and came at some very surprising
observations:

·
Firstly,
Zeta is closed form whereas chaotic billiards have no such integral expression
or otherwise.

·
Although
the statistics are indistinguishable between levels spacings distribution of
zeta and the chaotic billiards, it does not necessarily imply a common
underlying mechanism (except maybe pure randomness), as CLT’s are observed in
lots of completely unrelated situations and instances in the physical world.

·
I showed
that if I assumed a zero existed off the Critical line (which would render the
RH to be false), I would never be able to compute it because numerical noise
due to computer precision higher up the line and, even down to Planck scale and
beyond. In fact, arbitrary amonts of noise, would not allow me to find it, even
it existed.

·
The
numerical evidence of billions of roots on the line (Odlyzko, te Riele)
suggests that RH is True. Furthermore the latter bullet point implies it’s
unprovable (because you can’t prove its falsity) so it should become an axiom,
as I believe suggested (by Chaitin) from applying Gödel’s Theorem.

1.
However,
the accepted wisdom appears to be that the RH is true and is a theorem.
Furthermore, the mathematical constructions to prove RH seem to be based on
this assumption, namely that the Riemann operator aka Hilbert-Pólya Hermitian
operator exists. I believe it does not (physically) exist because if it existed
then RH would be mathematically provable. (Its assumed existence may lead to
self-inconsistency in these constructions and, perhaps, has been the reason why
the operator has proven to be so elusive for over 150 years. The question of
how many angels fit on the end of pin comes to mind if the angels themselves don’t
exist).

2.
I believe
that the Zeta series maps to a whole possibility of physical possibilities
(one-to-many map), whereas the series for sin(x) say maps to a unique Physics
(namely Euclidean geometry from 2500yrs ago). For sin(x), maths to physics is a
one-to-one map then. This makes (Log) Zeta noise limited whereas sine is signal
limited (i.e. its spectrum is very simple namely multiples of π)

Both these were points of contention that I had with Professor
Berry, albeit in a very friendly and cordial manner.

The
Zeta function is defined by analytically continuing to the whole complex
z-plane the well-known Euler’s product over primes:

ζ
(z) = 1 / ** Π** (1- p_{k}^{-z}) =1 + 1/2^{z} + 1/3^{z}
+ 1/4^{z} + 1/5^{z} + 1/6^{z} +…
Eqn (1)

"
primes p_{k}

Then to find any root z* s.t ζ (z*) = 0

**The
Root Sum Square (RSS) Statistic**

The
most revealing statistic of Log (zeta) is its Root Mean Square =√
(variance), which is a (normalised) incoherent sum of terms in the tail of
Log Zeta ^{9}.
The Log (Zeta) Root Mean Square (RMS) value describes the (noise or signal +
noise) distribution, and bounds it within probabilistic envelope because its
functional form is known in the asymptotic limit (i.e. where Zeta → 0, so
Log (Zeta)→ -∞). Note, the above expression is not convergent where
re(z) ≤ 1, but use of RMS is justified ^{10} in the Critical strip (The critical
strip is the region of all complex numbers where 0 ≤ re(z) ≤ 1) and
by mutual cross correlation of independent (trig Fourier) noise terms in the
tail only the coherent terms are left over (where each the sin and cosine
harmonics are the same) to give the Root Sum Square (RMS=RSS/n; for 'n' samples)

RSS
[|Σ p_{k}^{-z }|] = √ [{p_{M}^{-2re(z)}
+ p_{M+1}^{-2re(z)} + p_{M+2}^{-2re(z)} + p_{M+3}^{-2re(z)}
…..}]
Eqn (2)

On
the line, these are the stochastic sampling of all the possible physical
instances of exponent in N-bit or log (Zeta) at the zeros of Zeta, which I show
to be Rayleigh distributed^{11}. Above the Critical Line, I have shown
there is zero probability of computing roots there by simply showing the fact
that envelope of possible instances of Log(zeta) is only finite on the Line
where re(z)= ½ for these distributional forms. This is because RSS →
∞, as re(z) → ½ by comparison with the **Σ **1/n series.

The
RMS represents the distribution of instances [sampled computationally].
(However, there are examples of distribution such as they have no defined mean
or standard deviation which are used by mathematicians to test intuition because
they may be non-physical such as the Cauchy distribution. The standard
Cauchy has a probability density
function (pdf) =
1/p(1+x^{2})

Curve ® Rayleigh distribution as s®¥

One
can view the probabilistic distribution of computed instances of magnitude R=|log
ζ
(z)| (along
the x axis) in the previous figure, associated with every possible computed
instance of re(z) (real part of z)

While
converging to some root as iterative evaluations of updates of z, as re(z) ®½,
the above distribution curve moves to the right, so the envelope of possible
instances moves to the right as it becomes asymptotically Rayleigh.

During
this computation, the probabilistic envelope becomes infinite when the algorithm
is converging (locally) to a root,
ζ
(z)→ 0 therefore Real
(log ζ
(z))→
-∞ for
computed z.

The
computed roots potentially populate a non-zero (pdf) limiting distribution curve
(i.e. where R→
∞) corresponding to
RSS →
∞ viz.. re(z) ®
½;

At
other values where re(z) < ½ , a root find procedure (e.g. Newton-Raphson
type) will not work, because the probability of finding finite instances are
zero i.e. pdf(R) is undefined. Thus Re(z) =½ is the only possible candidate
domain for computable roots.

Note,
as the (Rayleigh) distributions move to the right, they actually become wider.
This is analogous to a signal becoming more and more smeared with noise, as a
convergence moves it to the right with each successive computation.

In the noise-limited case, this noise floor dominates and the coherency (of signal viz. Logζ (z)) is eventually destroyed causing a breakdown of the deterministic route search within the immediate neighbourhood of any root that may exist in this region. The computed Zeta function becomes non-analytic in this nbhd due to the indeterminacy of its derivative viz. its computed derivative becomes increasingly noisy as a root is approached.

**Central
Limit Theorems Revisited**

Furthermore,
I have reproduced the level spacings distribution (Berry/Odlyzko Bakerian
lecture) by CLT arguments based on Sonar theory, where it appears equivalent to the pairing of
(correlated) adjacent samples from a standard Rayleigh distribution (see 1^{st}
figure). The distance spacing between adjacent roots are, in effect, a Rayleigh^2 distribution, with appropriate normalisation that describe the
running mean spacings.

These
are magnitudes derived from the real and imaginary parts each mutually
independent and identical normally distributed in Log(Zeta), asymptotic values
around^{12}
zeros of Zeta (‘asymptotic’ here means as z → some root z0, or height T
→ ∞).

(Note,
An appropriate normalisation has been applied to RAYLEIGH^2 to compare with the
Odlyzko graphs as already shown in Fig 1 so that have an area of Probability =
1, but the essential shape appears exactly the same).

This
is because every half phase cycle of Z(t)^{13} on the CL, the two adjacent roots are
connected via a Taylor Series relationship but as the curve moves to the next
half cycle it passes through the one of these zeros to emerges on the other as
if it were randomly selected and not correlated with the previous half cycle by
Taylor Series (differentiability), but is in turn connected to the exit root to
the next cycle.

Z(t)
cycles along the Critical Line

The
crossing over of roots into the next half cycle passes through what is called
an omega number^{14}
to the other side, which can interpreted as an infinitely long string of all
possible random 1 and 0’s. The nearer we approach the root the more the N-bit
expansion (beyond this Quantum bits or ‘Qubits’), full omega at the root, but
compresses down again to a finite number of bit operations as we move away on
the other side. 'Randomness'
and 'Noise' may not be the same. However the term 'noise' here refers to a
non-periodic component, where there is a lack of coherency. For example, PI (=π) is completely deterministic (i.e. you can write
down a mathematical equation), yet it digits appear random so maybe it actually
contains the 'Sum of All Possible Signals' as a compact description. Pi and the
prime numbers are not “coin tossing” numbers as they are deterministic yet
they appear to have random properties. Infinity as a concept can be finitely
coded “Take a number, and add one repeatedly”, and pi
can be expressed as a mathematical formula or finite character (L27 alphabet)
expression.

The
Z(t) appears at first to be continuous and analytic on the large scale , but
around the non-trivial zeros I have shown physically non-analytic on the
micro-scale by the multitude of possibility of noise (fractal) in the N-bit of
Zeta viz. Log(Zeta); and here the (physical) closed form of (Log) Zeta share a
similar form as with the Feynmann integral ^{15} of many possible paths in Quantum
Mechanics:

Any
represented function F(s) in a computer will have errors due to numerical round
off and approximation but usually:

Nops

F(s)
[1 + Σ δ], should behave as F(s) when δ → 0.

Now
what happens if F(s)=ζ(s) [Zeta]?

Zeta
can be expressed in closed form (in the Critical Strip) with an integral factor
integrated between [-∞, +∞] allowing an infinite degree of freedom (Nops=∞) from an infinite number of arbitrary small ‘δ’ strip errors
at each integration step.

The
comparison of the GUE distribution of eigenvalues of Complex Hermitian matrices
appears to lack a physical mechanism, and I’m not sure the significance of all
this is understood but it appears to give similar qualitative behaviour as the
Modified RICE statistics. In particular, the latter is derived from families of
distributions that I’ve encountered in sonar ^{16}
(namely the ROC curves).

It would appear that the spacing distances between roots appears as not a deterministic mathematical structure after all (what I call ‘signal’ limited), but is a noise ‘footprint’ that emerges from pure randomness (what I call ‘noise’ limited). The noise (in the exponent Computed Log(Zeta) or the N-bit takes on an infinite bandwidth around a zero of Zeta, so CLT’s would always kick in there, and so would prevent computations of zeros off the Critical Line (assuming they exist) – a catch 22 argument. If it’s unprovable, you can’t prove it and you can’t disprove it by demonstration with numerical evidence, but combined with the observation that (non-trivial) solutions to date appear on the Critical Line, it must therefore be accepted as being True but unprovable. This seems to go against the accepted belief.

**Omega**

Chaitin
call this arbitrary and self-referential ‘noise’ an ‘Omega’ number ^{17}. I also
believe this ‘Omega’ number is equivalent to the ‘many’ physical possibilities
of randomness (arbitrary ‘noise’) of the ‘real world’ of measurements AND
COMPUTATION, whereas mathematicians seem to be searching for the unique
‘signal’, which only exists in the Platonic realm for zeta.

The
omega number only collapses down to a finite character set on the Critical Line
within an infinitely ^{18}
diverging (when The Root Mean Square RMS → ∞) noise envelope of Log
zeta, that translates to a solution only allowing physical solutions to be
computed there (assuming they exist, which they do as observed from the numerics!)
and nowhere else. Effectively this probability ‘repulsion’ mechanism does not
allow solutions to be computed off the line.

The computer itself has no concept of a unique limit point of convergence (to a zero), but instead computes from a set of possibilities. In fact, the tracking of the complex zeta around in phase can become critical in N-bit computation. The introduction of ‘omega’ noise in the nbhd of a zero, becomes more and more dominant the nearer we approach a zero (assuming continuity if indeed we can compute it off the line). The randomness of these bits/Qubits (that contribute to an overall RMS of all the possible values of Log(Zeta) series) frustrate any attempt of obtaining a computed solution of a zero off the Critical Line.

**Vibrating
Zeros**

In the Conrey graph, the consecutive spikes are reinforced in the pairwise correlation

X(T+d). X(T)

where T is the distance going up the critical line, where d is the mean level spacing. As can be easily checked on the Watkins graph, the subsequent spikes are shrunk in scale by this amount 1/ln(T/2p), in the asymptotic limit T ® ¥ so is normalised by this factor to reinforce the 'God spike'.

We
can imagine such inter-zero vibrations in the length of the level spacings as if they
were connecting springs, and has an intra-zero superposition of simple harmonic motions (the
part that satisfies Dirichlet Conditions .i.e. can be expressed completely as
Fourier Series) plus chaotic or 'noise' ( i.e. whose Fourier Series does not
converge) solutions where the Amplitudes represent the magnitude of the level
spacings.
This is appropriately normalised against the mean level spacing

Thus
the Riemann Hypothesis (RH) is unprovable. From the numerical evidence to date,
the Riemann Hypothesis can be regarded as a working hypothesis for practical
purposes. Furthermore using the above ‘noise’ arguments by probabilistic
restriction, it can be shown to be physically true but unprovable. This makes
it an axiom as opposed to a theorem.

All material on this site is property of Adrian Rifat.