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Unprovability of Riemann Hypothesis

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 distributed¹¹. 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+x²) ).

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 , the probability that iteration finds a root are zero, because the distribution has a finite probability of extent i.e. pdf(R)→ 0, as R→ ∞. However, as re(z) →½, instances of R from the distribution become infinite (around 37% are of them are greater than the RMS);

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 around¹² 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 number¹⁴ 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.

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 Root Sum 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

We can imagine such vibrations in the length of the level spacings as if they were connecting springs, and has 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 which shall represent a local noise level along the Critical Line, so that the mean signal-to-noise ratio i.e. <SNR>=1 in the level spacings and pair correlation graphs.

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.

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