Whilst not a mathematical proof other empirical evidence and theory appears to support my model that back the view that RH is (mathematically) undecidable, but is nevertheless True. These are:
· Recent work in the States regarding the fractal nature at the roots of Zeta (Castro & Woon) in its derivatives.
· Voronin's universality - any function can be arbitrarily close to the zeta function at some region up the line an infinite number of times. This means any signal can be encoded. For instance you could transmit/receive the entire contents of the Encyclopaedia Brittanica! The corollary of this is that the Zeta function encodes all (possible) theorems.
· Spectral analysing thermal heat ‘roll-off ‘ seen in the residue expansions around the roots of Zeta. ‘1/f’ noise (Planat) in the Chebyshev Psi function.
· Chaitin’s Theory on Information regarding ‘Omega number’, the way it contaminates mathematical proof and what he calls ‘thermodynamic epistemology’.
· The information-theoretic approach has lead to equivalent statements of RH in the literature [K. K. Nambiar] regarding the capacity of communications channel as defined by Shannon information and maximum entropy related phenomena.
· Discrete Log time value systems.
· Boltzmann Thermal Noise measurements in Sonar/Radar.
Whilst not very satisfactory in proving the RH, the Physics spawned from it is seen to be good and consistent with many different aspects including thermodynamics (in particular the quantisation of the Rayleigh/Jeans black body spectrum to suppress UV Catastrophe through finite extension of the possible signal because of the cutoff at the discontinuity) and questions in cosmology (dark matter, dark energy, black holes and the science of the Big Bang). The ripples caused by discontinuity are 'shimmers' in the fabric of space rather than waves in medium, and these Boltzmann 'spikes' appear to give rise to the essential features of scale.
Ramanujan Partition Numbers and the Boltzmann Information Spike
The tan function in the Boltzmann spike can be justified by the fact that the number of partitions7 for any given number n is related to exponential functions (as hyperbolic functions) and in the asymptotic limit n®¥, by the coordinate conversion as intuited in Section 4.
tan R = exp(Ön)/n Þ 2.tan(R).dR » dn /Ön asymptotically
This can be approximated linearly to cutoff at R = p/2
Ön is the normalisation factor.
Let T = exp(Ön)/n
Then ln T = Ön - ln n asymptotic approximation ln T ~ Ön
0ther Set Theoretic Reasons as to why The Riemann Hypothesis is Unprovable
Is RH True? Yes, but UNPROVABLE. Why? Here are two simple explanations that a laymen would understand simply based on logic, which avoids esoteric mathematical argument:
If the Riemann Zeta function encodes all knowledge and its definition is bounded,
then the tools to prove RH must exist within it i.e. If Ω
= Sum of All Knowledge, B = Tools, then
Ω so B È
(i.e. No added knowledge) i.e. we can't 'poke' it with anything outside the
Universe which is effectively the Riemann landscape.
2. z(z) = PA*(z - zi) infinite product of roots zi. Assume zk is off the Critical Line Re(z) = ½, then divide equation by (z-zk) for some index k still gives the same SUM of ALL SIGNALS. We therefore have two alternative stories YES/NO, 1 or 0 both equally correct.
In addition to the Probabilistic reasoning in Section 6 where power p=½ in RSS sum left over from pair-wise correlation c.f Harmonic series diverges.
There are known knowns. These are things we know that we know. Then there are known unknowns. That is to say, there are things that we know we don’t know. But there are also unknown unknowns. These are things we don’t know we don’t know.
Furthermore, perhaps true randomness is an ideal. For example, the digits of pi (symbol p) appear to be random, but they are clearly not random as they can be computed from a definite formula:
"There's a beauty to pi that keeps us looking at it ... The digits of pi are extremely random. They really have no pattern, and in mathematics that's really the same as saying they have every pattern." - Peter Borwein, 1996
As with sources of sound set against a noise background in the ocean, the digits of pi may be viewed as waves with signals as sources from deterministic series terms in an expression set against an emergent background ‘hiss’. They may appear random and so live on the far left hand side of the Montgomery/Dyson curve, but their characteristic SNR (per sinusoid) may be exceedingly small but not zero. However it could equally be argued that pi has a high SNR in the sense of high frequency spectral components so may actually exist to the infinite far right of the curve, although the digits themselves may appear random and therefore 'noisy' in a loose sense of the word, but may actually contain the 'sum of all signals'. Hence pi can be truly called a Transcendental number.
Pi is completely deterministic. 'Randomness' and 'Noise' may not be the same in this case. However the term 'noise' here refers to a non-periodic component, where there is a lack of coherency.
By contrast, Zeta appears to encode the complete spectrum of scale from the very small to the very large, as shown in the full outline of the Dyson curve. Maybe we can call it the ‘signature of God’.
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