Monday, April 28, 2014

Holistic Appreciation of the Riemann Zeros

I was reading in the past few days how Hugh Montgomery originally became interested in the Riemann zeros in an attempt to throw further light on the factorisation of imaginary numbers.

Montgomery initially believed that the zeros should be randomly distributed (in the same manner as the primes).

So in studying the Riemann zeros that were postulated to lie on an imaginary line, he was seeking experimental evidence to back up his initial hunch regarding their random nature.

However to his surprise he quickly found that this was not the case and that the zeros in fact tended to repel each other. So the observational evidence was very much against the notion of randomly distributed zeros.

Eventually Montgomery formed a conjecture (still unproven) as to the actual manner of distribution of the zeros. This was ultimately to lead to a chance meeting with the physicist Freeman Dyson, who could readily appreciate the relevance of the same distribution for the behaviour of excited energy states at the sub-atomic quantum level.

So this was to lead to the important realisation of the hitherto unexpected connection as between the Riemann zeros and quantum physics.

However from a holistic perspective it is pretty obvious why the Riemann zeros would not indeed be randomly distributed.

The very notion of random distribution is based on the assumption of independent events.

So for example if we repeatedly toss an unbiased coin we would expect the resulting number of heads and tails to be randomly distributed as each toss could be viewed as independent of - and thereby uninfluenced by - all other tosses.

However from a dynamic interactive perspective, the behaviour of the Riemann (non-trivial) zeros is directly complementary with the behaviour of the primes.

So therefore just as each prime represents the independent extreme of the number system (where a number has no factors other than itself and 1), the Riemann zeros by contrast express the interdependent number extreme.

Indeed I recently used this very fact to show how the frequency of the zeros is intimately related to the common factors of the composite numbers!

Thus we would not expect - by their very nature - that the Riemann zeros would be randomly distributed.

Rather rather like time series analysis in statistics they represent the smoothing out of the independent behaviour of each individual prime, so that each individual occurence can thereby be made fully compatible with the overall collective nature of the primes (where their interdependence with the natural number system is expressed).

Indeed an important clue as to this "smoothing" behaviour is given by the simple formula for estimation of the frequency of the primes.

As I have suggested an intimate complementary link exists as between the estimation of the frequency of primes and the common factors of the composite numbers.

So we can express the frequency of primes (up to n) as n/(log n – 1).

The corresponding frequency of the common factors of the composite numbers (up to n) is then given in a complementary manner as n(log n – 1).

Now the latter formula is directly linked with the formula for calculation of the frequency of non-trivial (Riemann) zeros.

However just as the primes as independent correspond with linear notions, the zeros (representing the corresponding collective interdependent of the primes) correspond with circular notions.

So we let n = t/2π

Then the formula for calculation of the frequency of the non-trivial zeros (up to t) on the imaginary line is given as

t/2π(log t/2π – 1).

Again the very fact that these zeros are postulated to lie on an imaginary line, directly suggests the notion of an interdependent identity for, from a holistic perspective, the imaginary notion represents an indirect manner of representing the notion of interdependence in a linear (i.e. independent) manner.   

Now what is remarkable about the formula for  the calculation of the frequency on non-trivial zeros, is that it is stunningly accurate (generally within 1 in absolute terms of the correct answer).

This contrasts sharply with the corresponding accuracy for calculation of the frequency of primes. Though this does indeed improve in relative terms as the value of n increases, the deviation from the actual frequency of primes likewise absolute terms.

However the non-trivial zeros do likewise have an independent identity. So, behaviour here is complementary to that of the primes.

Once again each individual prime has an independent identity (from a quantitative cardinal perspective).
However the overall collective behaviour of the primes has an interdependent identity (in being intimately related to the natural number system).

By contrast, in reverse manner, each individual non-trivial zeros has an interdependent identity in a qualitative holistic manner. However the collection of all trivial zeros has an independent identity in a quantitative fashion.  This indeed is why the non-trivial zeros can collectively be used to eliminate the deviations arising in the general estimation of prime number frequency.

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