Riemann Zeta Function and Odd Integer Values (1)

It has been well known since the time of Euler that the value of the Zeta Function for odd integers of s (i.e. s = 3, 5, 7, 9,...etc.) behaves in a very different manner than for corresponding even values. Euler, as we know, was able to prove that for any even integer s,

ζ(s) = k*(π^s) where k is a rational fraction.


However no such relationship characterises values of the Zeta Function for odd integers of s!


Indeed it took some time to prove that the first of these values for s = 3, is irrational (Apéry's constant) though it is not known if it is transcendental.
Also though it is known that other values of the Zeta function for odd integers values must likewise be irrational, little clarity can as yet be provided as to how this applies in specific cases.


Though of course there is a validity to the the quantitative attempt of attempting to provide a (Type 1) proof as to the precise status (as number type) that characterises Zeta Function results for the odd integers, inherently such an issue relates more to qualitative - rather than quantitative - considerations.


Now the first key indication that behaviour of the Zeta Function for odd integral values is quite distinct from corresponding behaviour with respect to the even integers is given through examination of the root structure for odd integer roots which reflects in quantitative terms corresponding interpretation of these same integers (as dimensional numbers).


Whereas for even numbered integers, roots can always be arranged in a complementary manner, this is never the case for odd numbered integers.

For example if we look at the 3 roots of 1 we have,

1, - 1/2 + {[(3^(1/2)]* i}/2 and - 1/2 - {[(3^(1/2)]* i}/2, which cannot be arranged in a complementary manner.

Thus the perfect matching of independent terms with interdependence through the direct complementary relationship as between roots, characterising the pure relation of linear to circular meaning (that defines the qualitative nature of π) is thereby missing with respect to odd integer roots and corresponding dimensional numbers.


When one looks more closely at the odd numbered roots one can see that + 1 is always one of these roots that stands in a sense alone. With respect to the other roots they always comprise conjugate pairs where the imaginary part is complementary with its partner (in the pairing).


There is also a significant clue as to what this entails in qualitative terms provided through the nature of higher psychospiritual contemplative development which involves the process of traversing these higher dimensions.


Basically the pattern keeps switching from differentiation (which always implies a degree of linear type understanding) to integration (where a new harmonious contemplative state is established).

So each even numbered stage (as the qualitative development with respect to such stage) represents the restoration of a new temporary equilibrium (characterised by the attainment of a contemplative state appropriate to such development).

However each odd numbered stage by contrast represents the breaking up of that (temporary) equilibrium is a new more refined experience of linear phenomena (which always however entails a degree of dislocation and asymmetry with respect to the harmony of the previous stage).

Then eventually with sufficient evolution in development, one again moves to the next even numbered stage and the restoration of a higher state of contemplative attainment.


Put more precisely in a qualitative mathematical manner, the dislocation and asymmetry associated with the odd numbered stages arises from a degree of inevitable confusion as between rational understanding (geared to linear interpretation) and irrational understanding (reflecting the corresponding attempt to apply the higher states of intuitive awareness already attained to such understanding).

In other words one tries to combine both rational and true intuitive appreciation - which is paradoxical to reason - in a coherent manner. However before a new integral state can be attained this is always necessarily associated with a degree of confusion as between both aspects (rational and irrational).


Now we have already mentioned how complementary type interpretation characterises the relationship as between RHS and LHS of the Riemann Zeta Function.

Interestingly when we look at the values of the Riemann Zeta Function for odd integers of s (negative) on the LHS (where s < 0), they are always represented as rational values.

This would therefore imply that the corresponding values of the Zeta Function for these odd integers (positive) would be thereby represented (in complementary fashion) by irrational values.


And these irrational values would be algebraic (rather than transcendental) in nature and necessarily apply in the case of all the odd integers.


Now the relationship of these values to π is still important.


In other words as the odd dimensional number increases an ever closer degree of complementarity characterises roots (belonging to the set of conjugate pairings).

What this means in psychospiritual terms is that the linear understanding characterising very high odd integer stages become so refined as to be almost transparent (indicating that it can now interpenetrate with intuition with a remarkable lack of confusion remaining).


Interestingly once again in complementary fashion, when this happens the rational values for associated negative integer values increase dramatically. What this implies in qualitative terms is that with independence and interdependence both successfully combined to a very significant degree in experience, that a very high level of (refined) rational activity becomes compatible with an advanced contemplative state.


So just as the Riemann Zeta Function has important physical implications, equally it has extremely important psychospiritual implications (which have not yet been addressed). And once again this unfortunate blockage in understanding is due to the mistaken emphasis of Conventional Mathematics on merely quantitative meaning!