The True Nature of the Zeta Zeros (1)

Once more, we return to this key issue with the attempt to provide a simple intuitively accessible explanation of the nature of the two sets of zeta zeros i.e. Zeta 1 and Zeta 2.

As perhaps, the latter set is easier to appreciate, we will start with the Zeta 2 zeros. Then through a complementary form of interpretation, the true nature of the better known Zeta 1 (i.e. Riemann) zeros can then be revealed in a coherent manner.


The Zeta 2 are intrinsically related to the ordinal nature of number. So corresponding for example to the cardinal notions of 1, 2, 3 we have the corresponding ordinal notions of 1st, 2nd and 3rd respectively..

However, whereas the cardinal in this context relate directly to the quantitative aspect of number, the ordinal notions, by contrast are directly associated with the corresponding qualitative aspect.

So once again, the cardinal relates the individual notion of each number as independent (in a quantitative manner).
By contrast, the ordinal relates to the collective notion of a group of numbers as interdependent with each other (in a qualitative manner).

However, crucially, conventional mathematical interpretation (in formal terms) is of a grossly reduced nature (i.e. where in every context, qualitative notions are necessarily reduced in quantitative terms).

Thus in the conventional treatment of ordinal number notions, their inherent qualitative nature is not explicitly recognised, but rather referred to in more neutral terms as representative of number rankings that correspond directly with cardinal notions.

Now from one legitimate perspective, this may indeed appear to be true.

Thus when we express ordinal notions with respect to the infinite number system (in linear fashion), no problem seemingly arises with 1st, 2nd and 3rd (as in our example) seeming to directly correspond in an unambiguous fashion with the absolute cardinal notions of 1, 2, and 3.

However this all subtly changes when we switch to the expression of ordinal notions within a finite group context, where they are now revealed to be of a strictly relative nature.

So for example within the (default) group of 3, 1st, 2nd and 3rd have a relative identity (that can be expressed in circular fashion). So we could represent the 3 ordinal positions as 3 equidistant points on the unit circle. However, depending on context, 1st, 2nd and 3rd could be equally identified with each point.

If, for example, we then consider the (default) group of 5, 1st, 2nd, 3rd, 4th and 5th now equally have a relative identity, that can again be expressed by 5 equidistant points on the unit circle.

However on reflection, it becomes clear that the very meaning of 1st, 2nd and 3rd in this latter group has now changed.

And there is no finite limit to such change, as we can keep increasing the size of the group with the relative meaning of the rankings likewise changing!

In fact this can readily be appreciated in conventional situations (though the enormous mathematical significance of this is not properly grasped).

For example if I told you that I entered a competition and came 2nd (out of 1000 entrants), you might indeed be impressed.

If however if in fact there had only been 2 entrants, finishing 2nd would not however have constituted much of an achievement.

So in forming judgement as regards ordinal number rankings, implicitly we acknowledge that the relative significance of such rankings changes (depending on the overall number of the group).

Thus the ordinal notion of 2nd (to give just one example) can be given an unlimited set of relative interpretations (depending on the number of members in the finite group to which it belongs).

And this of course equally applies to every ordinal notion which likewise can be given an unlimited set of relative interpretations.

Now the significance of the primes in this context is that the ordinal interpretation of its natural number members is always unique for such groups.

However, we still have the big problem of finding a satisfactory quantitative manner of uniquely expressing these relative ordinal notions!

This is where the Zeta 2 zeros come in!

By defining the unit circle in the complex plane, the various ordinal notions (unique for each prime group) can be simply obtained through obtaining the corresponding prime roots of 1 (easily obtained though the Euler Identity).

Thus is we take the simplest case, the prime group will be 2 which enables in this context consideration of both a 1st and 2nd member.

So the 1st and 2nd roots correspond to 1^1/2 and 1^2/2 respectively (in the Type 2 number system)
=  – 1 and + 1 respectively.

Now one of these roots i.e. + 1, is not unique and this is true in every case. Thus, here, the 2nd (in the case of 2) has an absolute rather than relative meaning (which is denoted by + 1).

What this means is that once the position of the 1st member is chosen, the position of the 2nd (in the case of 2) is then automatically known in an absolute manner.

So the various prime roots of 1 correspond to the simple equation, 1 – x^t  = 0 (where t is initially prime).

However one of these roots, i.e. 1 – x, = 0 is of an absolute nature and not unique!

Therefore to find the unique (i.e. truly relative) solutions we divide by 1 – x^t  by 1 – x and solve thereby for 

1 + x¹  + x² + ... + x^{t – 1} = 0 (where t is initially prime).

Now this can ultimately be extended for all natural numbers from 2 to t (due to the unique relationship between the primes and natural numbers).

These solutions constitute the Zeta 2 zeros.

So the important role of the Zeta 2 zeros is that they enable an (indirect) quantitative means of uniquely expressing the purely relative nature of ordinal rankings, except in the default case i.e. the 
t^th member of a group of t).

So remarkably they provide the means of consistently converting (in this context of ordinal numbers) from qualitative to corresponding quantitative expression.

Furthermore this connection is of a holistic - rather than strict - analytic nature.

Thus each root - and we must always include the non-unique root of 1 here as relative considerations can only be made with respect to what is initially understood as independent - has a certain (relative) independent identity (as separate) while the addition of roots displays (relative) interdependence (as combined).

Thus in the case of the two roots of 1, – 1 and + 1 are relatively independent of each other (in quantitative terms).

However the combined sum i.e. – 1  + 1 = 0 displays corresponding (relative) interdependence, which is strictly of a qualitative nature.  And this is exemplified by the fact that universally, the sum of roots of 1 (representing the natural number ordinal members of a group) = 0.


By a complementary form of understanding - and ultimately both approaches are completely complementary in a dynamic interactive manner -  we can now perhaps explain simply the key significance of the corresponding Zeta (i.e. Riemann) zeros.

As we know, from the cardinal perspective, the primes are the building blocks in quantitative terms of the natural number system.

Now once again we start off with recognition of the individual primes as fully independent numbers (in an absolute manner).

However on deeper reflection - just as we have already seen with the ordinals - the primes can be shown to possess a merely relative identity.

This appreciation comes through recognition of the - formally unrecognised - qualitative aspect of  multiplication.

For example we may initially understand the first  two primes i.e. 2 and 3 as number quantities in an absolute manner.

However when we multiply these two numbers (i.e. 2 * 3) a qualitative - as well as quantitative - transformation is involved.

The first clue to this comes from looking at the operation in a geometrical manner. So the representation of 2 * 3  would entail a rectangle (measured in square i.e. 2-dimensional units).

However, this qualitative transformation of a dimensional nature, is simply ignored in the conventional treatment of number multiplication. So from this reduced perspective 2 * 3 = 6 (i.e. 6¹).

In this way, we then form the utterly misleading impression that all the composite natural numbers can likewise be defined in absolute terms (in a merely quantitative manner).

However the more profound appreciation of the true nature of multiplication, entails both the notions of number independence (as quantitative) and number interdependence (as qualitative).

So for example if one lays out two rows of similar coins (with 3 in each row) clearly one must recognise the (quantitative) independence of each coin. However, for multiplication to validly take place, equally one must recognise that the coins in each row can be placed in mutual correspondence with each other (as possessing a shared common identity) So this latter recognition relates to the qualitative aspect. Therefore in the multiplication operation i.e. 2 * 3, 2 here defines the mutual correspondence (i.e. interdependent identity)  of the 3 coins in each row!

So all multiplication implicitly always entails this vital qualitative aspect (which in conventional mathematical terms remains completely unrecognised.

Thus quite simply we cannot reconcile the two operations of addition and multiplication without proper recognition of both the quantitative and qualitative aspects of number.


Now just as earlier in the case of  ordinal numbers we saw - how for example, the notion of 2nd could acquire an unlimited number of relative interpretations depending on the size of the number group to which it is related - likewise the cardinal notion of 2 can acquire an unlimited number of relative interpretations, through the process of multiplication.

Thus in the case of 2 * 2  = 4, 2 acquires a new relative identity as a unique factor of this composite number.

Then in the case of 2 * 3 = 6, 2 and indeed - a different context 3, likewise attain  new identities as unique factors of 6.

Thus as all prime numbers can in principle be combined without limit as factors of resulting natural numbers (entailing the qualitative notion of interdependence), therefore all primes possess unique relative identities that are ultimately without finite limit.

So just as the Zeta 2 zeros provide a unique means of indirectly expressing the unique inherent qualitative identity of ordinal numbers in an (indirect) quantitative manner, the Zeta 1 (Riemann) zeros, provide a corresponding unique means of expressing the inherent quantitative identity of cardinal numbers in an (indirect) qualitative manner.

Now you might remember how I defined the two aspects of the number system as Type 1 and Type 2 respectively.

Whereas the Type 1 related to a base quantitative notion (with respect to the default dimensional value of 1), the Type 2 related to a dimensional qualitative notion (with respect to the default base value of 1).

So the Zeta 1 zeros relate directly to dimensional values that express the hidden qualitative aspect of the primes (which arises through multiplication with other primes).

In this context, I  have mentioned before how a remarkably close relationship connects the frequency of the Zeta 1 (Riemann) non-trivial zeros with the corresponding frequency of the natural factors of the composite numbers.

Thus we can use knowledge of the frequency of the Zeta 1 zeros (up to t on the imaginary scale) to accurately predict the corresponding accumulated frequency of natural factors of the composites up to n (on the real scale) where n = /t/2π.

So the Zeta 1 (Riemann) zeros provide a remarkable way of expressing (indirectly) in precise numerical format, the hidden qualitative transformations that are involved through the multiplication of primes.

In this way, they express the purely relative nature of the primes (ultimately as pure energy states) which represents the opposite extreme to conventional quantitative notions where they appear as rigid and absolute.

And, as before, the nature of these zeros is strictly of a holistic nature.

In the natural number system we get a considerable amount of discontinuity in the movement as between the primes (with no factors other than themselves and 1) and the composite natural numbers (with 2 or more factors), which increases as we ascend the number scale.

The Zeta 1 zeros therefore represent a continual smoothing out of this discontinuity with locattions taken, so that in the identity of each zero, opposites are reconciled such as between notions of randomness and order and primes (without factors) with composite natural numbers (with factors).

So this reconciliation of opposites (in the identity of each zero) represents the qualitative aspect of number behaviour from this complementary perspective. Then the combination of all such zeros represents their quantitative aspect (which can be used to eliminate deviations in the general calculation of the frequency of primes).

I cannot stress however how far we have moved here from conventional mathematical understanding (based on mere analytic interpretation where the quantitative aspect is clearly divorced in absolute terms from its corresponding qualitative aspect).

Here we have arrived at the other extreme of pure holistic interpretation (where both quantitative and qualitative aspects are fully harmonised with each other in a truly relative manner).

And as it stands, there is no way whatsoever of grasping the true significance of the zeros (Zeta 1 and Zeta 2) from the conventional mathematical perspective (as it gives no formal recognition  to the holistic aspect of mathematical symbols).

In fact in their most profound sense they represent (both sets) a mysterious alchemy whereby we can convert consistently in two-way fashion from both qualitative to quantitative format (and quantitative to qualitative) with respect to the interpretation of number.