Monday, July 13, 2015

Zeta Zeros Made Simple (6)

The simplest way to understand the Riemann (Zeta 1) zeros is with respect to the dynamic notion of complementarity.

In Jungian terms, the complement of a psychological function, that is consciously revealed in personality, is its hidden shadow that remains unconscious.

So for example the (unconscious) complement of sensation (as consciously revealed) is intuition and vice versa in that the (unconscious) complement of intuition (as in turn consciously revealed) is sensation.

What this means is that these two functions are dynamically necessary for the expression of each other, which can only be achieved harmoniously when equal recognition is given to both conscious and unconscious aspects.

However when the conscious is solely recognised in explicit fashion, it creates an imbalance whereby it assumes a distorted absolute type significance, with its complementary aspect thereby remaining hidden in the unconscious (as its unrecognised shadow).

Now if we substitute the somewhat equivalent notions of analytic and holistic for conscious and unconscious respectively, then we can say that the Riemann (Zeta 1) zeros represent the holistic complement to the conventional analytic notion of the prime numbers.

Equally from the reverse perspective, the primes represent to holistic complement to the conventional analytic notion of the Riemann (Zeta 1) zeros.

Thus, just as the psychological functions, through which we interpret reality, have both conscious and unconscious aspects in dynamic interaction with each other, equally the number system has both analytic and holistic aspects in dynamic interaction with each other.

So once again, if we understand the primes with respect to their conventional analytic aspect, then the Riemann (Zeta 1) zeros represent the holistic complement to such understanding; and again from the opposite perspective, if the Riemann zeros now are understood with respect to their conventional analytic aspect, then the primes now represent in turn the corresponding holistic aspect.

So properly understood, all mathematical symbols can be given both (Type 1) analytic and (Type 2) holistic interpretations, with both continually switching through the dynamic interactive nature of experience.

However Conventional Mathematics is interpreted in a solely (Type 1) analytic type manner in absolute terms, which explicitly recognises the merely conscious aspect of rational understanding.

Quite simply however, the true relationship of the primes to the natural numbers (and natural numbers to the primes) cannot be properly understood in this manner.

So just as conscious and unconscious should be recognised in psychological terms as necessary partners, likewise this should also be true in terms of a comprehensive mathematical understanding with both analytic (Type 1) and holistic (Type 2) interpretations available for all its symbols and relationships.

We can in fact fruitfully probe further this relationship between the primes and Riemann (Zeta 1) zeros.

The primes are viewed in conventional terms as the independent building blocks of the natural number system.

So an independent building block in this sense has no factors!

Therefore the complement of this independent notion of prime numbers is corresponding notion of the composite numbers that are now comprised of two or more factors.

And in complementary terms we consider the natural number - rather than prime - factors of each composite number.

So on the one hand, we have the sequence of prime numbers that increases in a step like discontinuous fashion.

So we start with 2 to encounter the first prime. Then at 3 the step function again increases discontinuously by 1 (now yield an overall  frequency of 2 primes).

Then at 4, there is no change and we continue the function parallel to the x axis, before increasing once more by 1 at 5 (to now accumulate 3 primes). Then we have now change at 6 before increasing discontinuously by 1 at 7 (attaining a frequency of 4 primes).

And we can go on and on in this fashion to represent the exact frequency of primes.

Now we could use a similar type of step function to represent the complementary notion of natural number factors.

Now in measuring factors 1 is not included! With respect to the number itself (which is necessarily a factor) this is included only for composite numbers. Thus primes from this perspective have no factors. However a composite number such as 12 would have 5 natural number factors  i.e. 2, 3, 4, 6 and 12).

Thus once again we use the vertical axis to measure frequency. So we start with 2 (which is prime) and move along the axis horizontally to 3 (which is also prime) then at 4 we encounter 2 factors so we step up vertically by 2 at 4 (represent the accumulate frequency of natural number factors). Then we move from 2 (on the vertical axis) at 4 (on the horizontal axis) to 5 (which is prime) then we move horizontally across to 6 where we encounter 3 factors. So we step up by 3 at 6 (to now accumulate 5 factors on the vertical axis). Then we move from here horizontally, parallel to x axis, to 7 (which is prime) then to 8 (where again we step up by 3, to accumulate 8 factors on the vertical axis) and so on.

So we have on the one hand a step function to represent the exact frequency of primes (to any number) and on the other, a complementary step function to represent the exact frequency of natural number factors.

Now if we attempt to measure in the first case the general frequency of primes (up to any number) using a continuous smooth function, we could employ the simple formula  n/(log n – 1).

For example the exact frequency of primes to 1 million is 78,498.

The estimated frequency using our simple formula is 78,030 (which is already 99.4% accurate).

Now the corresponding formula n(log n – 1) can be used to accurately approximate the accumulated number of natural factors (up to a given number).

For example I manually counted 357 such factors up to 100 and the estimated measurement using the formula suggested is 361!

Thus whereas n/(log n – 1) provides a good general measurement of the frequency of primes (to a given number), the complementary formula, n(log n – 1) likewise provides a good general estimate of the accumulated frequency of natural number factors (to a given number).

Finally if we replace in this latter formula, n by t/2π, we now get the circular version t/2π{(log (t/2π) – 1}, which is the well known formula for calculation of the Riemann (Zeta 1) zeros to a given number.

In other words the Riemann (Zeta 1) zeros and the accumulated frequency of the natural number factors of composite numbers are intimately related.

Thus alternatively, to estimate the accumulated frequency of natural number factors to n, we estimate the frequency of trivial zeros to t (where n =  t/2π).

So just as the general formula can be used to smooth out as it were the discontinuous occurrence of actual primes, the trivial zeros themselves represent a complementary notion with respect to the general formula used for their estimation.

No comments:

Post a Comment