Meaning of Imaginary

I wish to show here as to how there is an intimate connection as between the Zeta 2 Function and the true mathematical meaning of the imaginary number i.

As I have stated on so many occasions all mathematical symbols are used in a reduced (i.e. merely quantitative) manner in conventional terms.

However the true context for such symbols is inherently dynamic and interactive, involving the two-way interplay of both quantitative (independent) and qualitative (interdependent) aspects.

In psychological terms quantitative recognition is of a (rational) conscious nature whereby objects are directly posited in experience.

Qualitative recognition - by contrast - is directly of an (intuitive) unconscious nature, whereby conscious recognition of objects is dynamically negated. So this negation of  specific quantitative features of objects is the very means by which corresponding qualitative recognition takes place in a holistic manner.

The very positing of an object as a distinctive unit in (conscious) experience implies the number 1.

Now in an important sense the entire natural number system can be seen as an extension of 1.

So for example 2 (in this quantitative fashion) = 1 + 1.

Then the next number 3 = 2 + 1 and so on.

Therefore by repeatedly adding 1 to each new member , we can continually extend the natural numbers to ultimately include any member of our choosing.

Thus from this quantitative perspective, + 1 can be seen as the essential building block of all (determinate) natural numbers.


However in the dynamics of experience the conscious continually interacts with the unconscious aspect. So the specific (independent) features of object recognition must be temporarily negated in an unconscious manner to enable holistic recognition of a qualitative nature to take place.

So once again, properly understood, number itself is of a dynamic interactive nature entailing aspects that are relatively independent and  interdependent with respect to each other.

This relativity with respect to number is of such an intimate nature that we have missed its significance completely.

For millennia now we have attempted to preserve the fiction that number can be successfully understood with respect to its - mere - quantitative characteristics.

However the very notion of relating numbers with each other implies interdependence (which is a qualitative notion).

So the cardinal notion of number (as an independent entity) has no strict meaning in the absence of its corresponding ordinal notion (which implies in any context the relationship between a group of numbers).

Thus cardinal and ordinal are quantitative and qualitative with respect to each other with a relative - rather than absolute - identity.

This is why we need two distinctive aspects of number that dynamically interact. When these are initially dealt with in relative isolation, I refer to them as Type 1 and Type 2 respectively. Then when they are combined as complementary I refer to this as Type 3 understanding of number.

This leads in turn to both Zeta 1 and Zeta 2 Functions (which again can initially be interpreted in relative isolation from each other). However comprehensive understanding of the Zeta Function (Zeta 3) requires that both aspects be fully combined in complementary manner whereby they are ultimately seen as identical with each other in an ineffable manner.


We have seen how from the Type 1 aspect how + 1 can serve (through addition) as the building block of all other natural numbers (2, 3, 4,.....). Once again this Type 1 (cardinal) appreciation relates to the standard quantitative interpretation of number (now however understood in a relative - rather than absolute - fashion).

What is fascinating is that from the perspective of the Type 2 (ordinal) aspect, all the numbers (2,3,4,...) can be combined in a  reverse manner, so that their sum = – 1.

What this entails is that in the context of the n roots of 1, the sum of the 2nd, 3rd 4th ... up to the nth root = – 1.

However though these roots do indeed have an indirect quantitative meaning in circular terms (i.e. with respect to the circle of unit radius in the complex plane), this relationship has a qualitative (holistic) rather than quantitative (analytic) significance.


Indeed this pertains to the central nature of the relationship as between whole and part.

From the standard (quantitative) perspective, the relationship between whole and part is reduced to  one-directional unambiguous interpretation (of a merely quantitative nature).

Thus from this perspective the parts are contained (quantitatively) in the whole.

However the true relationship is of much more refined subtle nature. So from the quantitative perspective, the parts are quantitatively contained in the whole; however from the complementary qualitative perspective, the whole is contained in each part.

So, when we recognise the Type 2 aspect of number we can quite literally how the whole (in the sum of the non-trivial ordinal roots of unity) = – 1.

So the qualitative aspect of number - literally - implies the temporary negation of the corresponding Type 1 aspect.

Thus where + 1 serves as the quantitative building block of the Type 1 (cardinal) aspect of number, thereby generating the natural numbers, from the complementary Type 2 (ordinal) aspect these natural numbers in turn serve as the building blocks for generating – 1.

In other words, when properly understood number keeps dynamically switching as between its Type 1 and Type 2 aspects in a paradoxical fashion implying both quantitative and qualitative interaction.


Now the key significance of the Zeta 2 Function, is that its non-trivial zeros directly correspond to the various roots of 1 (other than + 1). In other words the Zeta 2 Function is directly concerned with the holistic (qualitative) aspect of number behaviour, which when contrasted with the Type 1, is of a circular paradoxical nature (based on the complementarity of opposites).

So once we explicitly negate the Type 1 aspect (based on absolute either/or logic), we start generating a new type of circular understanding (based on a relative both/and logic).

Thus again using our crossroads analogy, when we employ a single frame of reference (in terms of the direction in which it is approached) a turn is unambiguously either left or right. So this represents linear (1-dimensional) interpretation based on a single frame of reference.

However when we allow the crossroads from two (opposite) directions, a turn is both left and right (depending on context). So we are now viewing left and right in a holistic manner (as interdependent) in a 2-dimensional manner. This two-way interdependence now leads to (circular) paradox, whereas previously each turn was viewed independently in an unambiguous (linear) manner.

As it stands the supreme limitation of Conventional Mathematics is that it is formally based solely on linear (1-dimensional) logic.  So when paradoxes arise (as for example with Cantor's theory of sets) it attempts to interpret these through the very same logical framework that generates the paradoxes in the first place.

When one appreciates its nature, one steadily grows in the realisation, that as it stands, Conventional Mathematics has no means for properly dealing (so as to avoid gross reductionism) with the fundamental notion of interdependence (which is qualitative in nature).

And the key relationship of the primes with the natural numbers (and the natural numbers with the primes) cannot be understood in the absence of the true notion of interdependence.

So the mystery of the number system comes down to the manner in which its quantitative and qualitative aspects are mediated - in relative two-way fashion - through the primes and natural numbers!


As we have seen interdependence at a minimum requires incorporating 2-dimensional - rather than 1-dimensional understanding.

In other words it requires both positive (+ 1) and negative (– 1) polarities in relation to each other through the complementarity of both quantitative and qualitative aspects of understanding.


Now the holistic aspect does find its way however into Conventional Mathematics in a fascinating reduced manner.

In other words the very use of the imaginary notion i, represents the attempt to express the (unconscious) holistic notion of negated (unitary) form in a reduced linear manner. So just as in quantitative terms, when a number is expressed with respect to the power of 2, we express it in 1-dimensional terms by obtaining its square root, likewise with – 1.

So from a qualitative perspective, the imaginary number i expresses the indirect attempt to express meaning which is properly of a holistic nature (in accordance with the 2nd dimension) in a reduced linear manner.

Now clearly this is extraordinarily useful in quantitative terms. However its equally important qualitative significance is completely missed in conventional mathematical terms.

For example the fact that Riemann's Zeta Function is defined with respect to the complex plane clearly implies both  real (analytic) and imaginary (holistic) aspects of understanding.

From a deeper perspective. it can then be seen that the Function is designed precisely to demonstrate the intimate links as between the complementary analytic (quantitative) and holistic (qualitative) aspects of number i.e. its cardinal and ordinal aspects.

And this key insight which then quickly unlocks the very nature of the Riemann Hypothesis (as the condition for the ultimate identity of both aspects) is not accessible from a conventional mathematical perspective.

So once again not alone can the Riemann Hypothesis neither be proved (nor disproved) within the present mathematical paradigm; more importantly, it cannot be properly understood from this limited perspective.

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 +.... +  s^{t – 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.

Precise Nature of Zeta 1 Zeros

I wish to probe further here as to the precise nature of the Zeta 1 non-trivial zeros.

We will illustrate with a simple example where we start with the Zeta 2 finite expression for the 2 non-trivial roots of the 3 roots of 1, which again is represented as,

1 + s¹ + s² = 0.

Once again we can give values to the 3 terms of the expression by substituting s with the 2 non-trivial roots of s
i.e. – .5 + .866i  and – .5 – .866i respectively.

So when we substitute the first value for s (– .5 + .866i ) , the three terms on the RHS of the equation are

1,  – .5 + .866i  and – .5 – .866i .

Then we sustitute the second value for s (– .5 – .866i ), the three terms on the RHS of the equation are 

 1,  – .5 – .866i  and – .5 + .866i   respectively.


So with t here - representing the 3 roots of 1 - the formula for the total number of terms generated for the 2 non-trivial roots = t * (t - 1) = 3 * 2 = 6.

It is apparent that we are referring here to the natural numbers as representing dimensional values (i.e. exponents or powers) rather than base quantities. In other words, number here corresponds to Type 2 rather than Type 1 interpretation.

Thus for example the t - 1 non-trivial roots (of the t roots of 1) would be represented in Type 2 terms through the Zeta 2 Function as,

1 + s¹ + s² + s³ +.... +  s^{t – 1} = 0 with again the total number of values generated on the RHS of the expression

= t * (t - 1).

If however we concern ourselves with respect to these dimensional natural numbers (i.e. from 1 to t) with roots corresponding to the average spacing between primes (with respect to those natural numbers), we obtain

(log t - 1) non-trivial roots.

Then when we continue with these roots in regular groupings of log t - 1, through all the t - 1 non-trivial roots of 1, the total number of possible values = t * (log t - 1).

These non-trivial roots of t, with respect to the Zeta 1 Function, relate to circular values i.e. drawn on the unit circle in the complex plane (where the sum = 0).

So again with respect to 1 + s¹ + s² = 0, as the expression for the 2 non-trivial roots,

we have for s = – .5 + .866i  

1  – .5 + .866i   – .5 – .866i  = 0 and for s = – .5 + .866i  

1 – .5 + .866i  – .5 + .866i   = 0.

However when we express these values on a linear scale the sum of terms ≠ 0

Now in the unit circle when r = 1, the circumference has length 2π.

Therefore to convert on the linear scale to single units we divide by 2π.

Therefore where we now consider non-trivial roots as the average gap between primes up to t where this average gap = log t/2π = 3, then t/2π = 20.0855 (with t = 126.20116).


Thus on this basis the number of non-trivial zeros up to t

= 20,0855 (3 – 1) = 40.171 (where the actual value is in fact 40).

 

So the heights in this case (with respect to the Zeta 1) actually correspond to linear type measurements with respect to circular Zeta 2 values relating to the circumference of the unit circle in the complex plane. To standardise this value in a linear fashion with respect to single linear units, we thereby divide by 2π.

One final important point relates to the fact that these linear values of t (for Zeta 1 non-trivial zeros) are measured on an imaginary - rather than real - number line.

The reason for this is that we are using a linear (quantitative) way of indirectly representing a circular type relationship (which inherently carries a holistic meaning of a qualitative nature).


And the appropriate way of carrying out this indirect linear translation is with respect to imaginary (rather than real) units.

Indeed when one properly understands the nature of the Zeta 1 non-trivial zeros, by definition, they must all lie on a straight line as the indirect linear representation of number interdependence which is inherently circular (of a holistic qualitative nature).

The True Holistic Nature of the Zeta Zeros

I mentioned previously the stunning accuracy associated with the formula for calculation of the number of trivial zeros up to any given height on the imaginary number line.

Once again this formula can be given as:

 t/2π{(log t/2π) – 1}

I then suggested in a previous blog entry that perhaps a slightly more accurate version can be given through the addition of 1,

i.e.  t/2π{(log t/2π) – 1} + 1.

Thus using this (amended) version the number of non-trivial zeros up to 100 i.e. where t = 100 on the imaginary line,

=  15.9154943...(2.7672931... – 1) + 1 = 29.127...

The actual number of zeros up to 100 = 29! so we can see in this case how the formula gives a surprisingly accurate answer.

And as I conveyed in "Stunning Accuracy" this accuracy remains even at the highest values of t for which non-trivial zeros have been yet calculated.

However though this formula has been explicitly formulated with respect to the Zeta 1 Function, once again strong complementary links can be shown to exist with respect to the Zeta 2 Function,

i.e. ζ₂(s) = 1 + s¹ + s² + s³ +……  = 1/(1 – s).

As we have seen in a qualified sense.  ζ₂(s) = 0 for non-trivial roots of the equation (where the number of roots of 1 is prime and we consider all t – 1 roots except 1).


In this case the t – 1 non-trivial roots correspond to the finite equation,

1 + s¹ + s² + s³ +.... +  s^{t – 1} = 0.

Then when we continue with regular cycles of t terms, these t – 1 roots will likewise correspond to the infinite equation,

1 + s¹ + s² + s³ +……  = 0.

In yesterday's blog entry, we saw how the number of possible values generated = t * (t – 1)

For example where t = 3, the two non-trivial roots correspond to the finite equation,

1 + s¹ + s² = 0.

So 3 * 2 possible values (= 6) can be generated in this case.

Firstly, we obtain the 3 values from substituting the root (– .5 + .866i ) as a value for s,

to obtain 1, – .5 + .866i  and – .5 – .866i respectively.

Then we substitute the root (– .5 –  .866i ) as a value for s,

to obtain 1, – .5  –  .866i  and – .5 +.866i respectively.

The finite Zeta 2 Function is expressed in terms of natural number powers of s (from 1 to t – 1).

However strictly the non-trivial zeros relate to values of t that are prime!

Now the average spread (or gap) between primes in the region of t = log t.

So when we calculate the value of t terms with respect to the log t roots of t, we get

= t * {log t  – 1}

However, as we have seen the roots of 1 are defined with respect to the unit circle with circumference 2π. 

Therefore to convert from circular to linear units of measurement we divide t by 2π!

So therefore, when expressed in linear terms, the number of possible values generated up to s^{t – 1} for frequency of prime (i.e. log t) roots

= t/2π{(log t/2π) – 1}

and this is identical with the formula for frequency of non-trivial zeros with respect to the Zeta 1 Function.

This provides the holistic means of understanding what the non-trivial zeros with respect to the Zeta 1 Function truly represent. In fact, from the dynamic interactive perspective, through which they are appropriately viewed, they represent the opposite (complementary) aspect of prime numbers, or - as I have expressed before - the perfect shadow counterpart number system to the primes.

We start by viewing each prime in linear terms as a specific locally defined independent number (i.e. with no factors other that itself and 1) which thereby serves as a unique building block for the (composite) natural numbers.

However here at the other extreme, we have a set of counterpart numbers in circular terms that possess - by contrast - a collective holistically defined identity as a set of numbers, that serve to perfectly reconcile the primes and natural numbers as fully interdependent with each other.

Now because the Zeta 1 zeros are indirectly expressed in a linear (imaginary) format, it is difficult from this perspective to appreciate their true holistic identity.

However it will help to recognise that this linear imaginary identity (in Zeta 1 terms) represents just an indirect way of translating their inherent circular holistic identity (in Zeta 2 terms).

As we know the frequency of the primes lessens as we ascend the (real) number line. In the region of 100 we would expect an average gap of less than 5 as between successive primes; then in the region of 1,000 it would be 7 (approx.) and then in the region of 1,000,000, 14 (approx.)

Now this average gap as between primes in the region of t is given as log t.

However, if for example in the region of 1,000, we expect an average gap of about 7 between each successive prime (as an independent number with no factors), then this equally entails 6 other composite numbers (expressing the interdependence of a unique set of prime factors).

As we have seen, whereas the independent aspect of number is expressed through the Type 1, the interdependent aspect is expressed through the Type 2 approach. And in the context of the Zeta Function these two aspects relate to the Zeta 1 and Zeta 2 Functions respectively.

Thus the interdependence attaching to the composite numbers is appropriately expressed through the Zeta 2 Function.

Thus in the region of 1,000 we would express the interdependence of  the 6 composite numbers by the finite equation which provides in the case of 7, the 6 non-trivial roots of 1,

 1 + s¹ + s² + s³+ s⁴ + s⁵ + s⁶ = 0

now with t = 1,000 in this case the 6 terms in s are given as log t – 1 .

Now there are seven distinct values in the equation and we can express this with respect to each of the 6 possible values (as roots) of s. thus in total we have

log t * (log t – 1) values.

However if we now continue up t0 t (in circular groups of 7) the number of possible values =

 t * (log t – 1).

However as these values are based on the circle of unit radius, to convert to linear terms we divide t by 2π.

In this format the number of possible values is

t/2π{(log t/2π) – 1}, which is the well-known formula for calculation of the frequency of Zeta 1 zeros.

Thus we can perhaps see here that the Zeta 1 zeros serve as an indirect quantitative representation (in a necessarily imaginary linear format) of the collective holistic nature of the number system (which is directly qualitative in nature).

So therefore though each individual Zeta 1 zero is quantitative in nature (of an imaginary nature), the true significance of these zeros is in their overall collective holistic nature (which is qualitative).

And the direct way of appreciating this qualitative aspect is through the Zeta 2 Function!

Reformulating the Riemann Hypothesis

In my last blog I was at pains to explain the significance of 1/2 in the context of the Zeta 2 Function.

So once again,

1/(1 – s) = 1 + s¹ + s² + s³ +……

Now when s = – 1, 1/(1 – s ) = 1 – 1 + 1 – 1 +……  = 1/2.

In this case, the value of s =  – 1, corresponds to the 2nd root of 1, with the values for the first two terms of the series continually repeating in cyclical groups of 2. So as the value of the infinite series keeps alternating as between 0 and 1, the value of 1/2 given by the formula represents the mean average of these two possible values.
Therefore, the series keeps alternating as between an even and odd number of terms.
So when the number of terms is even, the value of the infinite series corresponds to the value of the first two terms.

From a quantitative perspective this implies that that the value of the series = 1 – 1 = 0 (where positive and negative terms cancel out); from  a qualitative perspective it implies interpretation corresponding to the 2nd (of two dimensions) based on the complementarity of opposite poles i.e. circular both/and logic (which implies paradox). 

When however the series is odd, from a quantitative perspective, the value of the infinite series corresponds to the first term = 1; from a qualitative perspective this implies interpretation corresponding to the 1st (of two dimensions) i.e. linear either/or logic (that is unambiguous). 

I have illustrated the nature of these two types of understanding on countless occasions previously with respect to  assigning directions at a crossroads. 

When we apply a single frame of reference to movement e.g. "up" or "down" a road, the designation of  left and right turns is then unambiguous (as either left or right).

So interpretation here corresponds to linear logic based on just one independent  reference frame. 

However, when we consider both frames as interdependent, then interpretation corresponds to circular logic of a paradoxical nature (as both left and right depending on context).     


So insofar as we can consider relationships as independent based on isolated poles of reference, then linear logic (based on the 1st dimension)  is appropriate. However insofar as we consider relationships as interdependent, based on the dynamic interaction of opposite poles, then circular logic (related to the 2nd dimension) is now appropriate.

Thus at a minimum, to successfully deal with both issues of independence and interdependence with respect to the number system, we need 2-dimensional interpretation.

This in turn implies that with respect to numbers, both independence and interdependence are defined in a relative - rather than absolute - fashion.  


Therefore once again the the key problem with Conventional Mathematics is the fact that it is qualitatively defined in a merely 1-dimensional manner.

So from this linear rational perspective, no clear distinction can be made as between independence and interdependence. Clearly when one attempts to view numbers as absolutely independent, then all relationships between numbers (which imply qualitative notions of interdependence) can only be dealt with a reduced manner!         

As the ultimate nature of the number system implies intimate notions of interdependence (with respect to the relationship of the primes to the natural numbers, and the corresponding relationship of the natural numbers to the primes) quite simply it cannot be properly interpreted within the conventional mathematical paradigm (based solely on 1-dimensional  interpretation).

And once again this constitutes the key qualitative explanation as to why the Riemann Zeta Function remains undefined (where s = 1).

Because this interpretation is 1-dimensional and thereby absolute in nature, no meaningful relationship can be established as between analytic notions of number independence (on the one hand) and holistic notions of number interdependence (on the other) both of which are necessarily of a relative nature. 

So not alone can the Riemann Hypothesis neither be proved (nor disproved) from this perspective; more importantly it cannot even be properly interpreted in such limited terms!


So we have looked at interpretation of the value of the infinite Zeta 2 series in the simplest case where s = – 1.

This - as we have seen - corresponds with the 2 roots of 1 (where – 1 represents the single non-trivial root). Then the solution for the infinite equation = 0 (where we take successive groups of two terms).


Now we can extend this thinking to all prime numbered roots.

So for example in the case of the 3 roots of 1, two of these i.e. the two roots other than 1, – .5 + .866i and  – .5 - .866i, will be non-trivial.

So when we now sum the infinite Zeta 2 series in successive groups of 3 terms, once again the sum = 0. And this sum of all terms will be identical with the sum of the first 3 terms i.e. 1 + s¹ + s².

However this leaves two other possible situations where a finite solution to the series arises.

For example if we take 1 term the sum = 1. Now when we take 2 terms (where s¹= – .5 + .866i), the sum 

= 1 – .5 + .866i   = .5 + .866i.

However we could equally take  s¹  = – .5 + .866i in which case the sum of 1st two terms =  1 – .5 –.866i = .5 – .866i.

So the average of these two possible values for the second term = (.5 + .866i + .5 –.866i )/2 = 1/2.


Therefore in this way, the series can have 3 possible answers!

When we groups terms from the start with 3 in a group, the sum of the infinite series = 0.

When however we allow for one more term (than a multiple of 3) the sum of the infinite series = 1.

Finally when we allow for 2 more terms (than a multiple of 3) the (average) sum of the infinite series = 1/2.

Therefore if we now again average the value of the infinite series over the 3 possible answers in this case, the sum of the infinite series = 1/2.


So in this way, the value for the infinite series will in all cases = 1/2, where s represents the non-trivial prime numbered roots of 1 .

And these are what I define as the non-trivial solutions for the Zeta 2 Function!


Because both the Zeta 1 and Zeta 2 Functions are complementary, in dynamic interactive terms, this thereby provides the key explanation as to why all the non-trivial solutions for the Zeta 1 expression lie on the line through 1/2.

From a quantitative perspective, the significance of 1/2 is that it represents the midpoint of the additive identity (0) and the multiplicative identity (1).

From a corresponding qualitative perspective, the significance of 1/2 is that it represents the midpoint of both linear interpretation (based on isolated polar reference frames) and circular interpretation (requiring complementary reference frames).

Thus the very means of reconciling addition and multiplication (in quantitative terms) is thereby inseparable from the corresponding means of reconciling linear (analytic) with circular (holistic) type interpretation.


When appreciated in this light, the Riemann Hypothesis can be seen as the central condition for reconciling both addition and multiplication (in quantitative terms) through the corresponding reconcilation of both (linear) analytic and (circular) holistic interpretation (from a qualitative perspective).

In other words the Riemann Hypothesis implies that ultimately both quantitative and qualitative aspects of the number system are inseparable.