Friday, March 10, 2017

Ramanujan's Letter (6)

We return here to our main focus of concern regarding the need for a holistic circular conversion of each term - indirectly in a quantitative manner - in order to give an intuitive meaning to to results of the Riemann zeta function )for negative odd integer values of s).

And as we have seen, all these can be "converted" with respect to a central result, where s =  – 13.

113/{(1 –  e)/2} + 213/{(1 –  e)/2} + 313/{(1 –  e)/2} + 413/{(–  e8π )/2} + ...    =  – 1/12.

This can also be expressed as,

113/{(1 1– i )/2} + 213/{(1 1– 2i )/2} + 313/{(1 1– 3i )/2} + 413/{(1 – 1– 4i )/2} + ...    =  – 1/12.

However as we know, where negative even integer values of s are concerned in the Riemann zeta function, the results always = 0 (yielding the so-called trivial zeros).

So, for example, according to the Riemann zeta function when s = 2, we have,

1/1– 2 + 1/2– 2 + 1/3– 2 + 1/4– 2 + ... ,

 =  122 + 32 + 4+ ...    =  0,

i.e.   1 + 4 + 9 + 16 + ...  = 0.

But again, such a result is completely non-intuitive from the standard conventional linear perspective, where the sum of the series diverges to infinity.

So just as for negative odd integer values of s, an appropriate holistic circular conversion is required - indirectly in a quantitative manner - so that the numerical result then successfully accords with intuitive expectations.

What I suggest therefore is the following;

 (e2iπ – 1)/12  +  (e4iπ – 1)/2+  (e6iπ – 1)/3+  (e8iπ – 1)/4+ ...,

= (1 – 1)/12  +  (1 – 1)/2+  (1 – 1)/3+  (1 – 1)/4+ ... ,

= 0/1 + 0/4 + 0/9 + 0/16 + ....   = 0.

Now there is a direct connection here with the Type 2 aspect of the number system, which in a dynamic relative manner complements the corresponding Type 1 aspect.

So e2iπ  = 1,  i.e. 11,

     e4iπ   = 1,  i.e. 12,

     e6iπ   = 1,  i.e. 13,

     e8iπ   = 1,  i.e. 14, and so on.

Therefore we could equally write the holistic converted expression as,

(11 – 1)/12  +  (12 – 1)/2+  (13 – 1)/3+  (14 – 1)/4+ ...,

Thus what we have here is a perfect matching of each of the natural numbers from both the Type 1 and Type 2 perspectives (with the Type 2 appearing in the numerator and the Type 1 in the denominator respectively).

Therefore we are at the opposite holistic extreme here from the conventional analytic extreme of understanding.

Remember in conventional mathematical terms, that quantitative and qualitative aspects of understanding are abstracted from each other in an absolute manner so that the Type 2 aspect of the number system is completely ignored, being thereby reduced in every circumstance in a Type 1 (absolute) manner!

This therefore represents the analytic extreme of understanding i.e. where number is understood in a merely quantitative fashion.

However, from the holistic extreme, both quantitative (Type 1) and qualitative (Type 2) aspects of the number system are understood to be fully related to each other, as complementary aspects, in a dynamic interactive manner.

Thus when one perfectly understands the nature of number from this perspective, any separate notion of number, as formally existing, is eroded in a true intuitive appreciation of number as a pure state of energy (in psycho-physical terms).

So, quite literally one's experience of number in such an intuitive manner is - literally - nothing (i.e. 0) in formal terms.

Because of the great importance of this point, I will trace out now in more detail what this experiential 2-dimensional realisation (in negative terms) entails.

Now, again,  the standard 1-dimensional interpretation of number is one-directional where the (external) number object is clearly separated from its corresponding (internal) mental perception.

Thus in linear (1-dimensional) terms - which represents standard mathematical interpretation - the number object is explicitly given a single direction, where its is - literally - posited in an (external) conscious manner.

However with circular (2-dimensional) interpretation, both the (external) number objects and its corresponding (internal) mental perception are seen as complementary aspects of the same number experience. So both aspects, external and internal, have now a merely relative status with true appreciation of such relativity requiring that the external object aspects be dynamically negated (in an unconscious manner) thereby enabling smooth switching to its complementary internal aspect, which is thereby posited in a conscious manner. In like manner, one must then dynamically negate the (internal) number perception to enable switching to the (external) number object which is now again posited.

So in dynamic terms, a continual positing (conscious) and negating (unconscious) of both number aspects takes place, which is the very means by which the holistic interdependent nature of number is thereby appreciated.

Then as we approach the holistic limit of appreciation, switching as between both external and internal aspects is now so dynamic, with emphasis on ever-more refined unconscious negation, that we lose any separate notion of the identity of either aspect. And this is where the pure intuitive appreciation of number, as an energy state, is reached.

And this is precisely 2-dimensional in holistic terms, with both a positive (+) and negative (– ) pole.

Now there are many "higher" even dimensions (that are negative in sign). However they all entail an exact matching of complementary positive and negative poles resulting in ever more refined purely intuitive energy states. And this likewise concurs with an exact matching of Type 1 and Type 2 aspects of number (as suggested by the formula).

To write the converted holistic expression in more general terms for all trivial zeros results of the Riemann zeta function (where s is a negative even integer) we have the following:

(e2iπk – 1)/12k  +  (e4iπk – 1)/22k +  (e6iπk – 1)/32k +  (e8iπk – 1)/42k + ...,

where k = 1, 2, 3, 4, ...

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