Riemann Hypothesis &

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24 Νοε 2013 (πριν από 3 χρόνια και 8 μήνες)

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Riemann Hypothesis &
Signal Processing Paradigm


by

Adrian J Rifat

Introduction


A new physical paradigm is proposed based on signal processing within
Information Theory.




Has implications to both Number Theory and Physics.



In 1859, Bernhard Riemann postulated what is known as the Riemann
Hypothesis concerning the distribution of prime numbers, the information
building blocks of arithmetic. This has both mystified researchers and
frustrated attempts to prove/disprove it.



Riemann Hypothesis, Prime Numbers and ‘Universality’ from the very small
scale to the very large (at the interface of Quantum Mechanics & Chaos
Theory)


Ref. Bakerian Lecture 1987 (Berry).



Sum of All Possible Signals + Noise
predicts emergent properties of our Universe.



Based on Rice distribution used in Sonar. Further modulated by a
combinatorial sum which is a sum of all possible signals to give an
‘Information Spike of Everything’ (appropriately normalized with effective
SNR=¼ over a hologram) of infinite bandwidth.




Its characteristic signature also appears in Number Theory in the statistics
of the Riemann Zeros, and consequently the same process may explain the
distribution of prime numbers.



This gives credence to a belief that Information Theory underlies both
Mathematics and Physics
-

“The Unreasonable Effectiveness of
Mathematics” Wigner
.



Assumptions of the New Proposed
Paradigm of Fundamental Physics



PHYSICAL REALITY IS BUILT ON INFORMATION.



IN OUR OBSERVABLE UNIVERSE, INFORMATION CANNOT BE
CREATED OR DESTROYED.



ENERGY MAPS TO INFORMATION IN AN ESSENTIAL WAY VIZ. YOU
NEED ENERGY TO STORE INFORMATION. YOU ALSO NEED
INFORMATION TO DESCRIBE THE ENERGY. THEY ARE
SYNONOMOUS.



INFORMATION COMING TROUGH IS QUANTIZED TO AVOID UV
-

CATASTROPHE.



EVERYTHING CAN BE REPRESENTED AS SOME COMBINATION OF
SIGNALS + NOISE.



RICE distribution


The Signal + Noise are described by RICE statistics. The signal is assumed a sinusoid of
Amplitude A and a noise RMS value σ.












A
2
/2σ
2

is the mean signal
-
to
-
noise ratio (SNR 10log
10

(A
2
/2σ
2
) in decibels). σ is an RMS value for
Noise
.




I
0

is the Modified Bessel function of the 1st kind (n=0 of Zero order). Note if A=0, we have the
Rayleigh (Noise
-
limited) form.




Pure ‘noise’ in this particular study refers to non
-
periodicity or ‘chaos
-
like’, whereas in practical
sonar terms it can include unwanted signal as well. The signals are regarded as modes, that lend
themselves to Fourier Analysis. The ‘Noise’ is the Chaos that has no periodic frequency content
where period T




Pd Pfa




SNR

RMS Noise
-3
-2
-1
0
1
2
3
0
100
200
300
400
500
Noise
Signal + Noise
-3
-2
-1
0
1
2
3
0
100
200
300
400
500
S+N
Modulating Noise with Signal

What we know already about RH


States all the non
-
trivial zeros of Riemann Zeta function lie on the
Critical Line (Re(z) =½)
. Confounded Proof and remains mysterious.



Duality: Primes


婥Z潳
.



Central Limit Theorems (CLTs)

around the Zeros. Log(Zeta) Real and Imaginary components, identically
independent and Gaussian. Indicates some kind of probabilistic process going on based on spectral components.



Strong Computational Evidence

(Odlyzko) up to 10
20
th zero. All the first N=50billion+1 zeros lie on the line.



The Hilbert
-
Polyà Conjecture

motivates the
Hermitian Operator

approach. The non
-
trivial
zeros correspond to the spectrum of eigenvalues (energy levels) of a Hamiltonian governing a quantum mechanical system whose
underlying classical mechanics are chaotic as suggested by Berry. This spectral interpretation is supported by the Montgomery
-
Dy
son
observations of correlations between the zeros along the Critical Line (Re(z) =½).
Distribution ~ SINC(X) = [SIN(X)/X]



Riemann Zeta function is a fractal structure and by Voronin's
Univerality encodes All Possible Theorems

(Woon).



RMT

Theory predicts the moments of the Riemann Zeta function (Keating/Snaith). In fact, is same Combinatorial 2
-
d Young Tableaux
(OEIS A039622). (Possible connection here with entropy defined over a surface by planar partitions).



RAYLEIGH^2


Level spacings

Using Mk1 ‘eyeball’, looks like Maxwell
-
Boltzmann distribution?

KEY EQUATION


MRICE(R) = [1 + tan(R)]* RICE(R)




This is the SUM of ALL SIGNALS + NOISE


PLAY TUNES with the Equation. Vary SNR, σ
2



The NOISE is RAYLEIGH distributed. The mean signal
-
to
-
noise ratio is unity <SNR> = 1 x 1/4 = ¼


The quarter is due to a mysterious entropy factor.

Thus
the Holographic SNR = ¼, so plug in
A
2
/2σ
2

= ¼ into the
key equation e.g. σ
2

= ½, A=½ say.


Combinatorial = Entropy


Cosmological implications


Intuitively, if you can imagine all combinations of amplitude of one signal A,
two signals A +B, three signals A + B + C, so on; there is a rapidly
escalating progression of probability density function. Imagine all instances
of bit length L, i.e. 2L, then L tends to infinity we can range compress by
transformation tan R = L, so singularity occurs at R =
π

/2



In Sonar, oscillations of this kind are associated with discontinuity in time
series data, and their suppression (called side lobe suppression) uses
window functions to zero the boundaries at the ends of each time interval to
make the function continuously periodic.



The tangent function discontinuous at R =
π
/2, and by analytic continuation,
the amplitude in the distribution becomes negative beyond that (anti
-
matter,
inflation/accelerating universe?)




Analogous to complex roots of quadratic equations where non
-
zero
imaginary imply the curve does not touch or cross the x
-
axis (with
cosmological implications).

0
50
100
150
200
250
300
350
400
450
500
-100
0
100
200
300
400
500
600
Power Spectral Density Bins
MRICE distribution of sum of all signals + noise

Now take 1


Fourier Transform (Spike), we get the Montgomery
-
Dyson
Correlation graphs of normalised spacing between zeros (~2
π

/ln T). Compare
with ‘upside down’ spikes on Conrey graph from PNT in Chebyshev error term.


Sidelobes indicative of spikes (i.e. data discontinuities


Gibbs phenomena).

God Spike

¼

= Effective

Holographic
Signal
-
to
-
Noise

Montgomery
-
Dyson curve SNR=1/4, σ
2

= ½

0
0.5
1
1.5
2
2.5
3
0
0.2
0.4
0.6
0.8
1
1 - FT(Correlation)
Noise (Rayleigh)
Noise + Signal = RICE
Modulated RICE
Complex Hermitian stats = Zeros of Zeta
Montgomery
-
Dyson curve SNR=1/4, σ
2

= 0.97

0
0.5
1
1.5
2
2.5
3
0
0.2
0.4
0.6
0.8
1
1 - FT(Correlation)
Noise (Rayleigh)
Noise + Signal = RICE
Modulated RICE
Complex Hermitian stats = Zeros of Zeta
Montgomery
-
Dyson curve


Fourier Transform of the ‘Spike’ SNR=1/4, σ
2

= 0.97

0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0.7
0.75
0.8
0.85
0.9
0.95
1
1 - FT(Correlation)
Noise (Rayleigh)
Noise + Signal = RICE
Modulated RICE
Complex Hermitian stats = Zeros of Zeta
Montgomery
-
Dyson curve


Infinite Bandwidth SNR=1/4, σ
2

= 0.97

0
2
4
6
8
10
12
14
16
18
20
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
1 - FT(Correlation)
Noise (Rayleigh)
Noise + Signal = RICE
Modulated RICE
Complex Hermitian stats = Zeros of Zeta
Montgomery
-
Dyson curve


Forensic Survey SNR=1/4, σ
2

= ½

0
1
2
3
4
5
6
7
8
9
10
0.9
0.92
0.94
0.96
0.98
1
1.02
1 - FT(Correlation)
Noise (Rayleigh)
Noise + Signal = RICE
Modulated RICE
Complex Hermitian stats = Zeros of Zeta
Montgomery
-
Dyson curve


Forensic Survey SNR=1/4, σ
2

= 0.97

0
1
2
3
4
5
6
7
8
9
10
0.9
0.92
0.94
0.96
0.98
1
1.02
1 - FT(Correlation)
Noise (Rayleigh)
Noise + Signal = RICE
Modulated RICE
Complex Hermitian stats = Zeros of Zeta
Montgomery
-
Dyson curve


Conjoined SNR=1/4, σ
2

= ½

0.2
0.21
0.22
0.23
0.24
0.25
0.26
0.27
0.28
0.29
0.3
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.2
1 - FT(Correlation)
Noise (Rayleigh)
Noise + Signal = RICE
Modulated RICE
Complex Hermitian stats = Zeros of Zeta
Log Montgomery
-
Dyson curve SNR=1/4, σ
2

= 0.97

0.2
0.4
0.6
0.8
1
1.2
1.4
-25
-20
-15
-10
-5
0
dB [1 - FT(Correlation)]
Noise (Rayleigh)
Noise + Signal = RICE
Modulated RICE
Complex Hermitian stats = Zeros of Zeta
Log Montgomery
-
Dyson curve
-

zoom in SNR=1/4, σ
2

= 0.97

0.5
1
1.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
dB [1 - FT(Correlation)]
Noise (Rayleigh)
Noise + Signal = RICE
Modulated RICE
Complex Hermitian stats = Zeros of Zeta
Montgomery
-
Dyson


close up to zero SNR=1/4, σ
2

= ½

0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
1 - FT(Correlation)
Noise (Rayleigh)
Noise + Signal = RICE
Modulated RICE
Complex Hermitian stats = Zeros of Zeta
Montgomery
-
Dyson


close up to zero SNR=1/4, σ
2

= 0.97

0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
1 - FT(Correlation)
Noise (Rayleigh)
Noise + Signal = RICE
Modulated RICE
Complex Hermitian stats = Zeros of Zeta
Is RH True?


Yes, but UNPROVABLE. Why?


If the Riemann Zeta function encodes all knowledge, then the tools
to prove RH must exist within it i.e. If
Ω

= Sum of All Knowledge, B
= Tools, then


B


Ω

so B


Ω

=
Ω

(i.e. No added knowledge)

i.e. we can't ‘poke’
it with anything outside the Universe = Riemann landscape
.



(z) =
π

A*(z
-

z
i
) infinite product of roots z
i
. Assume z
k

is off the
Critical Line Re(z) =
½, then divide equation by



(z
-
zk) still gives the same SUM of ALL SIGNALS (cf 2*
¶R

=
¶R
).
We have two alternative stories YES/NO, 1 or 0.


Probabilistic reasoning
-

power p=
½ in RSS sum in ln[

(z)]
left over
from pair
-
wise correlation c.f Harmonic series diverges.

CONSEQUENCES

If it holds true, it may help shed light on the following:


Dark Energy and Dark Matter. = Spikes and Ripples.


Cosmological Horizon and Inflation.


Black Hole and galaxy formation.


The Incorporation of Thermodynamics and Time
Asymmetry.


Renormalization, Feynman Integration over all Paths &
The Casimir Effect.


Quantum Holography and quantization.


The true nature of randomness and why our Universe
evolved from ‘Nothing’.

Summary


Reality must be information
-
based down at some
Level X (whatever X may be).


Information Invariance


DVD or Video VCR?



The Sum of All Possible signals is the most
unconstrained definition. In its neat and tidy little
definition, it contains the most information possible.


Galaxy Formation. Dark Energy and Dark Matter co
-
exist as opposite sides of the same coin, whatever
the evolution.



This ¼ factor may have something to do with
holography. Remember, the entropy of a black hole
of Area A is actually A/4. The TAN factor is like a
scale compression ratio = projected length/(inverse
of) informational depth in the Far Field.


Summary


Area under spike. Renormalization, N. h = 1, to allow for
cutoff


sampling in discrete data bins. Semi
-
classical
limit h

0.


Analytic Continuity and negative (virtual) energy (Dirac)


Playing Tunes with Equation. Try σ
2

~ 1 with SNR =¼,
we get maximum ‘face
-
on’ area with about a 72/27 ratio.
Asymmetry. (To Calibrate Rayleigh
-
Rice curve, try σ
2

=
0.97 in the numerics, gives the exact energy ratio with
maximum possible information i.e. maximum entropy
taking


⡘椩帲⁡獳畭楮朠瑨攠䡯汯杲慰桩挠偲楮捩灬e

).


Optimal information packing FCC,



0.7404 =

/

18


THE END