Tuesday, June 11, 2013

Addendum on Formula for Zeta Zeros

I recommended before a slight amendment for the customary formula for calculation of zeta non-trivial zeros up to t (on the imaginary number scale).

So instead of  t/2π{(log t/2π) – 1}, I suggest

 t/2π{(log t/2π) – 1} + 1.

In particular this seems to be especially accurate for low valued estimates where we take the midpoint between zeros for our estimate with the 1st zero for example then identified with the midpoint between the 1st and 2nd zeros.

So the midpoint of the first two zeros = 14.134725 + 20.022040 = 17.0783825

Thus if we let t = 17.0783825 the formula is 2.7181{(log 2.7181 – 1} + 1 = .999348.

Therefore it is very striking indeed in this case that our calculated answer is almost exactly equal to 1 when the actual value on this basis = 1! 

I continued in this manner with next 6 midpoints with answers never deviating from actual results by more than .2

Then finally taking the mid-point between the 99th and 100th zeros (which on this basis should concur with the 99th zero)  I obtained the answer 99.12!

However with high valued estimates (where the gap between the zeros is very small) it is no longer of any real significance.

However I would suggest that the amended version of the formula is still a more accurate predictor of the actual number of zeta zeros up to a given height t (and indeed as I have stated before stunningly accurate).

For example the highest value given by Andrew Odlyzko in his tables is for the zero 1022 + 104.

This occurs at t = 1,370,919,909,931,995,309,568.33538975

Now using the (slightly) amended formula  

i.e. t/2π{(log t/2π) – 1} + 1 to predict the number of zeros we obtain,

10000000000000000010000.012798296 = 1022 + 104 + .012798296. So remarkably (when rounded) this estimate represents exactly the actual number of zeros resulting!

The deeper explanation for this stunning accuracy is that unlike prime number behaviour, which is locally independent in nature, the trivial zeros - by their very nature - strive to achieve complete interdependence with respect to the prime and natural numbers. 

Now once again such interdependence is best appreciated in a (Type 2) circular number context.

As we know for t > 1 the sum of the t roots of 1 = 0. What is not properly appreciated however is the qualitative holistic significance of this relationship. 

Thus complete balance is maintained as between each specific root as independent and the overall holistic sum of roots = 0, which thereby maintains the perfect collective group interdependence of all these roots.

So if we seek to mirror Odlyzko's tables in Zeta 2 terms for 1 + s1 + s2 + s3 +.... +  st – 1 = 0, where t = 1022 + 104, remarkably we can solve this equation with 1022 + 104  – 1 different non-trivial values as roots with the total sum of roots in all cases = – 1. Then, with the addition of the trivial root for 1 – s = 0, the sum all t roots of 1 = 0.

And this remarkable number behaviour - properly understood - represents the true holistic nature of the interdependence of the number system.

And what should also be obvious - though again its significance is not at all appreciated - is that the same behaviour exactly characterises prime number as well as natural number roots.

So put simply, for t > 1, we would always expect the sum of roots of 1 = 0 (irrespective of whether t is a prime or composite natural number).

Therefore  with respect to this holistic circular number behaviour, no distinction exists as between the prime and natural numbers.

As I have stated before the Zeta 1 Function represents an indirect quantitative means, on an imaginary linear scale,  of representing such number interdependence (which directly is of a holistic circular nature).    

So the non-trivial zeros on the imaginary number line are located so as to smooth out the purely local effects arising from the independence of primes (with respect to this system).

So whereas in real linear terms, the primes represent the independent extreme of the number system (serving as the building blocks of the natural number system), by contrast the Zeta 1 non-trivial zeros represent the interdependent extreme (whereby the primes and natural numbers are seen as identical). 

However such independence and interdependence is of a merely relative nature. Therefore when we change the context, the prime numbers attain a holistic interdependent identity and the non-trivial seros a corresponding independent identity respectively.

Once again all of these relationships can only be properly viewed within a dynamic interactive context.

Needless to say, the conventional paradigm as it currenly stands is totally unsuited to such dynamic interpretation.

This - above all else - is by far the most important issue in Mathematics and urgently needs addressing.

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