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Montgomery/Dyson Curve

This is an even more SURPRISING result.

The Montgomery Conjecture states that (Fourier Transform of) the paired Correlation3 of distribution of zeros of Zeta is exactly the same as that of Complex Hermitian Statistics, based on a wealth of numerical evidence [Odlyzko et al]. (This is the ‘sinc’ function, which is the Fourier Transform of the uniform ‘chirp’ or rect function).

In fact the curve shows 1-FFT{Correlation}

I’ve recently done some numerical experiments in MATLAB computing the correlations for:

·        Rayleigh ‘Noise’

·        Rayleigh ‘Noise’ + Signal = RICE

·        Beyond RICE [an empirical approximation used which is a modulated RICE function]

The graphs have been visually fitted to maintain the normalisation, so the essential shapes are always preserved, with only linear scaling in (x, y) axis.

The x-axis represents an ensemble average of level spacing. The level spacings represents the instantaneous frequency of zeta so its interpretation is spectral f inst = dΦ /dt /2π, whereΦ is the phase and ‘time’ t is measured along the y-axis.

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® ¥

On the graphs, the Montgomery-Dyson curve is shown in red, and the modulated RICE is shown in black.

One can easily derive the RICE4 distribution, but beyond that it is a difficult integral to solve so I’ve used an empirical solution that appears to match simulated data:

Now here is the ‘key’, the modulated RICE probability density function, I have defined as

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

This is the SUM of ALL SIGNALS + NOISE

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.

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

The phase becoming more random, and the combination of bits become ratio-like to obtain the tangent function defined against a normalised distance scale, giving the modulated RICE, a spike up of infinite detail that is singular (simple pole) at π/2 with scale compression.

In the ‘time’ domain, the shape of Modulated Rice looks like a spike, which I shall call a ‘God’ spike. [Note the graph should actually be of Amplitude or ÖPsd]. It is schematic, not drawn to scale. An interesting feature of this (1-sided) graph is that if you compare it (upside down) with the Conrey graph in the Watkins section of Fourier analysis and number theory of the Fourier Transform of the error term in the Prime Number Theorem (Refer to subject resource http://www.math.ex.ac.uk/~mwatkins/zeta/Ntfourier.htm).

A series of scaled harmonics is seen which appears fractal.

Now transform into the frequency domain by Fourier analysis. The structural detail of data fit is extraordinary on closer examination,

This is so important, that it should be replicated to reinforce or destroy the argument. Compare this to the moment graph γ2 in the Bakerian Lecture (Berry 1987).

The x-axis represents the mean level spacing between zeros. The y-axis is directly proportional to the Signal + Noise. Note the distribution is Rayleigh or ‘noise-limited’ as x is asymptotically x→0 near zero in the micro-scale, but becomes signal-like as x becomes large when we move into the macro-scale.

As x=1 we can see the cross over the root to the next half cycle and passing through the ‘omega’ number so the bandwidth of |Log(Zeta)| expands, because it is ‘noise’ limited here and therefore uncorrelated. In fact, the ‘noise’ can mean all possible outcomes of waves in Quantum Mechanics. This rises slightly with oscillation, with more modes, as we move into the macro-scale as the instantaneous phase of Zeta describes a local frequency. Then the next crossover becomes uncorrelated again at any integer number of average spacing, and so on with more ripples.

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

If one alters slightly the position or order of the singularity, the qualitative features break down so the Modulated RICE appears very special indeed. Also if the sign of the tangent is made positive, the features break down.

Some further forensic examination of the plots, are shown as follows. Please note the glints, and the qualitative behaviour:

The Montgomery Dyson graph, The RICE and The Modified RICE appear to blend, not just meet, together above x=1, and on closer inspection they seem to diverge away from each other. This is at around the nexus point x=1.177 (≈ π1/7) for the RICE curves. This number may have some significance.

The oscillatory shape of the Modified RICE appears to follow in fine detail that of the Montgomery-Dyson curve. The level of detail in the glint or spikes in the data, are completely coincidental with the Montgomery-Dyson. Again, THIS CANNOT BE A COINCIDENCE.

With rise in the first lobe, the Modified RICE graph also appears to have infinite bandwidth (B=∞) in the high frequency components. Is this the Casimir Effect taking place? Here, Quantum fluctuations in space, are reduced to standing waves between very close boundary plates (the order of atomic distance), to move further up the Dyson-Montgomery curve and become more signal limited. This infinite band can be removed by reducing the spike in the ‘tan’ function and is indeed why it is not seen on computers because the QM effects are not observable (limited to the bit), so the experimental data is discretized.

This is a purely physical observation and is no proof and I could be totally wrong, but certainly this demands investigation. Please observe the level of coincidence in the glints in the curves.

The curve has also been plotted logarithmically to zoom in on the left hand side, just to show how good the fit is.

Sensitivity analysis around SNR =1/4 using the decibel scale shows the divergence away from this value.

From these, I propose that the ‘vibrating drum’ is the statistics of (Rayleigh) Noise + the sum of All Signals. It seems to provide a VERY general description of physical reality, namely:

NOISE + ΣSIGNALS = NOISE + Σ-∞+∞cn. exp(inΦ)

[Where cn=½(an± i.bn) Fourier Coefficients]

We can then compute the correlation for all the modes. I believe one can generate all the observable statistics of Zeta from this.

This invites further investigation and comment.


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