Thursday, January 19, 2012

Further Considerations

I thought it would be interesting to calculate the value of t where the average spread as between prime numbers and non-trivial zeros would be identical.

This would be given by the equation,

log t = 2π/log(t/2π) (It has to be remembered that whereas t on the LHS of equation corresponds to a real scale (representing quantities), t on the RHS corresponds to an imaginary scale (representing dimensional values).

After a bit of iteration, I found that the value for t corresponds to 36.2 (approx). Here the average gap as between prime numbers = 3.56 (approx); equally the average gap as between non-trivial zeros is also = 3.56 (approx).

It is fascinating once more to recognise that we can provide simple rules for switching from the average spread for one distribution (e.g. as between primes) to the corresponding average spread for the other distribution (i.e. as between non-trivial zeros).

So if we start with the average spread as between primes (in the region of t) i.e. log t, firstly we invert it to obtain 1/log t. Now we rewrite this as 1/(log t/1).

Then we replace the numerator 1 with 2π and the corresponding 1 (as denominator of log t also by 2π) So now we have 2π/(log t/2π) which gives us the estimate of the average spread as between non-trivial zeros (also in the region of t).

This would imply that the two approaches are in a sense mirrors of each other. So we can use the distribution of the (real) primes to infer the corresponding distribution of the (imaginary) non-trivial zeros. Equally in reverse fashion we can use the distribution of the (imaginary) non-trivial zeros to infer the corresponding distribution for (real) primes.

One further insight strikes me here. As is well known the non-trivial zeros can be effectively used to eliminate deviations (from actual) of the predicted general distribution of (real) prime numbers.

Therefore in principle it should be possible in reverse fashion to use the (real) primes to likewise eliminate the deviations (from actual results) of the predicted general distribution of the (imaginary) non-trivial zeros.

Finally, it is quite clear from the above that both the primes and the non-trivial zeros are in fact interdependent with each other representing a real to imaginary (and imaginary to real) relationship.

Now this might be easy enough to appreciate in quantitative terms, but of course in a balanced interpretation we must also include the qualitative counterparts of these very notions.

Put another way, it implies that quantitative nature of the primes (and their relationship to the natural numbers) is ultimately indistinguishable from corresponding qualitative interpretation of the primes (and their relationship to the natural numbers).

So once again the Riemann Hypothesis serves as the necessary condition for the reconciliation of both types of meaning. And this perfect reconciliation can only take place in an ineffable manner (where quantitative and qualitative are no longer distinguishable).

So the quest to know the ultimate nature of the prime numbers (in quantitative terms) - rightly understood - is inseparable from the corresponding quest to know likewise its nature in qualitative terms (both aspects which are interdependent)

So true knowledge of this ultimate mysterious nature (quantitative and qualitative) takes place in an ineffable manner (through perfect contemplative spiritual realisation).