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.

• A2/2σ2 is the mean signal-to-noise ratio (SNR 10log10 (A2/2σ2) in decibels). σ is an RMS value for Noise.

• I0 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 500S+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 Zeros.

• 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 1020th 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-Dyson 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 spacingsUsing 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, σ2 • 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 A2/2σ2 = ¼ into the key equation e.g. σ2 = 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 (~2π /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= EffectiveHolographic Signal-to-Noise

Montgomery-Dyson curve SNR=1/4, σ2 = ½

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1 -

FT

(Cor

rela

tion)

Noise (Rayleigh)

Noise + Signal = RICE

Modulated RICE

Complex Hermitian stats = Zeros of Zeta

Montgomery-Dyson curve SNR=1/4, σ2 = 0.97

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1 -

FT

(Cor

rela

tion)

Noise (Rayleigh)

Noise + Signal = RICE

Modulated RICE

Complex Hermitian stats = Zeros of Zeta

Montgomery-Dyson curve – Fourier Transform of the ‘Spike’ SNR=1/4, σ2 = 0.97

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.7

0.75

0.8

0.85

0.9

0.95

11

- F

T(C

orre

latio

n)

Noise (Rayleigh)

Noise + Signal = RICE

Modulated RICE

Complex Hermitian stats = Zeros of Zeta

Montgomery-Dyson curve – Infinite Bandwidth SNR=1/4, σ2 = 0.97

0 2 4 6 8 10 12 14 16 18 200.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1 -

FT

(Cor

rela

tion)

Noise (Rayleigh)

Noise + Signal = RICE

Modulated RICE

Complex Hermitian stats = Zeros of Zeta

Montgomery-Dyson curve – Forensic Survey SNR=1/4, σ2 = ½

0 1 2 3 4 5 6 7 8 9 100.9

0.92

0.94

0.96

0.98

1

1.02

1 -

FT

(Cor

rela

tion)

Noise (Rayleigh)

Noise + Signal = RICE

Modulated RICE

Complex Hermitian stats = Zeros of Zeta

Montgomery-Dyson curve – Forensic Survey SNR=1/4, σ2 = 0.97

0 1 2 3 4 5 6 7 8 9 100.9

0.92

0.94

0.96

0.98

1

1.02

1 -

FT

(Cor

rela

tion)

Noise (Rayleigh)

Noise + Signal = RICE

Modulated RICE

Complex Hermitian stats = Zeros of Zeta

Montgomery-Dyson curve – Conjoined SNR=1/4, σ2 = ½

0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

1 -

FT

(Cor

rela

tion)

Noise (Rayleigh)

Noise + Signal = RICE

Modulated RICE

Complex Hermitian stats = Zeros of Zeta

Log Montgomery-Dyson curve SNR=1/4, σ2 = 0.97

0.2 0.4 0.6 0.8 1 1.2 1.4-25

-20

-15

-10

-5

0

dB [

1 -

FT

(Cor

rela

tion)

]

Noise (Rayleigh)

Noise + Signal = RICE

Modulated RICE

Complex Hermitian stats = Zeros of Zeta

Log Montgomery-Dyson curve - zoom in SNR=1/4, σ2 = 0.97

0.5 1 1.5-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

dB [

1 -

FT

(Cor

rela

tion)

]

Noise (Rayleigh)

Noise + Signal = RICE

Modulated RICE

Complex Hermitian stats = Zeros of Zeta

Montgomery-Dyson – close up to zero SNR=1/4, σ2 = ½

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

1 -

FT

(Cor

rela

tion)

Noise (Rayleigh)

Noise + Signal = RICEModulated RICE

Complex Hermitian stats = Zeros of Zeta

Montgomery-Dyson – close up to zero SNR=1/4, σ2 = 0.97

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

1 -

FT

(Cor

rela

tion)

Noise (Rayleigh)

Noise + Signal = RICEModulated 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 - zi) infinite product of roots zi. Assume zk 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 h0.

• Analytic Continuity and negative (virtual) energy (Dirac)• Playing Tunes with Equation. Try σ2 ~ 1 with SNR =¼,

we get maximum ‘face-on’ area with about a 72/27 ratio. Asymmetry. (To Calibrate Rayleigh-Rice curve, try σ2 = 0.97 in the numerics, gives the exact energy ratio with maximum possible information i.e. maximum entropy

taking (Xi)^2 assuming the Holographic Principle ).• Optimal information packing FCC,

0.7404 =/18

THE END